diff --git a/changelog b/changelog index 367608e..45c8aba 100644 --- a/changelog +++ b/changelog @@ -1,3 +1,8 @@ +20090407 tpd src/axiom-website/patches.html 20090407.01.tpd.patch +20090407 tpd src/doc/bookvol4 removed, rewritten as book/bookvol4 +20090407 tpd src/doc/bookvol1.pdf removed, regenerated during build +20090407 tpd src/doc/bookvol1 removed, rewritten as book/bookvol1 +20090407 tpd src/doc/book removed, rewritten as book/bookvol0 20090406 tpd src/axiom-website/patches.html 20090406.01.tpd.patch 20090406 tpd src/interp/setq.lisp move $dalymode flag to bookvol5 20090406 tpd src/input/unittest3.input unit test )set debug diff --git a/src/axiom-website/patches.html b/src/axiom-website/patches.html index d000ddd..8d8ddfd 100644 --- a/src/axiom-website/patches.html +++ b/src/axiom-website/patches.html @@ -1052,5 +1052,7 @@ index.html Axiom on Windows instructions
index.html Axiom on Windows as html
20090406.01.tpd.patch bookvol5 add )set debug
+20090407.01.tpd.patch +src/doc remove unused files
diff --git a/src/doc/book.pamphlet b/src/doc/book.pamphlet deleted file mode 100644 index 4fdb58c..0000000 --- a/src/doc/book.pamphlet +++ /dev/null @@ -1,67731 +0,0 @@ -\documentclass{book} -%\usepackage{axiom} -\usepackage{graphicx} -% struggle with latex figure-floating behavior -\renewcommand\floatpagefraction{.9} -\renewcommand\topfraction{.9} -\renewcommand\bottomfraction{.9} -\renewcommand\textfraction{.1} -\setcounter{totalnumber}{50} -\setcounter{topnumber}{50} -\setcounter{bottomnumber}{50} - -% spadcommands are the actual text that you type at the axiom prompt -\newcommand{\spadcommand}[1]% -{\begin{flushleft}{\tt #1}\end{flushleft}\vskip .1cm } - -% spadgraph are the actual text that you type at the axiom prompt for draw -\newcommand{\spadgraph}[1]% -{\begin{flushleft}{\tt #1}\end{flushleft}\vskip .1cm } - -% returnType is the type signature returned by the axiom interpreter -\newcommand{\returnType}[1]% -{\begin{flushright}{\tt #1}\end{flushright}\vskip .1cm} - -% spadsig gives the standard -> notation for signatures -\newcommand{\spadsig}[2]{{\sf #1 $\rightarrow$ #2}} - -% The book begins with some introductory material that is not really -% listed as a chapter. This creates a header similar to \chapter. -\newcommand{\pseudoChapter}[1]% -{\vskip .5in \noindent {\Large{\bf #1}}\vskip .5in} - -% The book begins with some introductory material that is not really -% listed as a section. This creates a header similar to \section. -\newcommand{\pseudoSection}[1]% -{\vskip .25in \noindent {\large{\bf #1}}\vskip .25in} - -% spadofFrom records the operation in the index and the domain in the index -\newcommand{\spadopFrom}[2]{\index{library!operations!#1 @\begingroup \string\tt{} #1 \endgroup}\index{#2}``{\tt #1}''} - -% spadfunFrom records the function name and domain in the index -\newcommand{\spadfunFrom}[2]{{\bf #1}\index{#1 @\begingroup \string\bf{} #1 \endgroup}\index{#2}} - -% special meanings for math characters -\newcommand{\N}{\mbox{\bbold N}} -\newcommand{\Natural}{\mbox{\bbold N}} -\newcommand{\Z}{\mbox{\bbold Z}} -\newcommand{\Integer}{\mbox{\bbold Z}} -\newcommand{\Rational}{\mbox{\bbold Q}} -\newcommand{\Q}{\mbox{\bbold Q}} -\newcommand{\Complex}{\mbox{\bbold C}} -\newcommand{\C}{{\mathcal C}} -\newcommand{\Real}{\mbox{\bbold R}} -\newcommand{\F}{{\mathcal F}} -\newcommand{\R}{{\mathcal R}} - -% draw a box around a text block -\newcommand\boxed[2]{% -\begin{center} -\begin{tabular}{|c|} -\hline -\begin{minipage}{#1} -\normalsize -{#2} -\end{minipage}\\ -\hline -\end{tabular} -\end{center}} - -\newcommand{\optArg}[1]{{{\tt [}{#1}{\tt ]}}} -\newcommand{\argDef}[1]{{\tt ({#1})}} -\newcommand{\funSyntax}[2]{{\bf #1}{\tt ({\small\it{#2}})}} -\newcommand{\funArgs}[1]{{\tt ({\small\it {#1}})}\newline} - -\newcommand{\condata}[4]{{\bf #1} {\bf #2} {\bf #3} {\bf #4}} - -\def\glossaryTerm#1{{\bf #1}\index{#1}} -\def\glossaryTermNoIndex#1{{\bf #1}} -\def\glossarySyntaxTerm#1{{\tt #1}\index{#1}} -\long\def\ourGloss#1#2{\par\pagebreak[3]{#1}\newline{#2}} -\def\csch{\mathop{\rm csch}\nolimits} - -\def\erf{\mathop{\rm erf}\nolimits} - -\def\zag#1#2{ - {{\hfill \left. {#1} \right|} - \over - {\left| {#2} \right. \hfill} - } -} - - -% these bitmaps are used by HyperDoc -\newdimen\commentWidth -\commentWidth=11pc -\newdimen\colGutterWidth -\colGutterWidth=1pc -\newdimen\baseLeftSkip -\baseLeftSkip=\commentWidth \advance\baseLeftSkip by \colGutterWidth -\newcommand\ExitBitmap{{\setlength{\unitlength}{0.01in}\begin{picture}(50,16)(0,0)\special{psfile=ps/exit.ps}\end{picture}}} -\newcommand\ReturnBitmap{{\setlength{\unitlength}{0.01in}\begin{picture}(50,16)(0,0)\special{psfile=ps/home.ps}\end{picture}}} -\newcommand\HelpBitmap{{\setlength{\unitlength}{0.01in}\begin{picture}(50,16)(0,0)\special{psfile=ps/help.ps}\end{picture}}} -\newcommand\UpBitmap{{\setlength{\unitlength}{0.01in}\begin{picture}(50,16)(0,0)\special{psfile=ps/up.ps}\end{picture}}} -\newcommand{\tpd}[5]{{\setlength{\unitlength}{0.01in}\begin{picture}(#1,#2)(#3,#4)\special{psfile=#5}\end{picture}}} - -\begin{document} -\begin{titlepage} -\center{\includegraphics{ps/axiomfront.ps}} -\vskip 0.1in -\includegraphics{ps/bluebayou.ps}\\ -\vskip 0.1in -{\Huge{The 30 Year Horizon}} -\vskip 0.1in -$$ -\begin{array}{lll} -Manuel\ Bronstein & William\ Burge & Timothy\ Daly \\ -James\ Davenport & Michael\ Dewar & Martin\ Dunstan \\ -Albrecht\ Fortenbacher & Patrizia\ Gianni & Johannes\ Grabmeier \\ -Jocelyn\ Guidry & Richard\ Jenks & Larry\ Lambe \\ -Michael\ Monagan & Scott\ Morrison & William\ Sit \\ -Jonathan\ Steinbach & Robert\ Sutor & Barry\ Trager \\ -Stephen\ Watt & Jim\ Wen & Clifton\ Williamson -\end{array} -$$ -\end{titlepage} -\pagenumbering{roman} -\tableofcontents -\vfill -\eject -\setlength{\parindent}{0em} -\setlength{\parskip}{1ex} -{\Large{\bf New Foreword}} -\vskip .25in - -On October 1, 2001 Axiom was withdrawn from the market and ended -life as a commercial product. -On September 3, 2002 Axiom was released under the Modified BSD -license, including this document. -On August 27, 2003 Axiom was released as free and open source -software available for download from the Free Software Foundation's -website, Savannah. - -Work on Axiom has had the generous support of the Center for -Algorithms and Interactive Scientific Computation (CAISS) at -City College of New York. Special thanks go to Dr. Gilbert -Baumslag for his support of the long term goal. - -This document is a complete ``re-implementation'' of the original Axiom -book by Jenks and Sutor. Virtually every line has been reviewed and -rewritten into the new Axiom pamphlet format. Changes were made to -reflect the new Axiom system. Additional material was added and some -previous examples were rewritten. This is intended to be a ``living'' -document with material referenced or gathered automatically from other -parts of the system documentation. Future plans include adding active -examples (moving graphics, in-line command prompts) using Active-DVI. - -Axiom has been in existence for over thirty years. It is estimated to -contain about three hundred man-years of research and has, as of -September 3, 2003, 143 people listed in the credits. All of these -people have contributed directly or indirectly to making Axiom -available. Axiom is being passed to the next generation. I'm looking -forward to future milestones. - -With that in mind I've introduced the theme of the ``30 year horizon''. -We must invent the tools that support the Computational Mathematician -working 30 years from now. How will research be done when every bit of -mathematical knowledge is online and instantly available? What happens -when we scale Axiom by a factor of 100, giving us 1.1 million domains? -How can we integrate theory with code? How will we integrate theorems -and proofs of the mathematics with space-time complexity proofs and -running code? What visualization tools are needed? How do we support -the conceptual structures and semantics of mathematics in effective -ways? How do we support results from the sciences? How do we teach -the next generation to be effective Computational Mathematicians? - -The ``30 year horizon'' is much nearer than it appears. - -\vskip .25in -%\noindent -Tim Daly\\ -CAISS, City College of New York\\ -November 10, 2003 ((iHy)) -\vfill -\eject -%\noindent -{\Large{\bf Foreword}} -\vskip .25in - -You are holding in your hands an unusual book. Winston Churchill once -said that the empires of the future will be empires of the mind. This -book might hold an electronic key to such an empire. - -When computers were young and slow, the emerging computer science -developed dreams of Artificial Intelligence and Automatic Theorem -Proving in which theorems can be proved by machines instead of -mathematicians. Now, when computer hardware has matured and become -cheaper and faster, there is not too much talk of putting the burden -of formulating and proving theorems on the computer's shoulders. -Moreover, even in those cases when computer programs do prove -theorems, or establish counter-examples (for example, the solution of -the four color problem, the non-existence of projective planes of -order 10, the disproof of the Mertens conjecture), humans carry most -of the burden in the form of programming and verification. - -It is the language of computer programming that has turned out to be -the crucial instrument of productivity in the evolution of scientific -computing. The original Artificial Intelligence efforts gave birth to -the first symbolic manipulation systems based on LISP. The first -complete symbolic manipulation or, as they are called now, computer -algebra packages tried to imbed the development programming and -execution of mathematical problems into a framework of familiar -symbolic notations, operations and conventions. In the third decade -of symbolic computations, a couple of these early systems---REDUCE and -MACSYMA---still hold their own among faithful users. - -Axiom was born in the mid-70's as a system called Scratchpad -developed by IBM researchers. Scratchpad/Axiom was born big---its -original platform was an IBM mainframe 3081, and later a 3090. The -system was growing and learning during the decade of the 80's, and its -development and progress influenced the field of computer algebra. -During this period, the first commercially available computer algebra -packages for mini and and microcomputers made their debut. By now, -our readers are aware of Mathematica, Maple, Derive, and Macsyma. -These systems (as well as a few special purpose computer algebra -packages in academia) emphasize ease of operation and standard -scientific conventions, and come with a prepared set of mathematical -solutions for typical tasks confronting an applied scientist or an -engineer. These features brought a recognition of the enormous -benefits of computer algebra to the widest circles of scientists and -engineers. - -The Scratchpad system took its time to blossom into the beautiful -Axiom product. There is no rival to this powerful environment in -its scope and, most importantly, in its structure and organization. -Axiom contains the basis for any comprehensive and elaborate -mathematical development. It gives the user all Foundation and -Algebra instruments necessary to develop a computer realization of -sophisticated mathematical objects in exactly the way a mathematician -would do it. Axiom is also the basis of a complete scientific -cyberspace---it provides an environment for mathematical objects used -in scientific computation, and the means of controlling and -communicating between these objects. Knowledge of only a few Axiom -language features and operating principles is all that is required to -make impressive progress in a given domain of interest. The system is -powerful. It is not an interactive interpretive environment operating -only in response to one line commands---it is a complete language with -rich syntax and a full compiler. Mathematics can be developed and -explored with ease by the user of Axiom. In fact, during -Axiom's growth cycle, many detailed mathematical domains were -constructed. Some of them are a part of Axiom's core and are -described in this book. For a bird's eye view of the algebra -hierarchy of Axiom, glance inside the book cover. - -The crucial strength of Axiom lies in its excellent structural -features and unlimited expandability---it is open, modular system -designed to support an ever growing number of facilities with minimal -increase in structural complexity. Its design also supports the -integration of other computation tools such as numerical software -libraries written in FORTRAN and C. While Axiom is already a -very powerful system, the prospect of scientists using the system to -develop their own fields of Science is truly exciting---the day is -still young for Axiom. - -Over the last several years Scratchpad/Axiom has scored many -successes in theoretical mathematics, mathematical physics, -combinatorics, digital signal processing, cryptography and parallel -processing. We have to confess that we enjoyed using -Scratchpad/Axiom. It provided us with an excellent environment for -our research, and allowed us to solve problems intractable on other -systems. We were able to prove new diophantine results for $\pi$; -establish the Grothendieck conjecture for certain classes of linear -differential equations; study the arithmetic properties of the -uniformization of hyperelliptic and other algebraic curves; construct -new factorization algorithms based on formal groups; within -Scratchpad/Axiom we were able to obtain new identities needed for -quantum field theory (elliptic genus formula and double scaling limit -for quantum gravity), and classify period relations for CM varieties -in terms of hypergeometric series. - -The Axiom system is now supported and distributed by NAG, the group -that is well known for its high quality software products for -numerical and statistical computations. The development of Axiom -in IBM was conducted at IBM T.J. Watson Research Center at Yorktown, -New York by a symbolic computation group headed by Richard D. Jenks. -Shmuel Winograd of IBM was instrumental in the progress of symbolic -research at IBM. - -This book opens the wonderful world of Axiom, guiding the reader -and user through Axiom's definitions, rules, applications and -interfaces. A variety of fully developed areas of mathematics are -presented as packages, and the user is well advised to take advantage -of the sophisticated realization of familiar mathematics. The -Axiom book is easy to read and the Axiom system is easy to use. -It possesses all the features required of a modern computer -environment (for example, windowing, integration of operating system -features, and interactive graphics). Axiom comes with a detailed -hypertext interface (HyperDoc), an elaborate browser, and complete -on-line documentation. The HyperDoc allows novices to solve their -problems in a straightforward way, by providing menus for step-by-step -interactive entry. - -The appearance of Axiom in the scientific market moves symbolic -computing into a higher plane, where scientists can formulate their -statements in their own language and receive computer assistance in -their proofs. Axiom's performance on workstations is truly -impressive, and users of Axiom will get more from them than we, the -early users, got from mainframes. Axiom provides a powerful -scientific environment for easy construction of mathematical tools and -algorithms; it is a symbolic manipulation system, and a high -performance numerical system, with full graphics capabilities. We -expect every (computer) power hungry scientist will want to take full -advantage of Axiom. - -\vskip .25in -%\noindent -David V. Chudnovsky \hfill Gregory V. Chudnovsky -\vfill -\eject -\pagenumbering{arabic} -\pseudoChapter{Introduction to Axiom} -\label{ugNewIntro} -\section{Introduction to Axiom} -Welcome to the world of Axiom. -We call Axiom a scientific computation system: -a self-contained toolbox designed to meet -your scientific programming needs, -from symbolics, to numerics, to graphics. - -This introduction is a quick overview of what Axiom offers. - -\subsection{Symbolic Computation} -Axiom provides a wide range of simple commands for symbolic -mathematical problem solving. Do you need to solve an equation, to -expand a series, or to obtain an integral? If so, just ask Axiom -to do it. - -Given $$\int\left({{1\over{(x^3 \ {(a+b x)}^{1/3})}}}\right)dx$$ -we would enter this into Axiom as: - -\spadcommand{integrate(1/(x**3 * (a+b*x)**(1/3)),x)} -which would give the result: -$$ -{\left( -\begin{array}{@{}l} -\displaystyle --{2 \ {b^2}\ {x^2}\ {\sqrt{3}}\ {\log \left({{{\root{3}\of{a}}\ {{\root{3}\of{{b -\ x}+ a}}^2}}+{{{\root{3}\of{a}}^2}\ {\root{3}\of{{b \ x}+ -a}}}+ a}\right)}}+ -\\ -\\ -\displaystyle -{4 \ {b^2}\ {x^2}\ {\sqrt{3}}\ {\log \left({{{{\root{3}\of{a}}^ -2}\ {\root{3}\of{{b \ x}+ a}}}- a}\right)}}+ -\\ -\\ -\displaystyle -{{12}\ {b^2}\ {x^2}\ {\arctan \left({{{2 \ {\sqrt{3}}\ {{\root{3}\of{a}}^ -2}\ {\root{3}\of{{b \ x}+ a}}}+{a \ {\sqrt{3}}}}\over{3 \ a}}\right)}}+ - -\\ -\\ -\displaystyle -{{\left({{12}\ b \ x}-{9 \ a}\right)}\ {\sqrt{3}}\ {\root{3}\of{a}}\ {{\root{3}\of{{b -\ x}+ a}}^2}} -\end{array} -\right)}\over{{18}\ {a^2}\ {x^2}\ {\sqrt{3}}\ {\root{3}\of{a}}} -$$ -\returnType{Type: Union(Expression Integer,...)} -Axiom provides state-of-the-art algebraic machinery to handle your -most advanced symbolic problems. For example, Axiom's integrator -gives you the answer when an answer exists. If one does not, it -provides a proof that there is no answer. Integration is just one of -a multitude of symbolic operations that Axiom provides. - -\subsection{Numeric Computation} -Axiom has a numerical library that includes operations for linear -algebra, solution of equations, and special functions. For many of -these operations, you can select any number of floating point digits -to be carried out in the computation. - -Solve $x^{49}-49x^4+9$ to 49 digits of accuracy. -First we need to change the default output length of numbers: - -\spadcommand{digits(49)} -and then we execute the command: - -\spadcommand{solve(x**49-49*x**4+9 = 0,1.e-49)} -$$ -\begin{array}{@{}l} -\displaystyle -\left[{x = -{0.6546536706904271136718122105095984761851224331 -556}}, \right. -\\ -\\ -\displaystyle -\left.{x ={1.086921395653859508493939035954893289009213388763}}, - \right. -\\ -\\ -\displaystyle -\left.{x ={0.654653670725527173969468606613676483536148760766 -1}}\right] -\end{array} -$$ - - -\returnType{Type: List Equation Polynomial Float} -The output of a computation can be converted to FORTRAN to be used -in a later numerical computation. -Besides floating point numbers, Axiom provides literally -dozens of kinds of numbers to compute with. -These range from various kinds of integers, to fractions, complex -numbers, quaternions, continued fractions, and to numbers represented -with an arbitrary base. - -What is $10$ to the $90$-th power in base $32$? - -\spadcommand{radix(10**90,32)} -returns: - -%\noindent -{\tt FMM3O955CSEIV0ILKH820CN3I7PICQU0OQMDOFV6TP000000000000000000 } -\returnType{Type: RadixExpansion 32} - -The AXIOM numerical library can be enhanced with a -substantial number of functions from the NAG library of numerical and -statistical algorithms. These functions will provide coverage of a wide -range of areas including roots of functions, Fourier transforms, quadrature, -differential equations, data approximation, non-linear optimization, linear -algebra, basic statistics, step-wise regression, analysis of variance, -time series analysis, mathematical programming, and special functions. -Contact the Numerical Algorithms Group Limited, Oxford, England. - -\subsection{Graphics} -You may often want to visualize a symbolic formula or draw -a graph from a set of numerical values. -To do this, you can call upon the Axiom -graphics capability. - -Draw $J_0(\sqrt{x^2+y^2})$ for $-20 \leq x,y \leq 20$. - -\spadcommand{draw(5*besselJ(0,sqrt(x**2+y**2)), x=-20..20, y=-20..20)} -\begin{figure}[htbp] -\includegraphics[bbllx=1, bblly=39, bburx=298, bbury=290]{ps/bessintr.ps} -\caption{$J_0(\sqrt{x^2+y^2})$ for $-20 \leq x,y \leq 20$} -\label{tpdhere} -\end{figure} - -Graphs in Axiom are interactive objects you can manipulate with -your mouse. Just click on the graph, and a control panel pops up. -Using this mouse and the control panel, you can translate, rotate, -zoom, change the coloring, lighting, shading, and perspective on the -picture. You can also generate a PostScript copy of your graph to -produce hard-copy output. - -\subsection{HyperDoc} - -\begin{figure}[htbp] -\includegraphics[bbllx=1, bblly=1, bburx=298, bbury=290]{ps/h-root.ps} -\caption{Hyperdoc opening menu} -\label{fig-intro-br} -\end{figure} - -HyperDoc presents you windows on the world of Axiom, -offering on-line help, examples, tutorials, a browser, and reference -material. HyperDoc gives you on-line access to this document in a -``hypertext'' format. Words that appear in a different font (for -example, {\tt Matrix}, {\bf factor}, and -{\it category}) are generally mouse-active; if you click on one -with your mouse, HyperDoc shows you a new window for that word. - -As another example of a HyperDoc facility, suppose that you want to -compute the roots of $x^{49} - 49x^4 + 9$ to 49 digits (as in our -previous example) and you don't know how to tell Axiom to do this. -The ``basic command'' facility of HyperDoc leads the way. Through the -series of HyperDoc windows shown in Figure \ref{fig-intro-br} on -page~\pageref{fig-intro-br} and the specified mouse clicks, you and -HyperDoc generate the correct command to issue to compute the answer. - -\subsection{Interactive Programming } -Axiom's interactive programming language lets you define your -own functions. A simple example of a user-defined function is one -that computes the successive Legendre polynomials. Axiom lets -you define these polynomials in a piece-wise way. - -The first Legendre polynomial. - -\spadcommand{p(0) == 1} -\returnType{Type: Void} -The second Legendre polynomial. - -\spadcommand{p(1) == x} -\returnType{Type: Void} -The $n$-th Legendre polynomial for $(n > 1)$. - -\spadcommand{p(n) == ((2*n-1)*x*p(n-1) - (n-1) * p(n-2))/n} -\returnType{Type: Void} - -In addition to letting you define simple functions like this, the -interactive language can be used to create entire application -packages. All the graphs in the Axiom images section were created by -programs written in the interactive language. - -The above definitions for $p$ do no computation---they simply -tell Axiom how to compute $p(k)$ for some positive integer -$k$. - -To actually get a value of a Legendre polynomial, you ask for it. -\index{Legendre polynomials} - -What is the tenth Legendre polynomial? - -\spadcommand{p(10)} -\begin{verbatim} - Compiling function p with type Integer -> Polynomial Fraction - Integer - Compiling function p as a recurrence relation. -\end{verbatim} -$$ -{{{46189} \over {256}} \ {x \sp {10}}} -{{{109395} \over {256}} \ {x \sp -8}}+{{{45045} \over {128}} \ {x \sp 6}} -{{{15015} \over {128}} \ {x \sp -4}}+{{{3465} \over {256}} \ {x \sp 2}} -{{63} \over {256}} -$$ -\returnType{Type: Polynomial Fraction Integer} -Axiom applies the above pieces for $p$ to obtain the value -of $p(10)$. But it does more: it creates an optimized, compiled -function for $p$. The function is formed by putting the pieces -together into a single piece of code. By {\it compiled}, we mean that -the function is translated into basic machine-code. By {\it -optimized}, we mean that certain transformations are performed on that -code to make it run faster. For $p$, Axiom actually -translates the original definition that is recursive (one that calls -itself) to one that is iterative (one that consists of a simple loop). - -What is the coefficient of $x^{90}$ in $p(90)$? - -\spadcommand{coefficient(p(90),x,90)} -$$ -{5688265542052017822223458237426581853561497449095175} \over -{77371252455336267181195264} -$$ -\returnType{Type: Polynomial Fraction Integer} - -In general, a user function is type-analyzed and compiled on first use. -Later, if you use it with a different kind of object, the function -is recompiled if necessary. - -\subsection{Data Structures} - -A variety of data structures are available for interactive use. These -include strings, lists, vectors, sets, multisets, and hash tables. A -particularly useful structure for interactive use is the infinite -stream: - -Create the infinite stream of derivatives of Legendre polynomials. - -\spadcommand{[D(p(i),x) for i in 1..]} -$$ -\begin{array}{@{}l} -\displaystyle -\left[ 1, {3 \ x}, {{{{15}\over 2}\ {x^2}}-{3 \over 2}}, - {{{{35}\over 2}\ {x^3}}-{{{15}\over 2}\ x}}, {{{{315}\over -8}\ {x^4}}-{{{105}\over 4}\ {x^2}}+{{15}\over 8}}, \right. -\\ -\\ -\displaystyle -\left.{{{{693}\over 8}\ {x^5}}-{{{315}\over 4}\ {x^3}}+{{{105}\over -8}\ x}}, {{{{3003}\over{16}}\ {x^6}}-{{{3465}\over{16}}\ {x^ -4}}+{{{945}\over{16}}\ {x^2}}-{{35}\over{16}}}, \right. -\\ -\\ -\displaystyle -\left.{{{{6435}\over{16}}\ {x^7}}-{{{9009}\over{16}}\ {x^5}}+ -{{{3465}\over{16}}\ {x^3}}-{{{315}\over{16}}\ x}}, \right. -\\ -\\ -\displaystyle -\left.{{{{109395}\over{128}}\ {x^8}}-{{{45045}\over{32}}\ {x^ -6}}+{{{45045}\over{64}}\ {x^4}}-{{{3465}\over{32}}\ {x^2}}+{{3 -15}\over{128}}}, \right. -\\ -\\ -\displaystyle -\left.{{{{230945}\over{128}}\ {x^9}}-{{{109395}\over{32}}\ {x^ -7}}+{{{135135}\over{64}}\ {x^5}}-{{{15015}\over{32}}\ {x^3}}+ -{{{3465}\over{128}}\ x}}, \ldots \right] -\end{array} -$$ -\returnType{Type: Stream Polynomial Fraction Integer} - - -Streams display only a few of their initial elements. Otherwise, they -are ``lazy'': they only compute elements when you ask for them. - -Data structures are an important component for building application -software. Advanced users can represent data for applications in -optimal fashion. In all, Axiom offers over forty kinds of -aggregate data structures, ranging from mutable structures (such as -cyclic lists and flexible arrays) to storage efficient structures -(such as bit vectors). As an example, streams are used as the -internal data structure for power series. - -What is the series expansion -of $\log(\cot(x))$ -about $x=\pi/2$? -%NOTE: The book has a different answer (see p6) - -\spadcommand{series(log(cot(x)),x = \%pi/2)} -$$ -\begin{array}{@{}l} -\displaystyle -{\log \left({{-{2 \ x}+ \pi}\over 2}\right)}+ -{{1 \over 3}\ {{\left(x -{\pi \over 2}\right)}^2}}+ -{{7 \over{90}}\ {{\left(x -{\pi \over 2}\right)}^4}}+ -{{{62}\over{2835}}\ {{\left(x -{\pi \over 2}\right)}^6}}+ -\\ -\\ -\displaystyle -{{{127}\over{18900}}\ {{\left(x -{\pi \over 2}\right)}^8}}+ -{{{146}\over{66825}}\ {{\left(x -{\pi \over 2}\right)}^{10}}}+ -{O \left({{\left(x -{\pi \over 2}\right)}^{11}}\right)} -\end{array} -$$ -\returnType{Type: GeneralUnivariatePowerSeries(Expression Integer,x,pi/2)} - -Series and streams make no attempt to compute {\it all} their -elements! Rather, they stand ready to deliver elements on demand. - -What is the coefficient of the $50$-th -term of this series? - -\spadcommand{coefficient(\%,50)} -$$ -{44590788901016030052447242300856550965644} \over -{7131469286438669111584090881309360354581359130859375} -$$ -\returnType{Type: Expression Integer} - -\subsection{Mathematical Structures} -Axiom also has many kinds of mathematical structures. These -range from simple ones (like polynomials and matrices) to more -esoteric ones (like ideals and Clifford algebras). Most structures -allow the construction of arbitrarily complicated ``types.'' - -Even a simple input expression can -result in a type with several levels. - -\spadcommand{matrix [ [x + \%i,0], [1,-2] ]} -$$ -\left[ -\begin{array}{cc} -{x+i} & 0 \\ -1 & -2 -\end{array} -\right] -$$ -\returnType{Type: Matrix Polynomial Complex Integer} - -The Axiom interpreter builds types in response to user input. -Often, the type of the result is changed in order to be applicable to -an operation. - -The inverse operation requires that elements of the above matrices -are fractions. - -\spadcommand{inverse(\%)} -$$ -\left[ -\begin{array}{cc} -{1 \over {x+i}} & 0 \\ -{1 \over {{2 \ x}+{2 \ i}}} & -{1 \over 2} -\end{array} -\right] -$$ -\returnType{Type: Union(Matrix Fraction Polynomial Complex Integer,...)} - -\subsection{Pattern Matching} - -A convenient facility for symbolic computation is ``pattern -matching.'' Suppose you have a trigonometric expression and you want -to transform it to some equivalent form. Use a $rule$ command to -describe the transformation rules you \index{rule} need. Then give -the rules a name and apply that name as a function to your -trigonometric expression. - -Introduce two rewrite rules. - -\spadcommand{sinCosExpandRules := rule\\ -\ \ sin(x+y) == sin(x)*cos(y) + sin(y)*cos(x)\\ -\ \ cos(x+y) == cos(x)*cos(y) - sin(x)*sin(y)\\ -\ \ sin(2*x) == 2*sin(x)*cos(x)\\ -\ \ cos(2*x) == cos(x)**2 - sin(x)**2 -} - -\begin{verbatim} - {sin(y + x) == cos(x)sin(y) + cos(y)sin(x), - cos(y + x) == - sin(x)sin(y) + cos(x)cos(y), - sin(2x) == 2cos(x)sin(x), - 2 2 - cos(2x) == - sin(x) + cos(x) } -\end{verbatim} -\returnType{Type: Ruleset(Integer,Integer,Expression Integer)} - -Apply the rules to a simple trigonometric expression. - -\spadcommand{sinCosExpandRules(sin(a+2*b+c))} -$$ -\begin{array}{@{}l} -\displaystyle -{{\left(-{{\cos \left({a}\right)}\ {{\sin \left({b}\right)}^2}}- -{2 \ {\cos \left({b}\right)}\ {\sin \left({a}\right)}\ {\sin -\left({b}\right)}}+{{\cos \left({a}\right)}\ {{\cos \left({b}\right)}^ -2}}\right)}\ {\sin \left({c}\right)}}- -\\ -\\ -\displaystyle -{{\cos \left({c}\right)}\ {\sin \left({a}\right)}\ {{\sin \left({b}\right)}^ -2}}+{2 \ {\cos \left({a}\right)}\ {\cos \left({b}\right)}\ {\cos -\left({c}\right)}\ {\sin \left({b}\right)}}+ -\\ -\\ -\displaystyle -{{{\cos \left({b}\right)}^2}\ {\cos \left({c}\right)}\ {\sin -\left({a}\right)}} -\end{array} -$$ -\returnType{Type: Expression Integer} - - -Using input files, you can create your own library of transformation -rules relevant to your applications, then selectively apply the rules -you need. - -\subsection{Polymorphic Algorithms} -All components of the Axiom algebra library are written in the -Axiom library language. This language is similar to the -interactive language except for protocols that authors are obliged to -follow. The library language permits you to write ``polymorphic -algorithms,'' algorithms defined to work in their most natural -settings and over a variety of types. - -Define a system of polynomial equations $S$. - -\spadcommand{S := [3*x**3 + y + 1 = 0,y**2 = 4]} -$$ -\left[ -{{y+{3 \ {x \sp 3}}+1}=0}, {{y \sp 2}=4} -\right] -$$ -\returnType{Type: List Equation Polynomial Integer} - -Solve the system $S$ using rational number arithmetic and -30 digits of accuracy. - -\spadcommand{solve(S,1/10**30)} -$$ -\left[ -{\left[ {y=-2}, {x={{1757879671211184245283070414507} \over -{2535301200456458802993406410752}}} -\right]}, - {\left[ {y=2}, {x=-1} -\right]} -\right] -$$ -\returnType{Type: List List Equation Polynomial Fraction Integer} - -Solve $S$ with the solutions expressed in radicals. - -\spadcommand{radicalSolve(S)} -$$ -\begin{array}{@{}l} -\displaystyle -\left[{\left[{y = 2}, {x = - 1}\right]}, {\left[{y = 2}, -{x ={{-{\sqrt{- 3}}+ 1}\over 2}}\right]}, \right. -\\ -\\ -\displaystyle -\left.{\left[{y = 2}, {x ={{{\sqrt{- 3}}+ 1}\over 2}}\right]}, - {\left[{y = - 2}, {x ={1 \over{\root{3}\of{3}}}}\right]}, - \right. -\\ -\\ -\displaystyle -\left.{\left[{y = - 2}, {x ={{{{\sqrt{- 1}}\ {\sqrt{3}}}- 1}\over{2 -\ {\root{3}\of{3}}}}}\right]}, {\left[{y = - 2}, {x ={{-{{\sqrt{- - 1}}\ {\sqrt{3}}}- 1}\over{2 \ {\root{3}\of{3}}}}}\right]}\right] -\end{array} -$$ -\returnType{Type: List List Equation Expression Integer} - -While these solutions look very different, the results were produced -by the same internal algorithm! The internal algorithm actually works -with equations over any ``field.'' Examples of fields are the -rational numbers, floating point numbers, rational functions, power -series, and general expressions involving radicals. - -\subsection{Extensibility} - -Users and system developers alike can augment the Axiom library, -all using one common language. Library code, like interpreter code, -is compiled into machine binary code for run-time efficiency. - -Using this language, you can create new computational types and new -algorithmic packages. All library code is polymorphic, described in -terms of a database of algebraic properties. By following the -language protocols, there is an automatic, guaranteed interaction -between your code and that of colleagues and system implementers. -\vfill\eject -\pseudoChapter{A Technical Introduction} -\label{ugTechIntro} -Axiom has both an {\it interactive language} for user -interactions and a {\it programming language} for building library -modules. Like Modula 2, \index{Modula 2} PASCAL, \index{PASCAL} -FORTRAN, \index{FORTRAN} and Ada, \index{Ada} the programming language -emphasizes strict type-checking. Unlike these languages, types in -Axiom are dynamic objects: they are created at run-time in -response to user commands. - -Here is the idea of the Axiom programming language in a -nutshell. Axiom types range from algebraic ones (like -polynomials, matrices, and power series) to data structures (like -lists, dictionaries, and input files). Types combine in any -meaningful way. You can build polynomials of matrices, matrices of -polynomials of power series, hash tables with symbolic keys and -rational function entries, and so on. - -{\it Categories} define algebraic properties to ensure mathematical -correctness. They ensure, for example, that matrices of polynomials -are OK, but matrices of input files are not. Through categories, -programs can discover that polynomials of continued fractions have a -commutative multiplication whereas polynomials of matrices do not. - -Categories allow algorithms to be defined in their most natural -setting. For example, an algorithm can be defined to solve polynomial -equations over {\it any} field. Likewise a greatest common divisor -can compute the ``gcd'' of two elements from {\it any} Euclidean -domain. Categories foil attempts to compute meaningless ``gcds'', for -example, of two hashtables. Categories also enable algorithms to be -compiled into machine code that can be run with arbitrary types. - -The Axiom interactive language is oriented towards ease-of-use. -The Axiom interpreter uses type-inferencing to deduce the type -of an object from user input. Type declarations can generally be -omitted for common types in the interactive language. - -So much for the nutshell. -Here are these basic ideas described by ten design principles: - -\subsection{Types are Defined by Abstract Datatype Programs} - -Basic types are called {\it domains of computation}, or, -simply, {\it domains.} -\index{domain} -Domains are defined by Axiom programs of the form: - -\begin{verbatim} -Name(...): Exports == Implementation -\end{verbatim} - -Each domain has a capitalized {\tt Name} that is used to refer to the -class of its members. For example, {\tt Integer} denotes ``the -class of integers,'' {\tt Float}, ``the class of floating point -numbers,'' and {\tt String}, ``the class of strings.'' - -The ``{\tt ...}'' part following {\tt Name} lists zero or more -parameters to the constructor. Some basic ones like {\tt Integer} take -no parameters. Others, like {\tt Matrix}, {\tt Polynomial} and -{\tt List}, take a single parameter that again must be a domain. For -example, {\tt Matrix(Integer)} denotes ``matrices over the integers,'' -{\tt Polynomial (Float)} denotes ``polynomial with floating point -coefficients,'' and {\tt List (Matrix (Polynomial (Integer)))} denotes -``lists of matrices of polynomials over the integers.'' There is no -restriction on the number or type of parameters of a domain -constructor. - -SquareMatrix(2,Integer) is an example of a domain constructor that accepts -both a particular data value as well as an integer. In this case the -number 2 specifies the number of rows and columns the square matrix -will contain. Elements of the matricies are integers. - -The {\tt Exports} part specifies operations for creating and -manipulating objects of the domain. For example, type -{\tt Integer} exports constants $0$ and $1$, and -operations \spadopFrom{+}{Integer}, \spadopFrom{-}{Integer}, and -\spadopFrom{*}{Integer}. While these operations are common, others -such as \spadfunFrom{odd?}{Integer} and \spadfunFrom{bit?}{Integer} -are not. In addition the Exports section can contain symbols that -represent properties that can be tested. For example, the Category -{\tt EntireRing} has the symbol {\tt noZeroDivisors} which asserts -that if a product is zero then one of the factors must be zero. - -The {\tt Implementation} part defines functions that implement the -exported operations of the domain. These functions are frequently -described in terms of another lower-level domain used to represent the -objects of the domain. Thus the operation of adding two vectors of -real numbers can be described and implemented using the addition -operation from {\tt Float}. - -\subsection{The Type of Basic Objects is a Domain or Subdomain} - -Every Axiom object belongs to a {\it unique} domain. The domain -of an object is also called its {\it type.} Thus the integer $7$ -has type {\tt Integer} and the string {\tt "daniel"} has type -{\tt String}. - -The type of an object, however, is not unique. The type of integer -$7$ is not only {\tt Integer} but {\tt NonNegativeInteger}, -{\tt PositiveInteger}, and possibly, in general, any other -``subdomain'' of the domain {\tt Integer}. A {\it subdomain} -\index{subdomain} is a domain with a ``membership predicate''. -{\tt PositiveInteger} is a subdomain of {\tt Integer} with the -predicate ``is the integer $> 0$?''. - -Subdomains with names are defined by abstract datatype programs -similar to those for domains. The {\it Export} part of a subdomain, -however, must list a subset of the exports of the domain. The {\tt -Implementation} part optionally gives special definitions for -subdomain objects. - -\subsection{Domains Have Types Called Categories} - -Domain and subdomains in Axiom are themselves objects that have -types. The type of a domain or subdomain is called a {\it category}. -\index{category} Categories are described by programs of the form: - -\begin{verbatim} -Name(...): Category == Exports -\end{verbatim} -The type of every category is the distinguished symbol {\tt Category.} -The category {\tt Name} is used to designate the class of domains of -that type. For example, category {\tt Ring} designates the class -of all rings. Like domains, categories can take zero or more -parameters as indicated by the ``{\tt ...}'' part following {\tt -Name.} Two examples are {\tt Module(R)} and -{\tt MatrixCategory(R,Row,Col)}. - -The {\tt Exports} part defines a set of operations. For example, -{\tt Ring} exports the operations \spadopFrom{0}{Ring}, -\spadopFrom{1}{Ring}, \spadopFrom{+}{Ring}, \spadopFrom{-}{Ring}, and -\spadopFrom{*}{Ring}. Many algebraic domains such as -{\tt Integer} and {\tt Polynomial (Float)} are rings. -{\tt String} and {\tt List (R)} (for any domain $R$) -are not. - -Categories serve to ensure the type-correctness. The definition of -matrices states {\tt Matrix(R: Ring)} requiring its single parameter -$R$ to be a ring. Thus a ``matrix of polynomials'' is allowed, -but ``matrix of lists'' is not. - -Categories say nothing about representation. Domains, which are -instances of category types, specify representations. - -\subsection{Operations Can Refer To Abstract Types} - -All operations have prescribed source and target types. Types can be -denoted by symbols that stand for domains, called ``symbolic -domains.'' The following lines of Axiom code use a symbolic -domain $R$: - -\begin{verbatim} -R: Ring -power: (R, NonNegativeInteger): R -> R -power(x, n) == x ** n -\end{verbatim} - -Line 1 declares the symbol $R$ to be a ring. Line 2 declares the -type of $power$ in terms of $R$. From the definition on -line 3, $power(3,2)$ produces 9 for $x = 3$ and $R =$ -{\tt Integer}. Also, $power(3.0,2)$ produces $9.0$ for -$x = 3.0$ and $R =$ {\tt Float}. -$power("oxford",2)$ however fails since $"oxford"$ has type -{\tt String} which is not a ring. - -Using symbolic domains, algorithms can be defined in their most -natural or general setting. - -\subsection{Categories Form Hierarchies} - -Categories form hierarchies (technically, directed-acyclic graphs). A -simplified hierarchical world of algebraic categories is shown below. -At the top of this world is {\tt SetCategory}, the class of -algebraic sets. The notions of parents, ancestors, and descendants is -clear. Thus ordered sets (domains of category {\tt OrderedSet}) -and rings are also algebraic sets. Likewise, fields and integral -domains are rings and algebraic sets. However fields and integral -domains are not ordered sets. - -\begin{verbatim} -SetCategory +---- Ring ---- IntegralDomain ---- Field - | - +---- Finite ---+ - | \ - +---- OrderedSet -----+ OrderedFinite -\end{verbatim} -\begin{center} -Figure 1. A simplified category hierarchy. -\end{center} - -\subsection{Domains Belong to Categories by Assertion} - -A category designates a class of domains. Which domains? You might -think that {\tt Ring} designates the class of all domains that -export $0$, $1$, \spadopFrom{+}{Integer}, -\spadopFrom{-}{Integer}, and \spadopFrom{*}{Integer}. But this is not -so. Each domain must {\it assert} which categories it belongs to. - -The {\tt Export} part of the definition for {\tt Integer} reads, -for example: - -\begin{verbatim} -Join(OrderedSet, IntegralDomain, ...) with ... -\end{verbatim} - -This definition asserts that {\tt Integer} is both an ordered set -and an integral domain. In fact, {\tt Integer} does not -explicitly export constants $0$ and $1$ and operations -\spadopFrom{+}{Ring}, \spadopFrom{-}{Ring} and \spadopFrom{*}{Ring} at -all: it inherits them all from $Ring$! Since -{\tt IntegralDomain} is a descendant of $Ring$, -{\tt Integer} is therefore also a ring. - -Assertions can be conditional. For example, {\tt Complex(R)} -defines its exports by: - -\begin{verbatim} -Ring with ... if R has Field then Field ... -\end{verbatim} -Thus {\tt Complex(Float)} is a field but {\tt Complex(Integer)} -is not since {\tt Integer} is not a field. - -You may wonder: ``Why not simply let the set of operations determine -whether a domain belongs to a given category?''. Axiom allows -operation names (for example, {\bf norm}) to have very different -meanings in different contexts. The meaning of an operation in -Axiom is determined by context. By associating operations with -categories, operation names can be reused whenever appropriate or -convenient to do so. As a simple example, the operation {\tt <} -might be used to denote lexicographic-comparison in an algorithm. -However, it is wrong to use the same {\tt <} with this definition -of absolute-value: $$abs(x) == if\ x < 0\ then -x\ else\ x$$ Such a -definition for {\tt abs} in Axiom is protected by context: -argument $x$ is required to be a member of a domain of category -{\tt OrderedSet}. - -\subsection{Packages Are Clusters of Polymorphic Operations} - -In Axiom, facilities for symbolic integration, solution of -equations, and the like are placed in ``packages''. A {\it package} -\index{package} is a special kind of domain: one whose exported -operations depend solely on the parameters of the constructor and/or -explicit domains. Packages, unlike Domains, do not specify the -representation. - -If you want to use Axiom, for example, to define some algorithms -for solving equations of polynomials over an arbitrary field $F$, -you can do so with a package of the form: - -\begin{verbatim} -MySolve(F: Field): Exports == Implementation -\end{verbatim} -where {\tt Exports} specifies the {\bf solve} operations -you wish to export from the domain and the {\tt Implementation} -defines functions for implementing your algorithms. Once Axiom has -compiled your package, your algorithms can then be used for any {\tt F}: -floating-point numbers, rational numbers, complex rational functions, -and power series, to name a few. - -\subsection{The Interpreter Builds Domains Dynamically} - -The Axiom interpreter reads user input then builds whatever types -it needs to perform the indicated computations. -For example, to create the matrix -$$M = \pmatrix{x^2+1&0\cr0&x / 2\cr}$$ -using the command: - -\spadcommand{M = [ [x**2+1,0],[0,x / 2] ]::Matrix(POLY(FRAC(INT)))} -$$ -M={\left[ -\begin{array}{cc} -x^2+1 & 0 \\ -0 & x/2 -\end{array} -\right]} -$$ -\returnType{Type: Matrix Polynomial Fraction Integer} -the interpreter first loads the modules {\tt Matrix}, -{\tt Polynomial}, {\tt Fraction}, and {\tt Integer} -from the library, then builds the {\it domain tower} ``matrices of -polynomials of rational numbers (i.e. fractions of integers)''. - -You can watch the loading process by first typing - -\spadcommand{)set message autoload on} -In addition to the named -domains above many additional domains and categories are loaded. -Most systems are preloaded with such common types. For efficiency -reasons the most common domains are preloaded but most (there are -more than 1100 domains, categories, and packages) are not. Once these -domains are loaded they are immediately available to the interpreter. - -Once a domain tower is built, it contains all the operations specific -to the type. Computation proceeds by calling operations that exist in -the tower. For example, suppose that the user asks to square the -above matrix. To do this, the function \spadopFrom{*}{Matrix} from -{\tt Matrix} is passed the matrix $M$ to compute $M * M$. -The function is also passed an environment containing $R$ -that, in this case, is {\tt Polynomial (Fraction (Integer))}. -This results in the successive calling of the \spadopFrom{*}{Fraction} -operations from {\tt Polynomial}, then from {\tt Fraction}, -and then finally from {\tt Integer}. - -Categories play a policing role in the building of domains. Because -the argument of {\tt Matrix} is required to be a {\tt Ring}, -Axiom will not build nonsensical types such as ``matrices of -input files''. - -\subsection{Axiom Code is Compiled} - -Axiom programs are statically compiled to machine code, then -placed into library modules. Categories provide an important role in -obtaining efficient object code by enabling: -\begin{itemize} -\item static type-checking at compile time; -\item fast linkage to operations in domain-valued parameters; -\item optimization techniques to be used for partially specified types -(operations for ``vectors of $R$'', for instance, can be open-coded even -though {\tt R} is unknown). -\end{itemize} - -\subsection{Axiom is Extensible} - -Users and system implementers alike use the Axiom language to -add facilities to the Axiom library. The entire Axiom -library is in fact written in the Axiom source code and -available for user modification and/or extension. - -Axiom's use of abstract datatypes clearly separates the exports -of a domain (what operations are defined) from its implementation (how -the objects are represented and operations are defined). Users of a -domain can thus only create and manipulate objects through these -exported operations. This allows implementers to ``remove and -replace'' parts of the library safely by newly upgraded (and, we hope, -correct) implementations without consequence to its users. - -Categories protect names by context, making the same names available -for use in other contexts. Categories also provide for code-economy. -Algorithms can be parameterized categorically to characterize their -correct and most general context. Once compiled, the same machine -code is applicable in all such contexts. - -Finally, Axiom provides an automatic, guaranteed interaction -between new and old code. For example: -\begin{itemize} -\item if you write a new algorithm that requires a parameter to be a -field, then your algorithm will work automatically with every field -defined in the system; past, present, or future. -\item if you introduce a new domain constructor that produces a field, -then the objects of that domain can be used as parameters to any algorithm -using field objects defined in the system; past, present, or future. -\end{itemize} - -These are the key ideas. For further information, we particularly -recommend your reading chapters 11, 12, and 13, where these ideas are -explained in greater detail. - -\section{Using Axiom as a Pocket Calculator} -At the simplest level Axiom can be used as a pocket calculator -where expressions involving numbers and operators are entered -directly in infix notation. In this sense the more advanced -features of the calculator can be regarded as operators (e.g -{\bf sin}, {\bf cos}, etc). - -\subsection{Basic Arithmetic} -An example of this might be to calculate the cosine of 2.45 (in radians). -To do this one would type: - -\spadcommand{cos 2.45} -$$ --{0.7702312540 473073417} -$$ -\returnType{Type: Float} - -Before proceeding any further it would be best to explain the previous -three lines. Firstly the text ``(1) {\tt ->} '' is part of the prompt that the -Axiom system provides when in interactive mode. The full prompt has other -text preceding this but it is not relevant here. The number in parenthesis -is the step number of the input which may be used to refer to the -{\sl results} of previous calculations. The step number appears at the start -of the second line to tell you which step the result belongs to. Since the -interpreter probably loaded numberous libraries to calculate the result given -above and listed each one in the prcess, there could easily be several pages -of text between your input and the answer. - -The last line contains the type of the result. The type {\tt Float} is used -to represent real numbers of arbitrary size and precision (where the user is -able to define how big arbitrary is -- the default is 20 digits but can be -as large as your computer system can handle). The type of the result can help -track down mistakes in your input if you don't get the answer you expected. - -Other arithmetic operations such as addition, subtraction, and multiplication -behave as expected: - -\spadcommand{6.93 * 4.1328} -$$ -28.640304 -$$ -\returnType{Type: Float} - -\spadcommand{6.93 / 4.1328} -$$ -1.6768292682 926829268 -$$ -\returnType{Type: Float} - -but integer division isn't quite so obvious. For example, if one types: - -\spadcommand{4/6} -$$ -2 \over 3 -$$ -\returnType{Type: Fraction Integer} - -a fractional result is obtained. The function used to display fractions -attempts to produce the most readable answer. In the example: - -\spadcommand{4/2} -$$ -2 -$$ -\returnType{Type: Fraction Integer} - -the result is stored as the fraction 2/1 but is displayed as the integer 2. -This fraction could be converted to type {\tt Integer} with no loss of -information but Axiom will not do so automatically. - -\subsection{Type Conversion} -To obtain the floating point value of a fraction one must convert -({\bf conversions} are applied by the user and -{\bf coercions} are applied automatically by the interpreter) the result -to type {\tt Float} using the ``::'' operator as follows: - -\spadcommand{(4.6)::Float} -$$ -4.6 -$$ -\returnType{Type: Float} - -Although Axiom can convert this back to a fraction it might not be the -same fraction you started with as due to rounding errors. For example, the -following conversion appears to be without error but others might not: - -\spadcommand{\%::Fraction Integer} -$$ -{23} \over 5 -$$ -\returnType{Type: Fraction Integer} - -where ``\%'' represents the previous {\it result} (not the calculation). - -Although Axiom has the ability to work with floating-point numbers to -a very high precision it must be remembered that calculations with these -numbers are {\bf not} exact. Since Axiom is a computer algebra package and -not a numerical solutions package this should not create too many problems. -The idea is that the user should use Axiom to do all the necessary symbolic -manipulation and only at the end should actual numerical results be extracted. - -If you bear in mind that Axiom appears to store expressions just as you have -typed them and does not perform any evalutation of them unless forced to then -programming in the system will be much easier. It means that anything you -ask Axiom to do (within reason) will be carried out with complete accuracy. - -In the previous examples the ``::'' operator was used to convert values from -one type to another. This type conversion is not possible for all values. -For instance, it is not possible to convert the number 3.4 to an integer -type since it can't be represented as an integer. The number 4.0 can be -converted to an integer type since it has no fractional part. - -Conversion from floating point values to integers is performed using the -functions {\bf round} and {\bf truncate}. The first of these rounds a -floating point number to the nearest integer while the other truncates -(i.e. removes the fractional part). Both functions return the result as a -{\bf floating point} number. To extract the fractional part of a floating -point number use the function {\bf fractionPart} but note that the sign -of the result depends on the sign of the argument. Axiom obtains the -fractional partof $x$ using $x - truncate(x)$: - -\spadcommand{round(3.77623)} -$$ -4.0 -$$ -\returnType{Type: Float} - -\spadcommand{round(-3.77623)} -$$ --{4.0} -$$ -\returnType{Type: Float} - -\spadcommand{truncate(9.235)} -$$ -9.0 -$$ -\returnType{Type: Float} - -\spadcommand{truncate(-9.654)} -$$ --{9.0} -$$ -\returnType{Type: Float} - -\spadcommand{fractionPart(-3.77623)} -$$ --{0.77623} -$$ -\returnType{Type: Float} - -\subsection{Useful Functions} -To obtain the absolute value of a number the {\bf abs} function can be used. -This takes a single argument which is usually an integer or a floating point -value but doesn't necessarily have to be. The sign of a value can be obtained -via the {\bf sign} function which rturns $-1$, $0$, or $1$ depending on the -sign of the argument. - -\spadcommand{abs(4)} -$$ -4 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{abs(-3)} -$$ -3 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{abs(-34254.12314)} -$$ -34254.12314 -$$ -\returnType{Type: Float} - -\spadcommand{sign(-49543.2345346)} -$$ --1 -$$ -\returnType{Type: Integer} - -\spadcommand{sign(0)} -$$ -0 -$$ -\returnType{Type: NonNegativeInteger} - -\spadcommand{sign(234235.42354)} -$$ -1 -$$ -\returnType{Type: PositiveInteger} - -Tests on values can be done using various functions which are generally more -efficient than using relational operators such as $=$ particularly if the -value is a matrix. Examples of some of these functions are: - -\spadcommand{positive?(-234)} -$$ -{\tt false} -$$ -\returnType{Type: Boolean} - -\spadcommand{negative?(-234)} -$$ -{\tt true} -$$ -\returnType{Type: Boolean} - -\spadcommand{zero?(42)} -$$ -{\tt false} -$$ -\returnType{Type: Boolean} - -\spadcommand{one?(1)} -$$ -{\tt true} -$$ -\returnType{Type: Boolean} - -\spadcommand{odd?(23)} -$$ -{\tt true} -$$ -\returnType{Type: Boolean} - -\spadcommand{odd?(9.435)} -$$ -{\tt false} -$$ -\returnType{Type: Boolean} - -\spadcommand{even?(-42)} -$$ -{\tt true} -$$ -\returnType{Type: Boolean} - -\spadcommand{prime?(37)} -$$ -{\tt true} -$$ -\returnType{Type: Boolean} - -\spadcommand{prime?(-37)} -$$ -{\tt false} -$$ -\returnType{Type: Boolean} - -Some other functions that are quite useful for manipulating numerical values -are: - -\begin{verbatim} -sin(x) Sine of x -cos(x) Cosine of x -tan(x) Tangent of x -asin(x) Arcsin of x -acos(x) Arccos of x -atan(x) Arctangent of x -gcd(x,y) Greatest common divisor of x and y -lcm(x,y) Lowest common multiple of x and y -max(x,y) Maximum of x and y -min(x,y) Minimum of x and y -factorial(x) Factorial of x -factor(x) Prime factors of x -divide(x,y) Quotient and remainder of x/y -\end{verbatim} - -Some simple infix and prefix operators: -\begin{verbatim} -+ Addition - Subtraction -- Numerical Negation ~ Logical Negation -/\ Conjunction (AND) \/ Disjunction (OR) -and Logical AND (/\) or Logical OR (\/) -not Logical Negation ** Exponentiation -* Multiplication / Division -quo Quotient rem Remainder -< less than > greater than -<= less than or equal >= greater than or equal -\end{verbatim} - -Some useful Axiom macros: -\begin{verbatim} -%i The square root of -1 -%e The base of the natural logarithm -%pi Pi -%infinity Infinity -%plusInfinity Positive Infinity -%minusInfinity Negative Infinity -\end{verbatim} - -\section{Using Axiom as a Symbolic Calculator} -In the previous section all the examples involved numbers and simple -functions. Also none of the expressions entered were assigned to anything. -In this section we will move on to simple algebra (i.e. expressions involving -symbols and other features available on more sophisticated calculators). - -\subsection{Expressions Involving Symbols} -Expressions involving symbols are entered just as they are written down, -for example: - -\spadcommand{xSquared := x**2} -$$ -x \sp 2 -$$ -\returnType{Type: Polynomial Integer} - -where the assignment operator ``:='' represents immediate assignment. Later -it will be seen that this form of assignment is not always desirable and -the use of the delayed assignment operator ``=='' will be introduced. The -type of the result is {\tt Polynomial Integer} which is used to represent -polynomials with integer coefficients. Some other examples along similar -lines are: - -\spadcommand{xDummy := 3.21*x**2} -$$ -{3.21} \ {x \sp 2} -$$ -\returnType{Type: Polynomial Float} - -\spadcommand{xDummy := x**2.5} -$$ -{x \sp 2} \ {\sqrt {x}} -$$ -\returnType{Type: Expression Float} - -\spadcommand{xDummy := x**3.3} -$$ -{x \sp 3} \ {{\root {{10}} \of {x}} \sp 3} -$$ -\returnType{Type: Expression Float} - -\spadcommand{xyDummy := x**2 - y**2} -$$ --{y \sp 2}+{x \sp 2} -$$ -\returnType{Type: Polynomial Integer} - -Given that we can define expressions involving symbols, how do we actually -compute the result when the symbols are assigned values? The answer is to -use the {\bf eval} function which takes an expression as its first argument -followed by a list of assignments. For example, to evaluate the expressions -{\bf XDummy} and {xyDummy} resulting from their respective assignments above -we type: - -\spadcommand{eval(xDummy,x=3)} -$$ -37.5405075985 29552193 -$$ -\returnType{Type: Expression Float} - -\spadcommand{eval(xyDummy, [x=3, y=2.1])} -$$ -4.59 -$$ -\returnType{Type: Polynomial Float} - -\subsection{Complex Numbers} -For many scientific calculations real numbers aren't sufficient and support -for complex numbers is also required. Complex numbers are handled in an -intuitive manner and Axiom, which uses the {\bf \%i} macro to represent -the square root of $-1$. Thus expressions involving complex numbers are -entered just like other expressions. - -\spadcommand{(2/3 + \%i)**3} -$$ --{{46} \over {27}}+{{1 \over 3} \ i} -$$ -\returnType{Type: Complex Fraction Integer} - -The real and imaginary parts of a complex number can be extracted using -the {\bf real} and {\bf imag} functions and the complex conjugate of a -number can be obtained using {\bf conjugate}: - -\spadcommand{real(3 + 2*\%i)} -$$ -3 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{imag(3+ 2*\%i)} -$$ -2 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{conjugate(3 + 2*\%i)} -$$ -3 -{2 \ i} -$$ -\returnType{Type: Complex Integer} - -The function {\bf factor} can also be applied to complex numbers but the -results aren't quite so obvious as for factoring integer: - -\spadcommand{144 + 24*\%i} -$$ -{144}+{{24} \ i} -$$ -\returnType{Type: Complex Integer} - -\subsection{Number Representations} -By default all numerical results are displayed in decimal with real numbers -shown to 20 significant figures. If the integer part of a number is longer -than 20 digits then nothing after the decimal point is shown and the integer -part is given in full. To alter the number of digits shown the function -{\bf digits} can be called. The result returned by this function is the -previous setting. For example, to find the value of $\pi$ to 40 digits -we type: - -\spadcommand{digits(40)} -$$ -20 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{\%pi::Float} -$$ -3.1415926535\ 8979323846\ 2643383279\ 502884197 -$$ -\returnType{Type: Float} - -As can be seen in the example above, there is a gap after every ten digits. -This can be changed using the {\bf outputSpacing} function where the argument -is the number of digits to be displayed before a space is inserted. If no -spaces are desired then use the value $0$. Two other functions controlling -the appearance of real numbers are {\bf outputFloating} and {\bf outputFixed}. -The former causes Axiom to display floating-point values in exponent notation -and the latter causes it to use fixed-point notation. For example: - -\spadcommand{outputFloating(); \%} -$$ -0.3141592653 5897932384 6264338327 9502884197 E 1 -$$ -\returnType{Type: Float} - -\spadcommand{outputFloating(3); 0.00345} -$$ -0.345 E -2 -$$ -\returnType{Type: Float} - -\spadcommand{outputFixed(); \%} -$$ -0.00345 -$$ -\returnType{Type: Float} - -\spadcommand{outputFixed(3); \%} -$$ -0.003 -$$ -\returnType{Type: Float} - -\spadcommand{outputGeneral(); \%} -$$ -0.00345 -$$ -\returnType{Type: Float} - -Note that the semicolon ``;'' in the examples above allows several -expressions to be entered on one line. The result of the last expression -is displayed. remember also that the percent symbol ``\%'' is used to -represent the result of a previous calculation. - -To display rational numbers in a base other than 10 the function {\bf radix} -is used. The first argument of this function is the expression to be -displayed and the second is the base to be used. - -\spadcommand{radix(10**10,32)} -$$ -{\rm 9A0NP00 } -$$ -\returnType{Type: RadixExpansion 32} - -\spadcommand{radix(3/21,5)} -$$ -0.{\overline {032412}} -$$ -\returnType{Type: RadixExpansion 5} - -Rational numbers can be represented as a repeated decimal expansion using -the {\bf decimal} function or as a continued fraction using -{\bf continuedFraction}. Any attempt to call these functions with irrational -values will fail. - -\spadcommand{decimal(22/7)} -$$ -3.{\overline {142857}} -$$ -\returnType{Type: DecimalExpansion} - -\spadcommand{continuedFraction(6543/210)} -$$ -{31}+ \zag{1}{6}+ \zag{1}{2}+ \zag{1}{1}+ \zag{1}{3} -$$ -\returnType{Type: ContinuedFraction Integer} - -Finally, partial fractions in compact and expanded form are available via the -functions {\bf partialFraction} and {\bf padicFraction} respectively. The -former takes two arguments, the first being the numerator of the fraction -and the second being the denominator. The latter function takes a fraction -and expands it further while the function {\bf compactFraction} does the -reverse: - -\spadcommand{partialFraction(234,40)} -$$ -6 -{3 \over {2 \sp 2}}+{3 \over 5} -$$ -\returnType{Type: PartialFraction Integer} - -\spadcommand{padicFraction(\%)} -$$ -6 -{1 \over 2} -{1 \over {2 \sp 2}}+{3 \over 5} -$$ -\returnType{Type: PartialFraction Integer} - -\spadcommand{compactFraction(\%)} -$$ -6 -{3 \over {2 \sp 2}}+{3 \over 5} -$$ -\returnType{Type: PartialFraction Integer} - -\spadcommand{padicFraction(234/40)} -$$ -{117} \over {20} -$$ -\returnType{Type: PartialFraction Fraction Integer} - -To extract parts of a partial fraction the function {\bf nthFractionalTerm} -is available and returns a partial fraction of one term. To decompose this -further the numerator can be obtained using {\bf firstNumer} and the -denominator with {\bf firstDenom}. The whole part of a partial fraction can -be retrieved using {\bf wholePart} and the number of fractional parts can -be found using the function {\bf numberOf FractionalTerms}: - -\spadcommand{t := partialFraction(234,40)} -$$ -6 -{3 \over {2 \sp 2}}+{3 \over 5} -$$ -\returnType{Type: PartialFraction Integer} - -\spadcommand{wholePart(t)} -$$ -6 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{numberOfFractionalTerms(t)} -$$ -2 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{p := nthFractionalTerm(t,1)} -$$ --{3 \over {2 \sp 2}} -$$ -\returnType{Type: PartialFraction Integer} - -\spadcommand{firstNumer(p)} -$$ --3 -$$ -\returnType{Type: Integer} - -\spadcommand{firstDenom(p)} -$$ -2 \sp 2 -$$ -\returnType{Type: Factored Integer} - -\subsection{Modular Arithmetic} -By using the type constructor {\tt PrimeField} it is possible to do -arithmetic modulo some prime number. For example, arithmetic module $7$ -can be performed as follows: - -\spadcommand{x : PrimeField 7 := 5} -$$ -5 -$$ -\returnType{Type: PrimeField 7} - -\spadcommand{x**5 + 6} -$$ -2 -$$ -\returnType{Type: PrimeField 7} - -\spadcommand{1/x} -$$ -3 -$$ -\returnType{Type: PrimeField 7} - -The first example should be read as: -\begin{center} -{\tt Let $x$ be of type PrimeField(7) and assign to it the value $5$} -\end{center} - -Note that it is only possible to invert non-zero values if the arithmetic -is performed modulo a prime number. Thus arithmetic modulo a non-prime -integer is possible but the reciprocal operation is undefined and will -generate an error. Attempting to use the {\tt PrimeField} type constructor -with a non-prime argument will generate an error. An example of non-prime -modulo arithmetic is: - -\spadcommand{y : IntegerMod 8 := 11} -$$ -3 -$$ -\returnType{Type: IntegerMod 8} - -\spadcommand{y*4 + 27} -$$ -7 -$$ -\returnType{Type: IntegerMod 8} - -Note that polynomials can be constructed in a similar way: - -\spadcommand{(3*a**4 + 27*a - 36)::Polynomial PrimeField 7} -$$ -{3 \ {a \sp 4}}+{6 \ a}+6 -$$ -\returnType{Type: Polynomial PrimeField 7} - -\section{General Points about Axiom} -\subsection{Computation Without Output} -It is sometimes desirable to enter an expression and prevent Axiom from -displaying the result. To do this the expression should be terminated with -a semicolon ``;''. In a previous section it was mentioned that a set of -expressions separated by semicolons would be evaluated and the result -of the last one displayed. Thus if a single expression is followed by a -semicolon no output will be produced (except for its type): - -\spadcommand{2 + 4*5;} -\returnType{Type: PositiveInteger} - -\subsection{Accessing Earlier Results} -The ``\%'' macro represents the result of the previous computation. The -``\%\%'' macro is available which takes a single integer argument. If the -argument is positive then it refers to the step number of the calculation -where the numbering begins from one and can be seen at the end of each -prompt (the number in parentheses). If the argument is negative then it -refers to previous results counting backwards from the last result. That is, -``\%\%(-1)'' is the same as ``\%''. The value of ``\%\%(0)'' is not defined and -will generate an error if requested. - -\subsection{Splitting Expressions Over Several Lines} -Although Axiom will quite happily accept expressions that are longer than -the width of the screen (just keep typing without pressing the {\bf Return} -key) it is often preferable to split the expression being entered at a point -where it would result in more readable input. To do this the underscore -``\_'' symbol is placed before the break point and then the {\bf Return} -key is pressed. The rest of the expression is typed on the next line, -can be preceeded by any number of whitespace chars, for example: -\begin{verbatim} -2_ -+_ -3 -\end{verbatim} -$$ -5 -$$ -\returnType{Type: PositiveInteger} - -The underscore symbol is an escape character and its presence alters the -meaning of the characters that follow it. As mentions above whitespace -following an underscore is ignored (the {\bf Return} key generates a -whitespace character). Any other character following an underscore loses -whatever special meaning it may have had. Thus one can create the -identifier ``a+b'' by typing ``a\_+b'' although this might lead to confusions. -Also note the result of the following example: - -\spadcommand{ThisIsAVeryLong\_\\ -VariableName} -$$ -ThisIsAVeryLongVariableName -$$ -\returnType{Type: Variable ThisIsAVeryLongVariableName} - -\subsection{Comments and Descriptions} -Comments and descriptions are really only of use in files of Axiom code but -can be used when the output of an interactive session is being spooled to -a file (via the system command {\bf )spool}). A comment begins with two -dashes ``- -'' and continues until the end of the line. Multi-line -comments are only possible if each individual line begins with two dashes. - -Descriptions are the same as comments except that the Axiom compiler will -include them in the object files produced and make them availabe to the -end user for documentation purposes. - -A description is placed {\bf before} a calculation begins with three -``+++'' signs and a description placed after a calculation begins with -two plus symbols ``++''. The so-called ``plus plus'' comments are used -within the algebra files and are processed by the compiler to add -to the documentation. The so-called ``minus minus'' comments are ignored -everywhere. - -\subsection{Control of Result Types} -In earlier sections the type of an expression was converted to another -via the ``::'' operator. However, this is not the only method for -converting between types and two other operators need to be introduced -and explained. - -The first operator is ``\$'' and is used to specify the package to be -used to calculate the result. Thus: - -\spadcommand{(2/3)\$Float} -$$ -0.6666666666\ 6666666667 -$$ -\returnType{Type: Float} - -tells Axiom to use the ``/'' operator from the {\tt Float} package to -evaluate the expression $2/3$. This does not necessarily mean that the -result will be of the same type as the domain from which the operator -was taken. In the following example the {\bf sign} operator is taken -from the {\tt Float} package but the result is of type {\tt Integer}. - -\spadcommand{sign(2.3)\$Float} -$$ -1 -$$ -\returnType{Type: Integer} - -The other operator is ``@'' which is used to tell Axiom what the desired -type of the result of the calculation is. In most situations all three -operators yield the same results but the example below should help -distinguish them. - -\spadcommand{(2 + 3)::String} -$$ -\mbox{\tt "5"} -$$ -\returnType{Type: String} - -\spadcommand{(2 + 3)@String} -\begin{verbatim} -An expression involving @ String actually evaluated to one of - type PositiveInteger . Perhaps you should use :: String . -\end{verbatim} - -\spadcommand{(2 + 3)\$String} -\begin{verbatim} - The function + is not implemented in String . -\end{verbatim} - -If an expression {\sl X} is converted using one of the three operators to -type {\sl T} the interpretations are: - -{\bf ::} means explicitly convert {\sl X} to type {\sl T} if possible. - -{\bf \$} means use the available operators for type {\sl T} to compute {\sl X}. - -{\bf @} means choose operators to compute {\sl X} so that the result is of -type {\sl T}. - -\section{Data Structures in Axiom} -This chapter is an overview of {\sl some} of the data structures provided -by Axiom. -\subsection{Lists} -The Axiom {\tt List} type constructor is used to create homogenous lists of -finite size. The notation for lists and the names of the functions that -operate over them are similar to those found in functional languages such -as ML. - -Lists can be created by placing a comma separated list of values inside -square brackets or if a list with just one element is desired then the -function {\bf list} is available: - -\spadcommand{[4]} -$$ -\left[ -4 -\right] -$$ -\returnType{Type: List PositiveInteger} - -\spadcommand{list(4)} -$$ -\left[ -4 -\right] -$$ -\returnType{Type: List PositiveInteger} - -\spadcommand{[1,2,3,5,7,11]} -$$ -\left[ -1, 2, 3, 5, 7, {11} -\right] -$$ -\returnType{Type: List PositiveInteger} - -The function {\bf append} takes two lists as arguments and returns the list -consisting of the second argument appended to the first. A single element -can be added to the front of a list using {\bf cons}: - -\spadcommand{append([1,2,3,5],[7,11])} -$$ -\left[ -1, 2, 3, 5, 7, {11} -\right] -$$ -\returnType{Type: List PositiveInteger} - -\spadcommand{cons(23,[65,42,19])} -$$ -\left[ -{23}, {65}, {42}, {19} -\right] -$$ -\returnType{Type: List PositiveInteger} - -Lists are accessed sequentially so if Axiom is asked for the value of the -twentieth element in the list it will move from the start of the list over -nineteen elements before it reaches the desired element. Each element of a -list is stored as a node consisting of the value of the element and a pointer -to the rest of the list. As a result the two main operations on a list are -called {\bf first} and {\bf rest}. Both of these functions take a second -optional argument which specifies the length of the first part of the list: - -\spadcommand{first([1,5,6,2,3])} -$$ -1 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{first([1,5,6,2,3],2)} -$$ -\left[ -1, 5 -\right] -$$ -\returnType{Type: List PositiveInteger} - -\spadcommand{rest([1,5,6,2,3])} -$$ -\left[ -5, 6, 2, 3 -\right] -$$ -\returnType{Type: List PositiveInteger} - -\spadcommand{rest([1,5,6,2,3],2)} -$$ -\left[ -6, 2, 3 -\right] -$$ -\returnType{Type: List PositiveInteger} - -Other functions are {\bf empty?} which tests to see if a list contains no -elements, {\bf member?} which tests to see if the first argument is a member -of the second, {\bf reverse} which reverses the order of the list, {\bf sort} -which sorts a list, and {\bf removeDuplicates} which removes any duplicates. -The length of a list can be obtained using the ``\#'' operator. - -\spadcommand{empty?([7,2,-1,2])} -$$ -{\tt false} -$$ -\returnType{Type: Boolean} - -\spadcommand{member?(-1,[7,2,-1,2])} -$$ -{\tt true} -$$ -\returnType{Type: Boolean} - -\spadcommand{reverse([7,2,-1,2])} -$$ -\left[ -2, -1, 2, 7 -\right] -$$ -\returnType{Type: List Integer} - -\spadcommand{sort([7,2,-1,2])} -$$ -\left[ --1, 2, 2, 7 -\right] -$$ -\returnType{Type: List Integer} - -\spadcommand{removeDuplicates([1,5,3,5,1,1,2])} -$$ -\left[ -1, 5, 3, 2 -\right] -$$ -\returnType{Type: List PositiveInteger} - -\spadcommand{\#[7,2,-1,2]} -$$ -4 -$$ -\returnType{Type: PositiveInteger} - -Lists in Axiom are mutable and so their contents (the elements and the links) -can be modified in place. Functions that operator over lists in this way have -names ending in the symbol ``!''. For example, {\bf concat!} takes two lists -as arguments and appends the second argument to the first (except when the -first argument is an empty list) and {\bf setrest!} changes the link -emanating from the first argument to point to the second argument: - -\spadcommand{u := [9,2,4,7]} -$$ -\left[ -9, 2, 4, 7 -\right] -$$ -\returnType{Type: List PositiveInteger} - -\spadcommand{concat!(u,[1,5,42]); u} -$$ -\left[ -9, 2, 4, 7, 1, 5, {42} -\right] -$$ -\returnType{Type: List PositiveInteger} - -\spadcommand{endOfu := rest(u,4)} -$$ -\left[ -1, 5, {42} -\right] -$$ -\returnType{Type: List PositiveInteger} - -\spadcommand{partOfu := rest(u,2)} -$$ -\left[ -4, 7, 1, 5, {42} -\right] -$$ -\returnType{Type: List PositiveInteger} - -\spadcommand{setrest!(endOfu,partOfu); u} -$$ -\left[ -9, 2, {\overline {4, 7, 1}} -\right] -$$ -\returnType{Type: List PositiveInteger} - -From this it can be seen that the lists returned by {\bf first} and {\bf rest} -are pointers to the original list and {\sl not} a copy. Thus great care must -be taken when dealing with lists in Axiom. - -Although the {\sl n}th element of the list {\sl l} can be obtained by -applying the {\bf first} function to $n-1$ applications of {\bf rest} -to {\sl l}, Axiom provides a more useful access method in the form of -the ``.'' operator: - -\spadcommand{u.3} -$$ -4 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{u.5} -$$ -1 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{u.6} -$$ -4 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{first rest rest u -- Same as u.3} -$$ -4 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{u.first} -$$ -9 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{u(3)} -$$ -4 -$$ -\returnType{Type: PositiveInteger} - -The operation {\sl u.i} is referred to as {\sl indexing into u} or -{\sl elting into u}. The latter term comes from the {\bf elt} function -which is used to extract elements (the first element of the list is at -index $1$). - -\spadcommand{elt(u,4)} -$$ -7 -$$ -\returnType{Type: PositiveInteger} - -If a list has no cycles then any attempt to access an element beyond the -end of the list will generate an error. However, in the example above there -was a cycle starting at the third element so the access to the sixth -element wrapped around to give the third element. Since lists are mutable it -is possible to modify elements directly: - -\spadcommand{u.3 := 42; u} -$$ -\left[ -9, 2, {\overline {{42}, 7, 1}} -\right] -$$ -\returnType{Type: List PositiveInteger} - -Other list operations are: -\spadcommand{L := [9,3,4,7]; \#L} -$$ -4 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{last(L)} -$$ -7 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{L.last} -$$ -7 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{L.(\#L - 1)} -$$ -4 -$$ -\returnType{Type: PositiveInteger} - -Note that using the ``\#'' operator on a list with cycles causes Axiom to -enter an infinite loop. - -Note that any operation on a list {\sl L} that returns a list ${\sl L}L^{'}$ -will, in general, be such that any changes to ${\sl L}L^{'}$ will have the -side-effect of altering {\sl L}. For example: - -\spadcommand{m := rest(L,2)} -$$ -\left[ -4, 7 -\right] -$$ -\returnType{Type: List PositiveInteger} - -\spadcommand{m.1 := 20; L} -$$ -\left[ -9, 3, {20}, 7 -\right] -$$ -\returnType{Type: List PositiveInteger} - -\spadcommand{n := L} -$$ -\left[ -9, 3, {20}, 7 -\right] -$$ -\returnType{Type: List PositiveInteger} - -\spadcommand{n.2 := 99; L} -$$ -\left[ -9, {99}, {20}, 7 -\right] -$$ -\returnType{Type: List PositiveInteger} - -\spadcommand{n} -$$ -\left[ -9, {99}, {20}, 7 -\right] -$$ -\returnType{Type: List PositiveInteger} - -Thus the only safe way of copying lists is to copy each element from one to -another and not use the assignment operator: - -\spadcommand{p := [i for i in n] -- Same as `p := copy(n)'} -$$ -\left[ -9, {99}, {20}, 7 -\right] -$$ -\returnType{Type: List PositiveInteger} - -\spadcommand{p.2 := 5; p} -$$ -\left[ -9, 5, {20}, 7 -\right] -$$ -\returnType{Type: List PositiveInteger} - -\spadcommand{n} -$$ -\left[ -9, {99}, {20}, 7 -\right] -$$ -\returnType{Type: List PositiveInteger} - -In the previous example a new way of constructing lists was given. This is -a powerful method which gives the reader more information about the contents -of the list than before and which is extremely flexible. The example - -\spadcommand{[i for i in 1..10]} -$$ -\left[ -1, 2, 3, 4, 5, 6, 7, 8, 9, {10} -\right] -$$ -\returnType{Type: List PositiveInteger} - -should be read as - -\begin{center} -``Using the expression {\sl i}, generate each element of the list by -iterating the symbol {\sl i} over the range of integers [1,10]'' -\end{center} - -To generate the list of the squares of the first ten elements we just use: - -\spadcommand{[i**2 for i in 1..10]} -$$ -\left[ -1, 4, 9, {16}, {25}, {36}, {49}, {64}, {81}, {100} -\right] -$$ -\returnType{Type: List PositiveInteger} - -For more complex lists we can apply a condition to the elements that are to -be placed into the list to obtain a list of even numbers between 0 and 11: - -\spadcommand{[i for i in 1..10 | even?(i)]} -$$ -\left[ -2, 4, 6, 8, {10} -\right] -$$ -\returnType{Type: List PositiveInteger} - -This example should be read as: -\begin{center} -``Using the expression {\sl i}, generate each element of the list -by iterating the symbol {\sl i} over the range of integers [1,10] such that -{\sl i} is even'' -\end{center} - -The following achieves the same result: - -\spadcommand{[i for i in 2..10 by 2]} -$$ -\left[ -2, 4, 6, 8, {10} -\right] -$$ -\returnType{Type: List PositiveInteger} - -\subsection{Segmented Lists} -A segmented list is one in which some of the elements are ranges of values. -The {\bf expand} function converts lists of this type into ordinary lists: - -\spadcommand{[1..10]} -$$ -\left[ -{1..{10}} -\right] -$$ -\returnType{Type: List Segment PositiveInteger} - -\spadcommand{[1..3,5,6,8..10]} -$$ -\left[ -{1..3}, {5..5}, {6..6}, {8..{10}} -\right] -$$ -\returnType{Type: List Segment PositiveInteger} - -\spadcommand{expand(\%)} -$$ -\left[ -1, 2, 3, 5, 6, 8, 9, {10} -\right] -$$ -\returnType{Type: List Integer} - -If the upper bound of a segment is omitted then a different type of -segmented list is obtained and expanding it will produce a stream (which -will be considered in the next section): - -\spadcommand{[1..]} -$$ -\left[ -{1..} -\right] -$$ -\returnType{Type: List UniversalSegment PositiveInteger} - -\spadcommand{expand(\%)} -$$ -\left[ -1, 2, 3, 4, 5, 6, 7, 8, 9, {10}, \ldots -\right] -$$ -\returnType{Type: Stream Integer} - -\subsection{Streams} -Streams are infinite lists which have the ability to calculate the next -element should it be required. For example, a stream of positive integers -and a list of prime numbers can be generated by: - -\spadcommand{[i for i in 1..]} -$$ -\left[ -1, 2, 3, 4, 5, 6, 7, 8, 9, {10}, \ldots -\right] -$$ -\returnType{Type: Stream PositiveInteger} - -\spadcommand{[i for i in 1.. | prime?(i)]} -$$ -\left[ -2, 3, 5, 7, {11}, {13}, {17}, {19}, {23}, {29}, -\ldots -\right] -$$ -\returnType{Type: Stream PositiveInteger} - -In each case the first few elements of the stream are calculated for display -purposes but the rest of the stream remains unevaluated. The value of items -in a stream are only calculated when they are needed which gives rise to -their alternative name of ``lazy lists''. - -Another method of creating streams is to use the {\bf generate(f,a)} function. -This applies its first argument repeatedly onto its second to produce the -stream $[a,f(a),f(f(a)),f(f(f(a)))\ldots]$. Given that the function -{\bf nextPrime} returns the lowest prime number greater than its argument we -can generate a stream of primes as follows: -\spadcommand{generate(nextPrime,2)\$Stream Integer} -$$ -\left[ -2, 3, 5, 7, {11}, {13}, {17}, {19}, {23}, {29}, -\ldots -\right] -$$ -\returnType{Type: Stream Integer} - -As a longer example a stream of Fibonacci numbers will be computed. The -Fibonacci numbers start at $1$ and each following number is the addition -of the two numbers that precede it so the Fibonacci sequence is: -$$1,1,2,3,5,8,\ldots$$. - -Since the generation of any Fibonacci number only relies on knowing the -previous two numbers we can look at the series through a window of two -elements. To create the series the window is placed at the start over -the values $[1,1]$ and their sum obtained. The window is now shifted to -the right by one position and the sum placed into the empty slot of the -window; the process is then repeated. To implement this we require a -function that takes a list of two elements (the current view of the window), -adds them, and outputs the new window. The result is the function -$[a,b]$~{\tt ->}~$[b,a+b]$: -\spadcommand{win : List Integer -> List Integer} -\returnType{Type: Void} - -\spadcommand{win(x) == [x.2, x.1 + x.2]} -\returnType{Type: Void} - -\spadcommand{win([1,1])} -$$ -\left[ -1, 2 -\right] -$$ -\returnType{Type: List Integer} - -\spadcommand{win(\%)} -$$ -\left[ -2, 3 -\right] -$$ -\returnType{Type: List Integer} - -Thus it can be seen that by repeatedly applying {\bf win} to the {\sl results} -of the previous invocation each element of the series is obtained. Clearly -{\bf win} is an ideal function to construct streams using the {\bf generate} -function: -\spadcommand{fibs := [generate(win,[1,1])]} -$$ -\left[ -{\left[ 1, 1 -\right]}, - {\left[ 1, 2 -\right]}, - {\left[ 2, 3 -\right]}, - {\left[ 3, 5 -\right]}, - {\left[ 5, 8 -\right]}, - {\left[ 8, {13} -\right]}, - {\left[ {13}, {21} -\right]}, - {\left[ {21}, {34} -\right]}, - {\left[ {34}, {55} -\right]}, - {\left[ {55}, {89} -\right]}, - \ldots -\right] -$$ -\returnType{Type: Stream List Integer} - -This isn't quite what is wanted -- we need to extract the first element of -each list and place that in our series: -\spadcommand{fibs := [i.1 for i in [generate(win,[1,1])] ]} -$$ -\left[ -1, 1, 2, 3, 5, 8, {13}, {21}, {34}, {55}, -\ldots -\right] -$$ -\returnType{Type: Stream Integer} - -Obtaining the 200th Fibonacci number is trivial: -\spadcommand{fibs.200} -$$ -280571172992510140037611932413038677189525 -$$ -\returnType{Type: PositiveInteger} - -One other function of interest is {\bf complete} which expands a finite -stream derived from an infinite one (and thus was still stored as an -infinite stream) to form a finite stream. - -\subsection{Arrays, Vectors, Strings, and Bits} -The simplest array data structure is the {\sl one-dimensional array} which -can be obtained by applying the {\bf oneDimensionalArray} function to a list: -\spadcommand{oneDimensionalArray([7,2,5,4,1,9])} -$$ -\left[ -7, 2, 5, 4, 1, 9 -\right] -$$ -\returnType{Type: OneDimensionalArray PositiveInteger} - -One-dimensional array are homogenous (all elements must have the same type) -and mutable (elements can be changed) like lists but unlike lists they are -constant in size and have uniform access times (it is just as quick to read -the last element of a one-dimensional array as it is to read the first; this -is not true for lists). - -Since these arrays are mutable all the warnings that apply to lists apply to -arrays. That is, it is possible to modify an element in a copy of an array -and change the original: -\spadcommand{x := oneDimensionalArray([7,2,5,4,1,9])} -$$ -\left[ -7, 2, 5, 4, 1, 9 -\right] -$$ -\returnType{Type: OneDimensionalArray PositiveInteger} - -\spadcommand{y := x} -$$ -\left[ -7, 2, 5, 4, 1, 9 -\right] -$$ -\returnType{Type: OneDimensionalArray PositiveInteger} - -\spadcommand{y.3 := 20 ; x} -$$ -\left[ -7, 2, {20}, 4, 1, 9 -\right] -$$ -\returnType{Type: OneDimensionalArray PositiveInteger} - -Note that because these arrays are of fixed size the {\bf concat!} function -cannot be applied to them without generating an error. If arrays of this -type are required use the {\bf FlexibleArray} constructor. - -One-dimensional arrays can be created using {\bf new} which specifies the size -of the array and the initial value for each of the elements. Other operations -that can be applied to one-dimensional arrays are {\bf map!} which applies -a mapping onto each element, {\bf swap!} which swaps two elements and -{\bf copyInto!(a,b,c)} which copies the array {\sl b} onto {\sl a} starting at -position {\sl c}. -\spadcommand{a : ARRAY1 PositiveInteger := new(10,3)} -$$ -\left[ -3, 3, 3, 3, 3, 3, 3, 3, 3, 3 -\right] -$$ -\returnType{Type: OneDimensionalArray PositiveInteger} - -(note that {\tt ARRAY1} is an abbreviation for the type -{\tt OneDimensionalArray}.) Other types based on one-dimensional arrays are -{\tt Vector}, {\tt String}, and {tt Bits}. - -\spadcommand{map!(i +-> i+1,a); a} -$$ -\left[ -4, 4, 4, 4, 4, 4, 4, 4, 4, 4 -\right] -$$ -\returnType{Type: OneDimensionalArray PositiveInteger} - -\spadcommand{b := oneDimensionalArray([2,3,4,5,6])} -$$ -\left[ -2, 3, 4, 5, 6 -\right] -$$ -\returnType{Type: OneDimensionalArray PositiveInteger} - -\spadcommand{swap!(b,2,3); b} -$$ -\left[ -2, 4, 3, 5, 6 -\right] -$$ -\returnType{Type: OneDimensionalArray PositiveInteger} - -\spadcommand{copyInto!(a,b,3)} -$$ -\left[ -4, 4, 2, 4, 3, 5, 6, 4, 4, 4 -\right] -$$ -\returnType{Type: OneDimensionalArray PositiveInteger} - -\spadcommand{a} -$$ -\left[ -4, 4, 2, 4, 3, 5, 6, 4, 4, 4 -\right] -$$ -\returnType{Type: OneDimensionalArray PositiveInteger} - -\spadcommand{vector([1/2,1/3,1/14])} -$$ -\left[ -{1 \over 2}, {1 \over 3}, {1 \over {14}} -\right] -$$ -\returnType{Type: Vector Fraction Integer} - -\spadcommand{"Hello, World"} -$$ -\mbox{\tt "Hello, World"} -$$ -\returnType{Type: String} - -\spadcommand{bits(8,true)} -$$ -\mbox{\tt "11111111"} -$$ -\returnType{Type: Bits} - -A vector is similar to a one-dimensional array except that if its -components belong to a ring then arithmetic operations are provided. - -\subsection{Flexible Arrays} -Flexible arrays are designed to provide the efficiency of one-dimensional -arrays while retaining the flexibility of lists. They are implemented by -allocating a fixed block of storage for the array. If the array needs to -be expanded then a larger block of storage is allocated and the contents -of the old block are copied into the new one. - -There are several operations that can be applied to this type, most of -which modify the array in place. As a result these functions all have -names ending in ``!''. The {\bf physicalLength} returns the actual length -of the array as stored in memory while the {\bf physicalLength!} allows this -value to be changed by the user. -\spadcommand{f : FARRAY INT := new(6,1)} -$$ -\left[ -1, 1, 1, 1, 1, 1 -\right] -$$ -\returnType{Type: FlexibleArray Integer} - -\spadcommand{f.1:=4; f.2:=3 ; f.3:=8 ; f.5:=2 ; f} -$$ -\left[ -4, 3, 8, 1, 2, 1 -\right] -$$ -\returnType{Type: FlexibleArray Integer} - -\spadcommand{insert!(42,f,3); f} -$$ -\left[ -4, 3, {42}, 8, 1, 2, 1 -\right] -$$ -\returnType{Type: FlexibleArray Integer} - -\spadcommand{insert!(28,f,8); f} -$$ -\left[ -4, 3, {42}, 8, 1, 2, 1, {28} -\right] -$$ -\returnType{Type: FlexibleArray Integer} - -\spadcommand{removeDuplicates!(f)} -$$ -\left[ -4, 3, {42}, 8, 1, 2, {28} -\right] -$$ -\returnType{Type: FlexibleArray Integer} - -\spadcommand{delete!(f,5)} -$$ -\left[ -4, 3, {42}, 8, 2, {28} -\right] -$$ -\returnType{Type: FlexibleArray Integer} - -\spadcommand{g:=f(3..5)} -$$ -\left[ -{42}, 8, 2 -\right] -$$ -\returnType{Type: FlexibleArray Integer} - -\spadcommand{g.2:=7; f} -$$ -\left[ -4, 3, {42}, 8, 2, {28} -\right] -$$ -\returnType{Type: FlexibleArray Integer} - -\spadcommand{insert!(g,f,1)} -$$ -\left[ -{42}, 7, 2, 4, 3, {42}, 8, 2, {28} -\right] -$$ -\returnType{Type: FlexibleArray Integer} - -\spadcommand{physicalLength(f)} -$$ -10 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{physicalLength!(f,20)} -$$ -\left[ -{42}, 7, 2, 4, 3, {42}, 8, 2, {28} -\right] -$$ -\returnType{Type: FlexibleArray Integer} - -\spadcommand{merge!(sort!(f),sort!(g))} -$$ -\left[ -2, 2, 2, 3, 4, 7, 7, 8, {28}, {42}, {42}, -{42} -\right] -$$ -\returnType{Type: FlexibleArray Integer} - -\spadcommand{shrinkable(false)\$FlexibleArray(Integer)} -$$ -{\tt true} -$$ -\returnType{Type: Boolean} - -There are several things to point out concerning these -examples. First, although flexible arrays are mutable, making copies -of these arrays creates separate entities. This can be seen by the -fact that the modification of element {\sl b.2} above did not alter -{\sl a}. Second, the {\bf merge!} function can take an extra argument -before the two arrays are merged. The argument is a comparison -function and defaults to ``{\tt <=}'' if omitted. Lastly, -{\bf shrinkable} tells the system whether or not to let flexible arrays -contract when elements are deleted from them. An explicit package -reference must be given as in the example above. - -\section{Functions, Choices, and Loops} -By now the reader should be able to construct simple one-line expressions -involving variables and different data structures. This section builds on -this knowledge and shows how to use iteration, make choices, and build -functions in Axiom. At the moment it is assumed that the reader has a rough -idea of how types are specified and constructed so that they can follow -the examples given. - -From this point on most examples will be taken from input files. - -\subsection{Reading Code from a File} -Input files contain code that will be fed to the command prompt. The -primary different between the command line and an input file is that -indentation matters. In an input file you can specify ``piles'' of code -by using indentation. - -The names of all input files in Axiom should end in ``.input'' otherwise -Axiom will refuse to read them. - -If an input file is named {\bf foo.input} you can feed the contents of -the file to the command prompt (as though you typed them) by writing: -{\bf )read foo.input}. - -It is good practice to start each input file with the {\bf )clear all} -command so that all functions and variables in the current environment -are erased. -\subsection{Blocks} -The Axiom constructs that provide looping, choices, and user-defined -functions all rely on the notion of blocks. A block is a sequence of -expressions which are evaluated in the order that they appear except -when it is modified by control expressions such as loops. To leave a -block prematurely use an expression of the form: -{\sl BoolExpr}~{\tt =>}~{\sl Expr} -where {\sl BoolExpr} is any Axiom expression that has type {\tt Boolean}. -The value and type of {\sl Expr} determines the value and type returned -by the block. - -If blocks are entered at the keyboard (as opposed to reading them from -a text file) then there is only one way of creating them. The syntax is: -$$( expression1 ; expression2; \ldots ; expressionN )$$ - -In an input file a block can be constructed as above or by placing all the -statements at the same indentation level. When indentation is used to -indicate program structure the block is called a {\sl pile}. As an example -of a simple block a list of three integers can be constructed using -parentheses: -\spadcommand{( a:=4; b:=1; c:=9; L:=[a,b,c])} -$$ -\left[ -4, 1, 9 -\right] -$$ -\returnType{Type: List PositiveInteger} - -Doing the same thing using piles in an input file you could type: -\begin{verbatim} -L := - a:=4 - b:=1 - c:=9 - [a,b,c] -\end{verbatim} -$$ -\left[ -4, 1, 9 -\right] -$$ -\returnType{Type: List PositiveInteger} - -Since blocks have a type and a value they can be used as arguments to -functions or as part of other expressions. It should be pointed out that -the following example is not recommended practice but helps to illustrate -the idea of blocks and their ability to return values: -\begin{verbatim} -sqrt(4.0 + - a:=3.0 - b:=1.0 - c:=a + b - c - ) -\end{verbatim} -$$ -2.8284271247\ 461900976 -$$ -\returnType{Type: Float} - -Note that indentation is {\bf extremely} important. If the example above -had the pile starting at ``a:='' moved left by two spaces so that the -``a'' was under the ``('' of the first line then the interpreter would -signal an error. Furthermore if the closing parenthesis ``)'' is moved -up to give -\begin{verbatim} -sqrt(4.0 + - a:=3.0 - b:=1.0 - c:=a + b - c) -\end{verbatim} -\begin{verbatim} - Line 1: sqrt(4.0 + - ....A - Error A: Missing mate. - Line 2: a:=3.0 - Line 3: b:=1.0 - Line 4: c:=a + b - Line 5: c) - .........AB - Error A: (from A up to B) Ignored. - Error B: Improper syntax. - Error B: syntax error at top level - Error B: Possibly missing a ) - 5 error(s) parsing -\end{verbatim} -then the parser will generate errors. If the parenthesis is shifted right -by several spaces so that it is in line with the ``c'' thus: -\begin{verbatim} -sqrt(4.0 + - a:=3.0 - b:=1.0 - c:=a + b - c - ) -\end{verbatim} -\begin{verbatim} - Line 1: sqrt(4.0 + - ....A - Error A: Missing mate. - Line 2: a:=3.0 - Line 3: b:=1.0 - Line 4: c:=a + b - Line 5: c - Line 6: ) - .........A - Error A: (from A up to A) Ignored. - Error A: Improper syntax. - Error A: syntax error at top level - Error A: Possibly missing a ) - 5 error(s) parsing -\end{verbatim} -a similar error will be raised. Finally, the ``)'' must be indented by -at least one space relative to the sqrt thus: -\begin{verbatim} -sqrt(4.0 + - a:=3.0 - b:=1.0 - c:=a + b - c - ) -\end{verbatim} -$$ -2.8284271247\ 461900976 -$$ -\returnType{Type: Float} -or an error will be generated. - -It can be seen that great care needs to be taken when constructing input -files consisting of piles of expressions. It would seem prudent to add -one pile at a time and check if it is acceptable before adding more, -particularly if piles are nested. However, it should be pointed out that -the use of piles as values for functions is not very readable and so -perhaps the delicate nature of their interpretation should deter programmers -from using them in these situations. Using piles should really be restricted -to constructing functions, etc. and a small amount of rewriting can remove -the need to use them as arguments. For example, the previous block could -easily be implemented as: -\begin{verbatim} -a:=3.0 -b:=1.0 -c:=a + b -sqrt(4.0 + c) -\end{verbatim} -\begin{verbatim} -a:=3.0 -\end{verbatim} -$$ -3.0 -$$ -\returnType{Type: Float} - -\begin{verbatim} -b:=1.0 -\end{verbatim} -$$ -1.0 -$$ -\returnType{Type: Float} - -\begin{verbatim} -c:=a + b -\end{verbatim} -$$ -4.0 -$$ -\returnType{Type: Float} - -\begin{verbatim} -sqrt(4.0 + c) -\end{verbatim} -$$ -2.8284271247\ 461900976 -$$ -\returnType{Type: Float} - -which achieves the same result and is easier to understand. Note that this -is still a pile but it is not as fragile as the previous version. -\subsection{Functions} -Definitions of functions in Axiom are quite simple providing two things -are observed. First, the type of the function must either be completely -specified or completely unspecified. Second, the body of the function is -assigned to the function identifier using the delayed assignment operator -``==''. - -To specify the type of something the ``:'' operator is used. Thus to define -a variable {\sl x} to be of type {\tt Fraction Integer} we enter: -\spadcommand{x : Fraction Integer} -\returnType{Type: Void} - -For functions the method is the same except that the arguments are -placed in parentheses and the return type is placed after the symbol -``{\tt ->}''. Some examples of function definitions taking zero, one, -two, or three arguments and returning a list of integers are: - -\spadcommand{f : () -> List Integer} -\returnType{Type: Void} - -\spadcommand{g : (Integer) -> List Integer} -\returnType{Type: Void} - -\spadcommand{h : (Integer, Integer) -> List Integer} -\returnType{Type: Void} - -\spadcommand{k : (Integer, Integer, Integer) -> List Integer} -\returnType{Type: Void} - -Now the actual function definitions might be: -\spadcommand{f() == [\ ]} -\returnType{Type: Void} - -\spadcommand{g(a) == [a]} -\returnType{Type: Void} - -\spadcommand{h(a,b) == [a,b]} -\returnType{Type: Void} - -\spadcommand{k(a,b,c) == [a,b,c]} -\returnType{Type: Void} - -with some invocations of these functions: -\spadcommand{f()} -\begin{verbatim} - Compiling function f with type () -> List Integer -\end{verbatim} -$$ -\left[\ -\right] -$$ -\returnType{Type: List Integer} - -\spadcommand{g(4)} -\begin{verbatim} - Compiling function g with type Integer -> List Integer -\end{verbatim} -$$ -\left[ -4 -\right] -$$ -\returnType{Type: List Integer} - -\spadcommand{h(2,9)} -\begin{verbatim} - Compiling function h with type (Integer,Integer) -> List Integer -\end{verbatim} -$$ -\left[ -2, 9 -\right] -$$ -\returnType{Type: List Integer} - -\spadcommand{k(-3,42,100)} -\begin{verbatim} - Compiling function k with type (Integer,Integer,Integer) -> List - Integer -\end{verbatim} -$$ -\left[ --3, {42}, {100} -\right] -$$ -\returnType{Type: List Integer} - -The value returned by a function is either the value of the last expression -evaluated or the result of a {\bf return} statement. For example, the -following are effectively the same: -\spadcommand{p : Integer -> Integer} -\returnType{Type: Void} - -\spadcommand{p x == (a:=1; b:=2; a+b+x)} -\returnType{Type: Void} - -\spadcommand{p x == (a:=1; b:=2; return(a+b+x))} -\returnType{Type: Void} - -Note that a block (pile) is assigned to the function identifier {\bf p} and -thus all the rules about blocks apply to function definitions. Also there was -only one argument so the parenthese are not needed. - -This is basically all that one needs to know about defining functions in -Axiom -- first specify the complete type and then assign a block to the -function name. The rest of this section is concerned with defining more -complex blocks than those in this section and as a result function definitions -will crop up continually particularly since they are a good way of testing -examples. Since the block structure is more complex we will use the {\bf pile} -notation and thus have to use input files to read the piles. - -\subsection{Choices} -Apart from the ``{\tt =>}'' operator that allows a block to exit before the end -Axiom provides the standard {\bf if-then-else} construct. The general -syntax is: -{\center{if {\sl BooleanExpr} then {\sl Expr1} else {\sl Expr2}}} - -where ``else {\sl Expr2}'' can be omitted. If the expression {\sl BooleanExpr} -evaluates to {\tt true} then {\sl Expr1} is executed otherwise {\sl Expr2} -(if present) will be executed. An example of piles and {\bf if-then-else} is: -(read from an input file) -\begin{verbatim} -h := 2.0 -if h > 3.1 then - 1.0 - else - z:= cos(h) - max(x,0.5) -\end{verbatim} -\begin{verbatim} -h := 2.0 -\end{verbatim} -$$ -2.0 -$$ -\returnType{Type: Float} - -\begin{verbatim} -if h > 3.1 then - 1.0 - else - z:= cos(h) - max(x,0.5) -\end{verbatim} -$$ -x -$$ -\returnType{Type: Polynomial Float} - -Note the indentation -- the ``else'' must be indented relative to the ``if'' -otherwise it will generate an error (Axiom will think there are two piles, -the second one beginning with ``else''). - -Any expression that has type {\tt Boolean} can be used as {\tt BooleanExpr} -and the most common will be those involving the relational operators ``$>$'', -``$<$'', and ``=''. Usually the type of an expression involving the equality -operator ``='' will be {\bf Boolean} but in those situations when it isn't -you may need to use the ``@'' operator to ensure that it is. - -\subsection{Loops} -Loops in Axiom are regarded as expressions containing another expression -called the {\sl loop body}. The loop body is executed zero or more times -depending on the kind of loop. Loops can be nested to any depth. - -\subsubsection{The {\tt repeat} loop} -The simplest kind of loop provided by Axiom is the {\bf repeat} loop. The -general syntax of this is: -{\center{{\bf repeat} {\sl loopBody}}} - -This will cause Axiom to execute {\sl loopBody} repeatedly until either a -{\bf break} or {\bf return} statement is encountered. If {\sl loopBody} -contains neither of these statements then it will loop forever. The -following piece of code will display the numbers from $1$ to $4$: -\begin{verbatim} -i:=1 -repeat - if i > 4 then break - output(i) - i:=i+1 -\end{verbatim} -\begin{verbatim} -i:=1 -\end{verbatim} -$$ -1 -$$ -\returnType{Type: PositiveInteger} - -\begin{verbatim} -repeat - if i > 4 then break - output(i) - i:=i+1 - - 1 - 2 - 3 - 4 -\end{verbatim} -\returnType{Type: Void} - -It was mentioned that loops will only be left when either a {\bf break} or -{\bf return} statement is encountered so why can't one use the ``{\tt =>}'' -operator? The reason is that the ``{\tt =>}'' operator tells Axiom to leave the -current block whereas {\bf break} leaves the current loop. The {\bf return} -statement leaves the current function. - -To skip the rest of a loop body and continue the next iteration of the loop -use the {\bf iterate} statement (the -- starts a comment in Axiom) -\begin{verbatim} -i := 0 -repeat - i := i + 1 - if i > 6 then break - -- Return to start if i is odd - if odd?(i) then iterate - output(i) -\end{verbatim} -\begin{verbatim} -i := 0 -\end{verbatim} -$$ -0 -$$ -\returnType{Type: NonNegativeInteger} - -\begin{verbatim} -repeat - i := i + 1 - if i > 6 then break - -- Return to start if i is odd - if odd?(i) then iterate - output(i) - - 2 - 4 - 6 -\end{verbatim} -\returnType{Type: Void} - -\subsubsection{The {\tt while} loop} -The while statement extends the basic {\bf repeat} loop to place the control -of leaving the loop at the start rather than have it buried in the middle. -Since the body of the loop is still part of a {\bf repeat} loop, {\bf break} -and ``{\tt =>}'' work in the same way as in the previous section. The general -syntax of a {\bf while} loop is: -{\center{while {\sl BoolExpr} repeat {\sl loopBody}}} - -As before, {\sl BoolExpr} must be an expression of type {\bf Boolean}. Before -the body of the loop is executed {\sl BoolExpr} is tested. If it evaluates to -{\tt true} then the loop body is entered otherwise the loop is terminated. -Multiple conditions can be applied using the logical operators such as -{\bf and} or by using several {\bf while} statements before the {\bf repeat}. -\begin{verbatim} -x:=1 -y:=1 -while x < 4 and y < 10 repeat - output [x,y] - x := x + 1 - y := y + 2 -\end{verbatim} -\begin{verbatim} -x:=1 -\end{verbatim} -$$ -1 -$$ -\returnType{Type: PositiveInteger} - -\begin{verbatim} -y:=1 -\end{verbatim} -$$ -1 -$$ -\returnType{Type: PositiveInteger} - -\begin{verbatim} -while x < 4 and y < 10 repeat - output [x,y] - x := x + 1 - y := y + 2 - - [1,1] - [2,3] - [3,5] -\end{verbatim} -\returnType{Type: Void} - -\begin{verbatim} -x:=1 -y:=1 -while x < 4 while y < 10 repeat - output [x,y] - x := x + 1 - y := y + 2 -\end{verbatim} -\begin{verbatim} -x:=1 -\end{verbatim} -$$ -1 -$$ -\returnType{Type: PositiveInteger} - -\begin{verbatim} -y:=1 -\end{verbatim} -$$ -1 -$$ -\returnType{Type: PositiveInteger} - -\begin{verbatim} -while x < 4 while y < 10 repeat - output [x,y] - x := x + 1 - y := y + 2 - - [1,1] - [2,3] - [3,5] -\end{verbatim} -\returnType{Type: Void} - -Note that the last example using two {\bf while} statements is {\sl not} a -nested loop but the following one is: -\begin{verbatim} -x:=1 -y:=1 -while x < 4 repeat - while y < 10 repeat - output [x,y] - x := x + 1 - y := y + 2 -\end{verbatim} -\begin{verbatim} -x:=1 -\end{verbatim} -$$ -1 -$$ -\returnType{Type: PositiveInteger} - -\begin{verbatim} -y:=1 -\end{verbatim} -$$ -1 -$$ -\returnType{Type: PositiveInteger} - -\begin{verbatim} -while x < 4 repeat - while y < 10 repeat - output [x,y] - x := x + 1 - y := y + 2 - - [1,1] - [2,3] - [3,5] - [4,7] - [5,9] -\end{verbatim} -\returnType{Type: Void} - -Suppose we that, given a matrix of arbitrary size, find the position and -value of the first negative element by examining the matrix in row-major -order: -\begin{verbatim} -m := matrix [ [ 21, 37, 53, 14 ],_ - [ 8, 22,-24, 16 ],_ - [ 2, 10, 15, 14 ],_ - [ 26, 33, 55,-13 ] ] - -lastrow := nrows(m) -lastcol := ncols(m) -r := 1 -while r <= lastrow repeat - c := 1 -- Index of first column - while c <= lastcol repeat - if elt(m,r,c) < 0 then - output [r,c,elt(m,r,c)] - r := lastrow - break -- Don't look any further - c := c + 1 - r := r + 1 -\end{verbatim} -\begin{verbatim} -m := matrix [ [ 21, 37, 53, 14 ],_ - [ 8, 22,-24, 16 ],_ - [ 2, 10, 15, 14 ],_ - [ 26, 33, 55,-13 ] ] -\end{verbatim} -$$ -\left[ -\begin{array}{cccc} -{21} & {37} & {53} & {14} \\ -8 & {22} & -{24} & {16} \\ -2 & {10} & {15} & {14} \\ -{26} & {33} & {55} & -{13} -\end{array} -\right] -$$ -\returnType{Type: Matrix Integer} - -\begin{verbatim} -lastrow := nrows(m) -\end{verbatim} -$$ -4 -$$ -\returnType{Type: PositiveInteger} - -\begin{verbatim} -lastcol := ncols(m) -\end{verbatim} -$$ -4 -$$ -\returnType{Type: PositiveInteger} - -\begin{verbatim} -r := 1 -\end{verbatim} -$$ -1 -$$ -\returnType{Type: PositiveInteger} - -\begin{verbatim} -while r <= lastrow repeat - c := 1 -- Index of first column - while c <= lastcol repeat - if elt(m,r,c) < 0 then - output [r,c,elt(m,r,c)] - r := lastrow - break -- Don't look any further - c := c + 1 - r := r + 1 - - [2,3,- 24] -\end{verbatim} -\returnType{Type: Void} - -\subsubsection{The {\tt for} loop} -The last loop statement of interest is the {\bf for} loop. There are two -ways of creating a {\bf for} loop. The first way uses either a list or -a segment: -\begin{center} -for {\sl var} in {\sl seg} repeat {\sl loopBody}\\ -for {\sl var} in {\sl list} repeat {\sl loopBody} -\end{center} -where {\sl var} is an index variable which is iterated over the values in -{\sl seg} or {\sl list}. The value {\sl seg} is a segment such as $1\ldots10$ -or $1\ldots$ and {\sl list} is a list of some type. For example: -\begin{verbatim} -for i in 1..10 repeat - ~prime?(i) => iterate - output(i) - - 2 - 3 - 5 - 7 -\end{verbatim} -\returnType{Type: Void} - -\begin{verbatim} -for w in ["This", "is", "your", "life!"] repeat - output(w) - - This - is - your - life! -\end{verbatim} -\returnType{Type: Void} - -The second form of the {\bf for} loop syntax includes a ``{\bf such that}'' -clause which must be of type {\bf Boolean}: -\begin{center} -for {\sl var} | {\sl BoolExpr} in {\sl seg} repeat {\sl loopBody}\\ -for {\sl var} | {\sl BoolExpr} in {\sl list} repeat {\sl loopBody} -\end{center} -Some examples are: -\begin{verbatim} -for i in 1..10 | prime?(i) repeat - output(i) - - 2 - 3 - 5 - 7 -\end{verbatim} -\returnType{Type: Void} - -\begin{verbatim} -for i in [1,2,3,4,5,6,7,8,9,10] | prime?(i) repeat - output(i) - - 2 - 3 - 5 - 7 -\end{verbatim} -\returnType{Type: Void} - -You can also use a {\bf while} clause: -\begin{verbatim} -for i in 1.. while i < 7 repeat - if even?(i) then output(i) - - 2 - 4 - 6 -\end{verbatim} -\returnType{Type: Void} - -Using the ``{\bf such that}'' clause makes this appear simpler: -\begin{verbatim} -for i in 1.. | even?(i) while i < 7 repeat - output(i) - - 2 - 4 - 6 -\end{verbatim} -\returnType{Type: Void} - -You can use multiple {\bf for} clauses to iterate over several sequences -in parallel: -\begin{verbatim} -for a in 1..4 for b in 5..8 repeat - output [a,b] - - [1,5] - [2,6] - [3,7] - [4,8] -\end{verbatim} -\returnType{Type: Void} - -As a general point it should be noted that any symbols referred to in the -``{\bf such that}'' and {\bf while} clauses must be pre-defined. This -either means that the symbols must have been defined in an outer level -(e.g. in an enclosing loop) or in a {\bf for} clause appearing before the -``{\bf such that}'' or {\bf while}. For example: -\begin{verbatim} -for a in 1..4 repeat - for b in 7..9 | prime?(a+b) repeat - output [a,b,a+b] - - [2,9,11] - [3,8,11] - [4,7,11] - [4,9,13] -\end{verbatim} -\returnType{Type: Void} - -Finally, the {\bf for} statement has a {\bf by} clause to specify the -step size. This makes it possible to iterate over the segment in -reverse order: -\begin{verbatim} -for a in 1..4 for b in 8..5 by -1 repeat - output [a,b] - - [1,8] - [2,7] - [3,6] - [4,5] -\end{verbatim} -\returnType{Type: Void} - -Note that without the ``by -1'' the segment 8..5 is empty so there is -nothing to iterate over and the loop exits immediately. - -\setcounter{chapter}{0} % Chapter 1 - -\hyphenation{ -multi-set -Uni-var-iate-Poly-nomial -Mul-ti-var-iate-Poly-nomial -Distributed-Mul-ti-var-iate-Poly-nomial -Homo-gen-eous-Distributed-Mul-ti-var-iate-Poly-nomial -New-Distributed-Mul-ti-var-iate-Poly-nomial -General-Distributed-Mul-ti-var-iate-Poly-nomial -} - -\chapter{An Overview of Axiom} -\begin{quote} -When we start cataloging the gains in tools sitting on a computer, the -benefits of software are amazing. But, if the benefits of software are -so great, why do we worry about making it easier -- don't the ends pay -for the means? We worry becuase making such software is extraordinarily -hard and almost no one can do it -- the detail is exhausting, the -creativity required is extreme, the hours of failure upon failure -requiring patience and persistence would tax anyone claiming to be -sane. Yet we require people with such characteristics be found and -employed and employed cheaply. - --- Christopher Alexander - -(from Patterns of Software by Richard Gabriel) - -\end{quote} -\label{ugIntro} - -Welcome to the Axiom environment for interactive computation and -problem solving. Consider this chapter a brief, whirlwind tour of the -Axiom world. We introduce you to Axiom's graphics and the -Axiom language. Then we give a sampling of the large variety of -facilities in the Axiom system, ranging from the various kinds -of numbers, to data types (like lists, arrays, and sets) and -mathematical objects (like matrices, integrals, and differential -equations). We conclude with the discussion of system commands and an -interactive ``undo.'' - -Before embarking on the tour, we need to brief those readers working -interactively with Axiom on some details. - -\section{Starting Up and Winding Down} -\label{ugIntroStart} -You need to know how to start the Axiom system and how to stop it. -We assume that Axiom has been correctly installed on your -machine (as described in another Axiom document). - -To begin using Axiom, issue the command {\bf axiom} to the -Axiom operating system shell. -\index{axiom @{\bf axiom}} There is a brief pause, some start-up -messages, and then one or more windows appear. - -If you are not running Axiom under the X Window System, there is -only one window (the console). At the lower left of the screen there -is a prompt that \index{prompt} looks like -\begin{verbatim} -(1) -> -\end{verbatim} - -When you want to enter input to Axiom, you do so on the same -line after the prompt. The ``1'' in ``(1)'', also called the equation -number, is the computation step number and is incremented -\index{step number} after you enter Axiom statements. -Note, however, that a system command such as {\tt )clear all} -may change the step number in other ways. We talk about step numbers -more when we discuss system commands and the workspace history facility. - -If you are running Axiom under the X Window System, there may be -two \index{X Window System} windows: the console window (as just -described) and the HyperDoc main menu. \index{Hyper@{HyperDoc}} -HyperDoc is a multiple-window hypertext system -that lets you \index{window} view Axiom documentation and -examples on-line, execute Axiom expressions, and generate -graphics. If you are in a graphical windowing environment, it is -usually started automatically when Axiom begins. If it is not -running, issue {\tt )hd} to start it. We discuss the basics of -HyperDoc in Chapter \ref{ugHyper} on page~\pageref{ugHyper}. - -To interrupt an Axiom computation, hold down the \index{interrupt} -{\bf Ctrl} (control) key and press {\bf c}. This brings you back to -the Axiom prompt. - -\boxed{4.6in}{ -\vskip 0.1cm -To exit from Axiom, move to the console window, \index{stopping -@{stopping Axiom}} type {\tt )quit} \index{exiting @{exiting -Axiom}} at the input prompt and press the {\bf Enter} key. -You will probably be prompted with the following -message: -\begin{center} -Please enter {\bf y} or {\bf yes} if you really want to leave the \\ -interactive environment and return to the operating system -\end{center} -You should respond {\bf yes}, for example, to exit Axiom.\\ -} - -We are purposely vague in describing exactly what your screen looks -like or what messages Axiom displays. Axiom runs on a number of -different machines, operating systems and window environments, and -these differences all affect the physical look of the system. You can -also change the way that Axiom behaves via {\it system commands} -described later in this chapter and in Appendix A. -System commands are special commands, like {\tt )set}, that begin with -a closing parenthesis and are used to change your environment. For -example, you can set a system variable so that you are not prompted -for confirmation when you want to leave Axiom. - -\subsection{Clef} -\label{ugAvailCLEF} -If you are using Axiom under the X Window System, the -\index{Clef} \index{command line editor} Clef command -line editor is probably available and installed. With this editor you -can recall previous lines with the up and down arrow keys. To move -forward and backward on a line, use the right and left arrows. You -can use the {\bf Insert} key to toggle insert mode on or off. When -you are in insert mode, the cursor appears as a large block and if you -type anything, the characters are inserted into the line without -deleting the previous ones. - -If you press the {\bf Home} key, the cursor moves to the beginning of -the line and if you press the {\bf End} key, the cursor moves to the -end of the line. Pressing {\bf Ctrl-End} deletes all the text from -the cursor to the end of the line. - -Clef also provides Axiom operation name completion for -\index{operation name completion} a limited set of operations. If you -enter a few letters and then press the {\bf Tab} key, Clef tries to -use those letters as the prefix of an Axiom operation name. If -a name appears and it is not what you want, press {\bf Tab} again to -see another name. - -You are ready to begin your journey into the world of Axiom. - -\section{Typographic Conventions} -\label{ugIntroTypo} -In this document we have followed these typographical conventions: -\begin{itemize} -% -\item Categories, domains and packages are displayed in this font: -{\tt Ring}, {\tt Integer}, {\tt DiophantineSolutionPackage}. -% -\item Prefix operators, infix operators, and punctuation symbols in -the Axiom language are displayed in the text like this: -{\tt +}, {\tt \$}, {\tt +->}. -% -\item Axiom expressions or expression fragments are displayed in this font:\\ -{\tt inc(x) == x + 1}. -% -\item For clarity of presentation, \TeX{} is often used to format expressions\\ -$g(x)=x^2+1$. -% -\item Function names and HyperDoc button names are displayed in the text in -this font: -{\bf factor}, {\bf integrate}, {\bf Lighting}. -% -\item Italics are used for emphasis and for words defined in the glossary: \\ -{\it category}. -\end{itemize} - -This document contains over 2500 examples of Axiom input and output. All -examples were run though Axiom and their output was created in \TeX{} -form by the Axiom {\tt TexFormat} package. We have deleted system -messages from the example output if those messages are not important -for the discussions in which the examples appear. - -\section{The Axiom Language} -\label{ugIntroExpressions} -The Axiom language is a rich language for performing interactive -computations and for building components of the Axiom library. -Here we present only some basic aspects of the language that you need -to know for the rest of this chapter. Our discussion here is -intentionally informal, with details unveiled on an ``as needed'' -basis. For more information on a particular construct, we suggest you -consult the index. - -\subsection{Arithmetic Expressions} -\label{ugIntroArithmetic} -For arithmetic expressions, use the ``{\tt +}'' and ``{\tt -}'' operator -as in mathematics. Use ``{\tt *}'' for multiplication, and ``{\tt **}'' -for exponentiation. To create a fraction, use ``{\tt /}''. When an -expression contains several operators, those of highest -{\it precedence} are evaluated first. For arithmetic operators, -``{\tt **}'' has highest precedence, ``{\tt *}'' and ``{\tt /}'' have the -next highest precedence, and ``{\tt +}'' and ``{\tt -}'' have the lowest -precedence. - -Axiom puts implicit parentheses around operations of higher -precedence, and groups those of equal precedence from left to right. -\spadcommand{1 + 2 - 3 / 4 * 3 ** 2 - 1} -$$ --{{19} \over 4} -$$ -\returnType{Type: Fraction Integer} - -The above expression is equivalent to this. -\spadcommand{((1 + 2) - ((3 / 4) * (3 ** 2))) - 1} -$$ --{{19} \over 4} -$$ -\returnType{Type: Fraction Integer} - -If an expression contains subexpressions enclosed in parentheses, -the parenthesized subexpressions are evaluated first (from left to -right, from inside out). -\spadcommand{1 + 2 - 3/ (4 * 3 ** (2 - 1))} -$$ -{11} \over 4 -$$ -\returnType{Type: Fraction Integer} - -\subsection{Previous Results} -\label{ugIntroPrevious} -Use the percent sign ``{\tt \%}'' to refer to the last result. -\index{result!previous} Also, use ``{\tt \%\%}' to refer to -previous results. \index{percentpercent@{\%\%}} ``{\tt \%\%(-1)}'' is -equivalent to ``{\tt \%}'', ``{\tt \%\%(-2)}'' returns the next to -the last result, and so on. ``{\tt \%\%(1)}'' returns the result from -step number 1, ``{\tt \%\%(2)}'' returns the result from step number 2, -and so on. ``{\tt \%\%(0)}'' is not defined. - -This is ten to the tenth power. -\spadcommand{10 ** 10} -$$ -10000000000 -$$ -\returnType{Type: PositiveInteger} - -This is the last result minus one. -\spadcommand{\% - 1} -$$ -9999999999 -$$ -\returnType{Type: PositiveInteger} - -This is the last result. -\spadcommand{\%\%(-1)} -$$ -9999999999 -$$ -\returnType{Type: PositiveInteger} - -This is the result from step number 1. -\spadcommand{\%\%(1)} -$$ -10000000000 -$$ -\returnType{Type: PositiveInteger} - -\subsection{Some Types} -\label{ugIntroTypes} -Everything in Axiom has a type. The type determines what operations -you can perform on an object and how the object can be used. -Chapter~\ref{ugTypes} on page~\pageref{ugTypes} is dedicated to the -interactive use of types. Several of the final chapters discuss how -types are built and how they are organized in the Axiom library. - -Positive integers are given type {\bf PositiveInteger}. -\spadcommand{8} -$$ -8 -$$ -\returnType{Type: PositiveInteger} - -Negative ones are given type {\bf Integer}. This fine -distinction is helpful to the Axiom interpreter. - -\spadcommand{-8} -$$ --8 -$$ -\returnType{Type: Integer} - -Here a positive integer exponent gives a polynomial result. -\spadcommand{x**8} -$$ -x \sp 8 -$$ -\returnType{Type: Polynomial Integer} - -Here a negative integer exponent produces a fraction. -\spadcommand{x**(-8)} -$$ -1 \over {x \sp 8} -$$ -\returnType{Type: Fraction Polynomial Integer} - -\subsection{Symbols, Variables, Assignments, and Declarations} -\label{ugIntroAssign} -A {\it symbol} is a literal used for the input of things like -the ``variables'' in polynomials and power series. - -We use the three symbols $x$, $y$, and $z$ in -entering this polynomial. -\spadcommand{(x - y*z)**2} -$$ -{{y \sp 2} \ {z \sp 2}} -{2 \ x \ y \ z}+{x \sp 2} -$$ -\returnType{Type: Polynomial Integer} - -A symbol has a name beginning with an uppercase or lowercase -alphabetic \index{symbol!naming} character, ``{\tt \%}'', or -``{\tt !}''. Successive characters (if any) can be any of the -above, digits, or ``{\tt ?}''. Case is distinguished: the symbol -{\tt points} is different from the symbol {\tt Points}. - -A symbol can also be used in Axiom as a {\it variable}. A variable -refers to a value. To {\sl assign} a value to a variable, -\index{variable!naming} the operator ``{\tt :=}'' \index{assignment} -is used.\footnote{Axiom actually has two forms of assignment: -{\it immediate} assignment, as discussed here, and {\it delayed -assignment}. See Section \ref{ugLangAssign} on page~\pageref{ugLangAssign} -for details.} A variable initially has no restrictions on the kinds -of \index{declaration} values to which it can refer. - -This assignment gives the value $4$ (an integer) to -a variable named $x$. -\spadcommand{x := 4} -$$ -4 -$$ -\returnType{Type: PositiveInteger} - -This gives the value $z + 3/5$ (a polynomial) to $x$. -\spadcommand{x := z + 3/5} -$$ -z+{3 \over 5} -$$ -\returnType{Type: Polynomial Fraction Integer} - -To restrict the types of objects that can be assigned to a variable, -use a {\it declaration} -\spadcommand{y : Integer} -\returnType{Type: Void} - -After a variable is declared to be of some type, only values -of that type can be assigned to that variable. -\spadcommand{y := 89} -$$ -89 -$$ -\returnType{Type: Integer} - -The declaration for $y$ forces values assigned to $y$ to -be converted to integer values. -\spadcommand{y := sin \%pi} -$$ -0 -$$ -\returnType{Type: Integer} - -If no such conversion is possible, -Axiom refuses to assign a value to $y$. -\spadcommand{y := 2/3} -\begin{verbatim} - Cannot convert right-hand side of assignment - 2 - - - 3 - - to an object of the type Integer of the left-hand side. -\end{verbatim} - -A type declaration can also be given together with an assignment. -The declaration can assist Axiom in choosing the correct -operations to apply. -\spadcommand{f : Float := 2/3} -$$ -0.6666666666\ 6666666667 -$$ -\returnType{Type: Float} - -Any number of expressions can be given on input line. -Just separate them by semicolons. -Only the result of evaluating the last expression is displayed. - -These two expressions have the same effect as -the previous single expression. - -\spadcommand{f : Float; f := 2/3} -$$ -0.6666666666\ 6666666667 -$$ -\returnType{Type: Float} - -The type of a symbol is either {\tt Symbol} -or {\tt Variable({\it name})} where {\it name} is the name -of the symbol. - -By default, the interpreter -gives this symbol the type {\tt Variable(q)}. - -\spadcommand{q} -$$ -q -$$ -\returnType{Type: Variable q} - -When multiple symbols are involved, {\tt Symbol} is used. -\spadcommand{[q, r]} -$$ -\left[ -q, r -\right] -$$ -\returnType{Type: List OrderedVariableList [q,r]} - -What happens when you try to use a symbol that is the name of a variable? -\spadcommand{f} -$$ -0.6666666666\ 6666666667 -$$ -\returnType{Type: Float} - -Use a single quote ``{\tt '}'' before \index{quote} the name to get the symbol. - -\spadcommand{'f} -$$ -f -$$ -\returnType{Type: Variable f} - -Quoting a name creates a symbol by preventing evaluation of the name -as a variable. Experience will teach you when you are most likely -going to need to use a quote. We try to point out the location of -such trouble spots. - -\subsection{Conversion} -\label{ugIntroConversion} -Objects of one type can usually be ``converted'' to objects of several -other types. To {\sl convert} an object to a new type, use the ``{\tt ::}'' -infix operator.\footnote{Conversion is discussed in detail in -\ref{ugTypesConvert} on page~\pageref{ugTypesConvert}.} For example, -to display an object, it is necessary to convert the object to type -{\tt OutputForm}. - -This produces a polynomial with rational number coefficients. - -\spadcommand{p := r**2 + 2/3} -$$ -{r \sp 2}+{2 \over 3} -$$ -\returnType{Type: Polynomial Fraction Integer} - -Create a quotient of polynomials with integer coefficients -by using ``{\tt ::}''. - -\spadcommand{p :: Fraction Polynomial Integer } -$$ -{{3 \ {r \sp 2}}+2} \over 3 -$$ -\returnType{Type: Fraction Polynomial Integer} - -Some conversions can be performed automatically when Axiom tries -to evaluate your input. Others conversions must be explicitly -requested. - -\subsection{Calling Functions} -\label{ugIntroCallFun} -As we saw earlier, when you want to add or subtract two values, you -place the arithmetic operator ``{\tt +}'' or ``{\tt -}'' between the two -arguments denoting the values. To use most other Axiom -operations, however, you use another syntax: \index{function!calling} -write the name of the operation first, then an open parenthesis, then -each of the arguments separated by commas, and, finally, a closing -parenthesis. If the operation takes only one argument and the -argument is a number or a symbol, you can omit the parentheses. - -This calls the operation {\bf factor} with the single integer argument $120$. - -\spadcommand{factor(120)} -$$ -{2 \sp 3} \ 3 \ 5 -$$ -\returnType{Type: Factored Integer} - -This is a call to {\bf divide} with the two integer arguments -$125$ and $7$. -\spadcommand{divide(125,7)} -$$ -\left[ -{quotient={17}}, {remainder=6} -\right] -$$ -\returnType{Type: Record(quotient: Integer, remainder: Integer)} - -This calls {\bf quatern} with four floating-point arguments. -\spadcommand{quatern(3.4,5.6,2.9,0.1)} -$$ -{3.4}+{{5.6} \ i}+{{2.9} \ j}+{{0.1} \ k} -$$ -\returnType{Type: Quaternion Float} - -This is the same as {\bf factorial}(10). -\spadcommand{factorial 10} -$$ -3628800 -$$ -\returnType{Type: PositiveInteger} - -An operations that returns a {\tt Boolean} value (that is, -{\tt true} or {\tt false}) frequently has a name suffixed with -a question mark (``?''). For example, the {\bf even?} -operation returns {\tt true} if its integer argument is an even -number, {\tt false} otherwise. - -An operation that can be destructive on one or more arguments -usually has a name ending in a exclamation point (``!''). -This actually means that it is {\it allowed} to update its -arguments but it is not {\it required} to do so. For example, -the underlying representation of a collection type may not allow -the very last element to removed and so an empty object may be -returned instead. Therefore, it is important that you use the -object returned by the operation and not rely on a physical -change having occurred within the object. Usually, destructive -operations are provided for efficiency reasons. - -\subsection{Some Predefined Macros} -\label{ugIntroMacros} -Axiom provides several macros for your convenience.\footnote{See -\ref{ugUserMacros} on page~\pageref{ugUserMacros} for a discussion on -how to write your own macros.} Macros are names -\index{macro!predefined} (or forms) that expand to larger expressions -for commonly used values. - -\begin{center} -\begin{tabular}{ll} -{\it \%i} & The square root of -1. \\ -{\it \%e} & The base of the natural logarithm. \\ -{\it \%pi} & $\pi$. \\ -{\it \%infinity} & $\infty$. \\ -{\it \%plusInfinity} & $+\infty$. \\ -{\it \%minusInfinity} & $-\infty$. -\end{tabular} -\end{center} -\index{\%i} -\index{\%e} -\index{\%pi} -\index{pi@{$\pi$ (= \%pi)}} -\index{\%infinity} -\index{infinity@{$\infty$ (= \%infinity)}} -\index{\%plusInfinity} -\index{\%minusInfinity} - -To display all the macros (along with anything you have -defined in the workspace), issue the system command {\tt )display all}. - -\subsection{Long Lines} -\label{ugIntroLong} -When you enter Axiom expressions from your keyboard, there will -be times when they are too long to fit on one line. Axiom does -not care how long your lines are, so you can let them continue from -the right margin to the left side of the next line. - -Alternatively, you may want to enter several shorter lines and have -Axiom glue them together. To get this glue, put an underscore -(\_) at the end of each line you wish to continue. - -\begin{verbatim} -2_ -+_ -3 -\end{verbatim} -is the same as if you had entered -\begin{verbatim} -2+3 -\end{verbatim} - -Axiom statements in an input file -(see Section \ref{ugInOutIn} on page~\pageref{ugInOutIn}), -can use indentation to indicate the program structure . -(see Section \ref{ugLangBlocks} on page~\pageref{ugLangBlocks}). - -\subsection{Comments} -\label{ugIntroComments} -Comment statements begin with two consecutive hyphens or two -consecutive plus signs and continue until the end of the line. - -The comment beginning with ``{\tt --}'' is ignored by Axiom. -\spadcommand{2 + 3 -- this is rather simple, no?} -$$ -5 -$$ -\returnType{Type: PositiveInteger} - -There is no way to write long multi-line comments other than starting -each line with ``{\tt --}'' or ``{\tt ++}''. - -\section{Numbers} -\label{ugIntroNumbers} -Axiom distinguishes very carefully between different kinds of -numbers, how they are represented and what their properties are. Here -are a sampling of some of these kinds of numbers and some things you -can do with them. - -Integer arithmetic is always exact. -\spadcommand{11**13 * 13**11 * 17**7 - 19**5 * 23**3} -$$ -25387751112538918594666224484237298 -$$ -\returnType{Type: PositiveInteger} - -Integers can be represented in factored form. -\spadcommand{factor 643238070748569023720594412551704344145570763243} -$$ -{{11} \sp {13}} \ {{13} \sp {11}} \ {{17} \sp 7} \ {{19} \sp 5} \ {{23} -\sp 3} \ {{29} \sp 2} -$$ -\returnType{Type: Factored Integer} - -Results stay factored when you do arithmetic. -Note that the $12$ is automatically factored for you. -\spadcommand{\% * 12} -\index{radix} -$$ -{2 \sp 2} \ 3 \ {{11} \sp {13}} \ {{13} \sp {11}} \ {{17} \sp 7} \ {{19} -\sp 5} \ {{23} \sp 3} \ {{29} \sp 2} -$$ -\returnType{Type: Factored Integer} - -Integers can also be displayed to bases other than 10. -This is an integer in base 11. -\spadcommand{radix(25937424601,11)} -$$ -10000000000 -$$ -\returnType{Type: RadixExpansion 11} - -Roman numerals are also available for those special occasions. -\index{Roman numerals} - -\spadcommand{roman(1992)} -$$ -{\rm MCMXCII } -$$ -\returnType{Type: RomanNumeral} - -Rational number arithmetic is also exact. - -\spadcommand{r := 10 + 9/2 + 8/3 + 7/4 + 6/5 + 5/6 + 4/7 + 3/8 + 2/9} -$$ -{55739} \over {2520} -$$ -\returnType{Type: Fraction Integer} - -To factor fractions, you have to pmap {\bf factor} onto the numerator -and denominator. - -\spadcommand{map(factor,r)} -$$ -{{139} \ {401}} \over {{2 \sp 3} \ {3 \sp 2} \ 5 \ 7} -$$ -\returnType{Type: Fraction Factored Integer} - -{\tt SingleInteger} refers to machine word-length integers. - -In English, this expression means ``$11$ as a small integer''. -\spadcommand{11@SingleInteger} -$$ -11 -$$ -\returnType{Type: SingleInteger} - -Machine double-precision floating-point numbers are also available for -numeric and graphical applications. -\spadcommand{123.21@DoubleFloat} -$$ -123.21000000000001 -$$ -\returnType{Type: DoubleFloat} - -The normal floating-point type in Axiom, {\tt Float}, is a -software implementation of floating-point numbers in which the -exponent and the mantissa may have any number of digits. -The types {\tt Complex(Float)} and -{\tt Complex(DoubleFloat)} are the corresponding software -implementations of complex floating-point numbers. - -This is a floating-point approximation to about twenty digits. -\index{floating point} The ``{\tt ::}'' is used here to change from -one kind of object (here, a rational number) to another (a -floating-point number). - -\spadcommand{r :: Float} -$$ -22.1186507936 50793651 -$$ -\returnType{Type: Float} - -Use \spadfunFrom{digits}{Float} to change the number of digits in -the representation. -This operation returns the previous value so you can reset it -later. -\spadcommand{digits(22)} -$$ -20 -$$ -\returnType{Type: PositiveInteger} - -To $22$ digits of precision, the number -$e^{\pi {\sqrt {163.0}}}$ appears to be an integer. -\spadcommand{exp(\%pi * sqrt 163.0)} -$$ -26253741 2640768744.0 -$$ -\returnType{Type: Float} - -Increase the precision to forty digits and try again. -\spadcommand{digits(40); exp(\%pi * sqrt 163.0)} -$$ -26253741\ 2640768743.9999999999\ 9925007259\ 76 -$$ -\returnType{Type: Float} - -Here are complex numbers with rational numbers as real and -\index{complex numbers} imaginary parts. -\spadcommand{(2/3 + \%i)**3} -$$ --{{46} \over {27}}+{{1 \over 3} \ i} -$$ -\returnType{Type: Complex Fraction Integer} - -The standard operations on complex numbers are available. -\spadcommand{conjugate \% } -$$ --{{46} \over {27}} -{{1 \over 3} \ i} -$$ -\returnType{Type: Complex Fraction Integer} - -You can factor complex integers. -\spadcommand{factor(89 - 23 * \%i)} -$$ --{{\left( 1+i -\right)} -\ {{\left( 2+i -\right)} -\sp 2} \ {{\left( 3+{2 \ i} -\right)} -\sp 2}} -$$ -\returnType{Type: Factored Complex Integer} - -Complex numbers with floating point parts are also available. -\spadcommand{exp(\%pi/4.0 * \%i)} -$$ -{0.7071067811\ 8654752440\ 0844362104\ 8490392849} + -$$ -$$ -{{0.7071067811\ 8654752440\ 0844362104\ 8490392848} \ i} -$$ -\returnType{Type: Complex Float} - -The real and imaginary parts can be symbolic. -\spadcommand{complex(u,v)} -$$ -u+{v \ i} -$$ -\returnType{Type: Complex Polynomial Integer} - -Of course, you can do complex arithmetic with these also. -\spadcommand{\% ** 2} -$$ --{v \sp 2}+{u \sp 2}+{2 \ u \ v \ i} -$$ -\returnType{Type: Complex Polynomial Integer} - -Every rational number has an exact representation as a -repeating decimal expansion -\spadcommand{decimal(1/352)} -$$ -0.{00284}{\overline {09}} -$$ -\returnType{Type: DecimalExpansion} - -A rational number can also be expressed as a continued fraction. - -\spadcommand{continuedFraction(6543/210)} -$$ -{31}+ \zag{1}{6}+ \zag{1}{2}+ \zag{1}{1}+ \zag{1}{3} -$$ -\returnType{Type: ContinuedFraction Integer} - -Also, partial fractions can be used and can be displayed in a -\index{partial fraction} -compact format -\index{fraction!partial} -\spadcommand{partialFraction(1,factorial(10))} -$$ -{{159} \over {2 \sp 8}} -{{23} \over {3 \sp 4}} -{{12} \over {5 \sp 2}}+{1 -\over 7} -$$ -\returnType{Type: PartialFraction Integer} - -or expanded format. -\spadcommand{padicFraction(\%)} -$$ -{1 \over 2}+{1 \over {2 \sp 4}}+{1 \over {2 \sp 5}}+{1 \over {2 \sp 6}}+{1 -\over {2 \sp 7}}+{1 \over {2 \sp 8}} -{2 \over {3 \sp 2}} -{1 \over {3 \sp -3}} -{2 \over {3 \sp 4}} -{2 \over 5} -{2 \over {5 \sp 2}}+{1 \over 7} -$$ -\returnType{Type: PartialFraction Integer} - -Like integers, bases (radices) other than ten can be used for rational -numbers. -Here we use base eight. -\spadcommand{radix(4/7, 8)} -$$ -0.{\overline 4} -$$ -\returnType{Type: RadixExpansion 8} - -Of course, there are complex versions of these as well. -Axiom decides to make the result a complex rational number. -\spadcommand{\% + 2/3*\%i} -$$ -{4 \over 7}+{{2 \over 3} \ i} -$$ -\returnType{Type: Complex Fraction Integer} - -You can also use Axiom to manipulate fractional powers. -\index{radical} -\spadcommand{(5 + sqrt 63 + sqrt 847)**(1/3)} -$$ -\root {3} \of {{{{14} \ {\sqrt {7}}}+5}} -$$ -\returnType{Type: AlgebraicNumber} - -You can also compute with integers modulo a prime. -\spadcommand{x : PrimeField 7 := 5} -$$ -5 -$$ -\returnType{Type: PrimeField 7} - -Arithmetic is then done modulo $7$. -\spadcommand{x**3} -$$ -6 -$$ -\returnType{Type: PrimeField 7} - -Since $7$ is prime, you can invert nonzero values. -\spadcommand{1/x} -$$ -3 -$$ -\returnType{Type: PrimeField 7} - -You can also compute modulo an integer that is not a prime. -\spadcommand{y : IntegerMod 6 := 5} -$$ -5 -$$ -\returnType{Type: IntegerMod 6} - -All of the usual arithmetic operations are available. -\spadcommand{y**3} -$$ -5 -$$ -\returnType{Type: IntegerMod 6} - -Inversion is not available if the modulus is not a prime number. -Modular arithmetic and prime fields are discussed in Section -\ref{ugxProblemFinitePrime} on page~\pageref{ugxProblemFinitePrime}. - -\spadcommand{1/y} -\begin{verbatim} - There are 12 exposed and 13 unexposed library operations named / - having 2 argument(s) but none was determined to be applicable. - Use HyperDoc Browse, or issue - )display op / - to learn more about the available operations. Perhaps - package-calling the operation or using coercions on the arguments - will allow you to apply the operation. - - Cannot find a definition or applicable library operation named / - with argument type(s) - PositiveInteger - IntegerMod 6 - - Perhaps you should use "@" to indicate the required return type, - or "$" to specify which version of the function you need. -\end{verbatim} - -This defines $a$ to be an algebraic number, that is, -a root of a polynomial equation. -\spadcommand{a := rootOf(a**5 + a**3 + a**2 + 3,a)} -$$ -a -$$ -\returnType{Type: Expression Integer} - -Computations with $a$ are reduced according to the polynomial equation. -\spadcommand{(a + 1)**10} -$$ --{{85} \ {a \sp 4}} -{{264} \ {a \sp 3}} -{{378} \ {a \sp 2}} -{{458} \ -a} -{287} -$$ -\returnType{Type: Expression Integer} - -Define $b$ to be an algebraic number involving $a$. -\spadcommand{b := rootOf(b**4 + a,b)} -$$ -b -$$ -\returnType{Type: Expression Integer} - -Do some arithmetic. -\spadcommand{2/(b - 1)} -$$ -2 \over {b -1} -$$ -\returnType{Type: Expression Integer} - -To expand and simplify this, call {\it ratDenom} -to rationalize the denominator. -\spadcommand{ratDenom(\%)} -$$ -\begin{array}{@{}l} -\displaystyle -{{\left({a^4}-{a^3}+{2 \ {a^2}}- a + 1 \right)}\ {b^3}}+{{\left({a^ -4}-{a^3}+{2 \ {a^2}}- a + 1 \right)}\ {b^2}}+ -\\ -\\ -\displaystyle -{{\left({a^4}-{a^3}+{2 \ {a^2}}- a + 1 \right)}\ b}+{a^4}-{a^ -3}+{2 \ {a^2}}- a + 1 -\end{array} -$$ - -\returnType{Type: Expression Integer} - -If we do this, we should get $b$. -\spadcommand{2/\%+1} -$$ -{\left( -\begin{array}{@{}l} -\displaystyle -{{\left({a^4}-{a^3}+{2 \ {a^2}}- a + 1 \right)}\ {b^3}}+{{\left({a^ -4}-{a^3}+{2 \ {a^2}}- a + 1 \right)}\ {b^2}}+ -\\ -\\ -\displaystyle -{{\left({a^4}-{a^3}+{2 \ {a^2}}- a + 1 \right)}\ b}+{a^4}-{a^ -3}+{2 \ {a^2}}- a + 3 -\end{array} -\right)}\over{\left( -\begin{array}{@{}l} -\displaystyle -{{\left({a^4}-{a^3}+{2 \ {a^2}}- a + 1 \right)}\ {b^3}}+{{\left({a^ -4}-{a^3}+{2 \ {a^2}}- a + 1 \right)}\ {b^2}}+ -\\ -\\ -\displaystyle -{{\left({a^4}-{a^3}+{2 \ {a^2}}- a + 1 \right)}\ b}+{a^4}-{a^ -3}+{2 \ {a^2}}- a + 1 -\end{array} -\right)} -$$ -\returnType{Type: Expression Integer} - -But we need to rationalize the denominator again. - -\spadcommand{ratDenom(\%)} -$$ -b -$$ -\returnType{Type: Expression Integer} - -Types {\tt Quaternion} and {\tt Octonion} are also available. -Multiplication of quaternions is non-commutative, as expected. - -\spadcommand{q:=quatern(1,2,3,4)*quatern(5,6,7,8) - quatern(5,6,7,8)*quatern(1,2,3,4)} -$$ --{8 \ i}+{{16} \ j} -{8 \ k} -$$ -\returnType{Type: Quaternion Integer} - -\section{Data Structures} -\label{ugIntroCollect} -Axiom has a large variety of data structures available. Many -data structures are particularly useful for interactive computation -and others are useful for building applications. The data structures -of Axiom are organized into {\sl category hierarchies}. - -A {\it list} \footnote{Lists are discussed in Section \ref{ListXmpPage} on -page~\pageref{ListXmpPage}} is the most commonly used data structure in -Axiom for holding objects all of the same type. The name {\it list} is -short for ``linked-list of nodes.'' Each node consists of a value -(\spadfunFrom{first}{List}) and a link (\spadfunFrom{rest}{List}) that -points to the next node, or to a distinguished value denoting the -empty list. To get to, say, the third element, Axiom starts at the -front of the list, then traverses across two links to the third node. - -Write a list of elements using square brackets with commas separating -the elements. -\spadcommand{u := [1,-7,11]} -$$ -\left[ -1, -7, {11} -\right] -$$ -\returnType{Type: List Integer} - -This is the value at the third node. Alternatively, you can say $u.3$. -\spadcommand{first rest rest u} -$$ -11 -$$ -\returnType{Type: PositiveInteger} - -Many operations are defined on lists, such as: {\bf empty?}, to test -that a list has no elements; {\bf cons}$(x,l)$, to create a new list -with {\bf first} element $x$ and {\bf rest} $l$; {\bf reverse}, to -create a new list with elements in reverse order; and {\bf sort}, to -arrange elements in order. - -An important point about lists is that they are ``mutable'': their -constituent elements and links can be changed ``in place.'' -To do this, use any of the operations whose names end with the -character ``{\tt !}''. - -The operation \spadfunFrom{concat!}{List}$(u,v)$ replaces the -last link of the list $u$ to point to some other list $v$. -Since $u$ refers to the original list, this change is seen by $u$. -\spadcommand{concat!(u,[9,1,3,-4]); u} -$$ -\left[ -1, -7, {11}, 9, 1, 3, -4 -\right] -$$ -\returnType{Type: List Integer} - -A {\it cyclic list} is a list with a ``cycle'': \index{list!cyclic} a -link pointing back to an earlier node of the list. \index{cyclic -list} To create a cycle, first get a node somewhere down the list. -\spadcommand{lastnode := rest(u,3)} -$$ -\left[ -9, 1, 3, -4 -\right] -$$ -\returnType{Type: List Integer} - -Use \spadfunFrom{setrest!}{List} to change the link emanating from -that node to point back to an earlier part of the list. - -\spadcommand{setrest!(lastnode,rest(u,2)); u} -$$ -\left[ -1, -7, {\overline {{11}, 9}} -\right] -$$ -\returnType{Type: List Integer} - -A {\it stream} is a structure that (potentially) has an infinite -number of distinct elements. Think of a stream as an -``infinite list'' where elements are computed successively. -\footnote{Streams are discussed in Section{StreamXmpPage} on -page~\pageref{StreamXmpPage}} - -Create an infinite stream of factored integers. Only a certain number -of initial elements are computed and displayed. - -\spadcommand{[factor(i) for i in 2.. by 2]} -$$ -\left[ -2, {2 \sp 2}, {2 \ 3}, {2 \sp 3}, {2 \ 5}, {{2 \sp 2} \ 3}, - {2 \ 7}, {2 \sp 4}, {2 \ {3 \sp 2}}, {{2 \sp 2} \ 5}, -\ldots -\right] -$$ -\returnType{Type: Stream Factored Integer} - -Axiom represents streams by a collection of already-computed -elements together with a function to compute the next element ``on -demand.'' Asking for the $n$-th element causes elements -$1$ through $n$ to be evaluated. -\spadcommand{\%.36} -$$ -{2 \sp 3} \ {3 \sp 2} -$$ -\returnType{Type: Factored Integer} - -Streams can also be finite or cyclic. -They are implemented by a linked list structure similar to lists -and have many of the same operations. -For example, {\bf first} and {\bf rest} are used to access -elements and successive nodes of a stream. - -A {\it one-dimensional array} is another data structure used to hold -objects of the same type \footnote{OnedimensionalArray is discussed in -Section \ref{OneDimensionalArrayXmpPage} on -page~\pageref{OneDimensionalArrayXmpPage}}. Unlike lists, -one-dimensional arrays are inflexible---they are -\index{array!one-dimensional} implemented using a fixed block of -storage. Their advantage is that they give quick and equal access -time to any element. - -A simple way to create a one-dimensional array is to apply the -operation {\bf oneDimensionalArray} to a list of elements. -\spadcommand{a := oneDimensionalArray [1, -7, 3, 3/2]} -$$ -\left[ -1, -7, 3, {3 \over 2} -\right] -$$ -\returnType{Type: OneDimensionalArray Fraction Integer} - -One-dimensional arrays are also mutable: you can change their -constituent elements ``in place.'' -\spadcommand{a.3 := 11; a} -$$ -\left[ -1, -7, {11}, {3 \over 2} -\right] -$$ -\returnType{Type: OneDimensionalArray Fraction Integer} - -However, one-dimensional arrays are not flexible structures. -You cannot destructively {\bf concat!} them together. -\spadcommand{concat!(a,oneDimensionalArray [1,-2])} -\begin{verbatim} - There are 5 exposed and 0 unexposed library operations named concat! - having 2 argument(s) but none was determined to be applicable. - Use HyperDoc Browse, or issue - )display op concat! - to learn more about the available operations. Perhaps - package-calling the operation or using coercions on the arguments - will allow you to apply the operation. - - Cannot find a definition or applicable library operation named - concat! with argument type(s) - OneDimensionalArray Fraction Integer - OneDimensionalArray Integer - - Perhaps you should use "@" to indicate the required return type, - or "$" to specify which version of the function you need. -\end{verbatim} - -Examples of datatypes similar to {\tt OneDimensionalArray} -are: {\tt Vector} (vectors are mathematical structures -implemented by one-dimensional arrays), {\tt String} (arrays -of ``characters,'' represented by byte vectors), and -{\tt Bits} (represented by ``bit vectors''). - -A vector of 32 bits, each representing the {\bf Boolean} value -${\tt true}$. -\spadcommand{bits(32,true)} -$$ -\mbox{\tt "11111111111111111111111111111111"} -$$ -\returnType{Type: Bits} - -A {\it flexible array} \footnote{FlexibleArray is discussed in Section -\ref{FlexibleArrayXmpPage} on page~\pageref{FlexibleArrayXmpPage}} is -a cross between a list \index{array!flexible} and a one-dimensional -array. Like a one-dimensional array, a flexible array occupies a fixed -block of storage. Its block of storage, however, has room to expand. -When it gets full, it grows (a new, larger block of storage is -allocated); when it has too much room, it contracts. - -Create a flexible array of three elements. -\spadcommand{f := flexibleArray [2, 7, -5]} -$$ -\left[ -2, 7, -5 -\right] -$$ -\returnType{Type: FlexibleArray Integer} - -Insert some elements between the second and third elements. -\spadcommand{insert!(flexibleArray [11, -3],f,2)} -$$ -\left[ -2, {11}, -3, 7, -5 -\right] -$$ -\returnType{Type: FlexibleArray Integer} - -Flexible arrays are used to implement ``heaps.'' A {\it heap} is an -example of a data structure called a {\it priority queue}, where -elements are ordered with respect to one another. A heap -\footnote{Heap is discussed in Section \ref{HeapXmpPage} on -page~\pageref{HeapXmpPage}} is organized so as to optimize insertion -and extraction of maximum elements. The {\bf extract!} operation -returns the maximum element of the heap, after destructively removing -that element and reorganizing the heap so that the next maximum -element is ready to be delivered. - -An easy way to create a heap is to apply the operation {\it heap} -to a list of values. -\spadcommand{h := heap [-4,7,11,3,4,-7]} -$$ -\left[ -{11}, 4, 7, -4, 3, -7 -\right] -$$ -\returnType{Type: Heap Integer} - -This loop extracts elements one-at-a-time from $h$ until the heap -is exhausted, returning the elements as a list in the order they were -extracted. -\spadcommand{[extract!(h) while not empty?(h)]} -$$ -\left[ -{11}, 7, 4, 3, -4, -7 -\right] -$$ -\returnType{Type: List Integer} - -A {\it binary tree} is a ``tree'' with at most two branches -\index{tree} per node: it is either empty, or else is a node -consisting of a value, and a left and right subtree (again, binary -trees). \footnote{BinarySearchTrees are discussed in Section -\ref{BinarySearchTreeXmpPage} on page~\pageref{BinarySearchTreeXmpPage}} -Examples of binary tree types are {\tt BinarySearchTree}, -{\tt PendantTree}, {\tt TournamentTree}, and {\tt BalancedBinaryTree}. - -A {\it binary search tree} is a binary tree such that, -\index{tree!binary search} for each node, the value of the node is -\index{binary search tree} greater than all values (if any) in the -left subtree, and less than or equal all values (if any) in the right -subtree. -\spadcommand{binarySearchTree [5,3,2,9,4,7,11]} -$$ -\left[ -{\left[ 2, 3, 4 -\right]}, - 5, {\left[ 7, 9, {11} -\right]} -\right] -$$ -\returnType{Type: BinarySearchTree PositiveInteger} - -A {\it balanced binary tree} is useful for doing modular computations. -\index{balanced binary tree} Given a list $lm$ of moduli, -\index{tree!balanced binary} {\bf modTree}$(a,lm)$ produces -a balanced binary tree with the values $a \bmod m$ at its leaves. -\spadcommand{modTree(8,[2,3,5,7])} -$$ -\left[ -0, 2, 3, 1 -\right] -$$ -\returnType{Type: List Integer} - -A {\it set} is a collection of elements where duplication and order is -irrelevant. \footnote{Sets are discussed in Section \ref{SetXmpPage} -on page~\pageref{SetXmpPage}} Sets are always finite and have no -corresponding structure like streams for infinite collections. - -Create sets using braces ``\{`` and ``\}'' rather than brackets. - -\spadcommand{fs := set[1/3,4/5,-1/3,4/5]} -$$ -\left\{ --{1 \over 3}, {1 \over 3}, {4 \over 5} -\right\} -$$ -\returnType{Type: Set Fraction Integer} - -A {\it multiset} is a set that keeps track of the number of duplicate -values. \footnote{Multisets are discussed in Section -\ref{MultisetXmpPage} on page~\pageref{MultisetXmpPage}} - -For all the primes $p$ between 2 and 1000, find the -distribution of $p \bmod 5$. -\spadcommand{multiset [x rem 5 for x in primes(2,1000)]} -$$ -\left\{ -0, {{42} \mbox{\rm : } 3}, {{40} \mbox{\rm : } 1}, {{38} \mbox{\rm : -} 4}, {{47} \mbox{\rm : } 2} -\right\} -$$ -\returnType{Type: Multiset Integer} - -A {\it table} is conceptually a set of ``key--value'' pairs and is a -generalization of a multiset. For examples of tables, see -{\tt AssociationList}, {\tt HashTable}, {\tt KeyedAccessFile}, -{\tt Library}, {\tt SparseTable}, {\tt StringTable}, and {\tt Table}. The -domain {\tt Table(Key, Entry)} provides a general-purpose type for -tables with {\it values} of type $Entry$ indexed by {\it keys} of type -$Key$. - -Compute the above distribution of primes using tables. First, let -$t$ denote an empty table of keys and values, each of type {\tt Integer}. -\spadcommand{t : Table(Integer,Integer) := empty()} -$$ -{\rm table}() -$$ -\returnType{Type: Table(Integer,Integer)} - -We define a function {\bf howMany} to return the number of values -of a given modulus $k$ seen so far. It calls -{\bf search}$(k,t)$ which returns the number of values -stored under the key $k$ in table $t$, or {\tt ``failed''} -if no such value is yet stored in $t$ under $k$. - -In English, this says ``Define $howMany(k)$ as follows. -First, let $n$ be the value of {\it search}$(k,t)$. -Then, if $n$ has the value $"failed"$, return the value -$1$; otherwise return $n + 1$.'' -\spadcommand{howMany(k) == (n:=search(k,t); n case "failed" => 1; n+1)} -\returnType{Type: Void} - -Run through the primes to create the table, then print the table. -The expression {\tt t.m := howMany(m)} updates the value in table $t$ -stored under key $m$. -\spadcommand{for p in primes(2,1000) repeat (m:= p rem 5; t.m:= howMany(m)); t} -\begin{verbatim} - Compiling function howMany with type Integer -> Integer -\end{verbatim} -$$ -{\rm table } -\left( -{{2={47}}, {4={38}}, {1={40}}, {3={42}}, {0=1}} -\right) -$$ -\returnType{Type: Table(Integer,Integer)} - -A {\it record} is an example of an inhomogeneous collection of -objects.\footnote{See \ref{ugTypesRecords} on -page~\pageref{ugTypesRecords} for details.} A record consists of a -set of named {\it selectors} that can be used to access its -components. \index{Record@{\sf Record}} - -Declare that $daniel$ can only be -assigned a record with two prescribed fields. -\spadcommand{daniel : Record(age : Integer, salary : Float)} -\returnType{Type: Void} - -Give $daniel$ a value, using square brackets to enclose the values of -the fields. -\spadcommand{daniel := [28, 32005.12]} -$$ -\left[ -{age={28}}, {salary={32005.12}} -\right] -$$ -\returnType{Type: Record(age: Integer,salary: Float)} - -Give $daniel$ a raise. -\spadcommand{daniel.salary := 35000; daniel} -$$ -\left[ -{age={28}}, {salary={35000.0}} -\right] -$$ -\returnType{Type: Record(age: Integer,salary: Float)} - -A {\it union} is a data structure used when objects have multiple -types.\footnote{See \ref{ugTypesUnions} on -page~\pageref{ugTypesUnions} for details.} \index{Union@{\sf Union}} - -Let $dog$ be either an integer or a string value. -\spadcommand{dog: Union(licenseNumber: Integer, name: String)} -\returnType{Type: Void} - -Give $dog$ a name. -\spadcommand{dog := "Whisper"} -$$ -\mbox{\tt "Whisper"} -$$ -\returnType{Type: Union(name: String,...)} - -All told, there are over forty different data structures in Axiom. -Using the domain constructors described in Chapter \ref{ugDomains} on -page~\pageref{ugDomains}, you can add your own data structure or -extend an existing one. Choosing the right data structure for your -application may be the key to obtaining good performance. - -\section{Expanding to Higher Dimensions} -\label{ugIntroTwoDim} -To get higher dimensional aggregates, you can create one-dimensional -aggregates with elements that are themselves aggregates, for example, -lists of lists, one-dimensional arrays of lists of multisets, and so -on. For applications requiring two-dimensional homogeneous -aggregates, you will likely find {\it two-dimensional arrays} -\index{matrix} and {\it matrices} most useful. -\index{array!two-dimensional} - -The entries in {\tt TwoDimensionalArray} and {\tt Matrix} objects are -all the same type, except that those for {\tt Matrix} must belong to a -{\tt Ring}. You create and access elements in roughly the same way. -Since matrices have an understood algebraic structure, certain -algebraic operations are available for matrices but not for arrays. -Because of this, we limit our discussion here to {\tt Matrix}, that -can be regarded as an extension of {\tt TwoDimensionalArray}. See {\tt -TwoDimensionalArray} for more information about arrays. For more -information about Axiom's linear algebra facilities, see {\tt Matrix}, -{\tt Permanent}, {\tt SquareMatrix}, {\tt Vector}, see Section -\ref{ugProblemEigen} on page~\pageref{ugProblemEigen} (computation of -eigenvalues and eigenvectors), and Section \ref{ugProblemLinPolEqn} on -page~\pageref{ugProblemLinPolEqn} (solution of linear and polynomial -equations). - -You can create a matrix from a list of lists, \index{matrix!creating} -where each of the inner lists represents a row of the matrix. -\spadcommand{m := matrix([ [1,2], [3,4] ])} -$$ -\left[ -\begin{array}{cc} -1 & 2 \\ -3 & 4 -\end{array} -\right] -$$ -\returnType{Type: Matrix Integer} - -The ``collections'' construct (see \ref{ugLangIts} on -page~\pageref{ugLangIts}) is useful for creating matrices whose -entries are given by formulas. \index{matrix!Hilbert} -\spadcommand{matrix([ [1/(i + j - x) for i in 1..4] for j in 1..4])} -$$ -\left[ -\begin{array}{cccc} --{1 \over {x -2}} & -{1 \over {x -3}} & -{1 \over {x -4}} & -{1 \over {x -5}} \\ --{1 \over {x -3}} & -{1 \over {x -4}} & -{1 \over {x -5}} & -{1 \over {x -6}} \\ --{1 \over {x -4}} & -{1 \over {x -5}} & -{1 \over {x -6}} & -{1 \over {x -7}} \\ --{1 \over {x -5}} & -{1 \over {x -6}} & -{1 \over {x -7}} & -{1 \over {x -8}} -\end{array} -\right] -$$ -\returnType{Type: Matrix Fraction Polynomial Integer} - -Let $vm$ denote the three by three Vandermonde matrix. -\spadcommand{vm := matrix [ [1,1,1], [x,y,z], [x*x,y*y,z*z] ]} -$$ -\left[ -\begin{array}{ccc} -1 & 1 & 1 \\ -x & y & z \\ -{x \sp 2} & {y \sp 2} & {z \sp 2} -\end{array} -\right] -$$ -\returnType{Type: Matrix Polynomial Integer} - -Use this syntax to extract an entry in the matrix. - -\spadcommand{vm(3,3)} -$$ -z \sp 2 -$$ -\returnType{Type: Polynomial Integer} - -You can also pull out a {\bf row} or a {\bf column}. - -\spadcommand{column(vm,2)} -$$ -\left[ -1, y, {y \sp 2} -\right] -$$ -\returnType{Type: Vector Polynomial Integer} - -You can do arithmetic. - -\spadcommand{vm * vm} -$$ -\left[ -\begin{array}{ccc} -{{x \sp 2}+x+1} & {{y \sp 2}+y+1} & {{z \sp 2}+z+1} \\ -{{{x \sp 2} \ z}+{x \ y}+x} & {{{y \sp 2} \ z}+{y \sp 2}+x} & {{z \sp -3}+{y \ z}+x} \\ -{{{x \sp 2} \ {z \sp 2}}+{x \ {y \sp 2}}+{x \sp 2}} & {{{y \sp 2} \ {z \sp -2}}+{y \sp 3}+{x \sp 2}} & {{z \sp 4}+{{y \sp 2} \ z}+{x \sp 2}} -\end{array} -\right] -$$ -\returnType{Type: Matrix Polynomial Integer} - -You can perform operations such as -{\bf transpose}, {\bf trace}, and {\bf determinant}. -\spadcommand{factor determinant vm} -$$ -{\left( y -x -\right)} -\ {\left( z -y -\right)} -\ {\left( z -x -\right)} -$$ -\returnType{Type: Factored Polynomial Integer} - -\section{Writing Your Own Functions} -\label{ugIntroYou} -Axiom provides you with a very large library of predefined -operations and objects to compute with. You can use the Axiom -library of constructors to create new objects dynamically of quite -arbitrary complexity. For example, you can make lists of matrices of -fractions of polynomials with complex floating point numbers as -coefficients. Moreover, the library provides a wealth of operations -that allow you to create and manipulate these objects. - -For many applications, you need to interact with the interpreter and -write some Axiom programs to tackle your application. -Axiom allows you to write functions interactively, -\index{function} thereby effectively extending the system library. -Here we give a few simple examples, leaving the details to -Chapter \ref{ugUser} on page~\pageref{ugUser}. - -We begin by looking at several ways that you can define the -``factorial'' function in Axiom. The first way is to give a -\index{function!piece-wise definition} piece-wise definition of the -function. \index{piece-wise function definition} This method is best -for a general recurrence relation since the pieces are gathered -together and compiled into an efficient iterative function. -Furthermore, enough previously computed values are automatically saved -so that a subsequent call to the function can pick up from where it -left off. - -Define the value of {\bf fact} at $0$. -\spadcommand{fact(0) == 1} -\returnType{Type: Void} - -Define the value of {\bf fact}(n) for general $n$. -\spadcommand{fact(n) == n*fact(n-1)} -\returnType{Type: Void} - -Ask for the value at $50$. The resulting function created by -Axiom computes the value by iteration. - -\spadcommand{fact(50)} -\begin{verbatim} - Compiling function fact with type Integer -> Integer - Compiling function fact as a recurrence relation. -\end{verbatim} -$$ -30414093201713378043612608166064768844377641568960512000000000000 -$$ -\returnType{Type: PositiveInteger} - -A second definition uses an {\tt if-then-else} and recursion. -\spadcommand{fac(n) == if n < 3 then n else n * fac(n - 1)} -\returnType{Type: Void} - -This function is less efficient than the previous version since -each iteration involves a recursive function call. -\spadcommand{fac(50)} -$$ -30414093201713378043612608166064768844377641568960512000000000000 -$$ -\returnType{Type: PositiveInteger} - -A third version directly uses iteration. -\spadcommand{fa(n) == (a := 1; for i in 2..n repeat a := a*i; a)} -\returnType{Type: Void} - -This is the least space-consumptive version. -\spadcommand{fa(50)} -\begin{verbatim} - Compiling function fac with type Integer -> Integer -\end{verbatim} -$$ -30414093201713378043612608166064768844377641568960512000000000000 -$$ -\returnType{Type: PositiveInteger} - -A final version appears to construct a large list and then reduces over -it with multiplication. -\spadcommand{f(n) == reduce(*,[i for i in 2..n])} -\returnType{Type: Void} - -In fact, the resulting computation is optimized into an efficient -iteration loop equivalent to that of the third version. -\spadcommand{f(50)} -\begin{verbatim} -Compiling function f with type - PositiveInteger -> PositiveInteger -\end{verbatim} -$$ -30414093201713378043612608166064768844377641568960512000000000000 -$$ -\returnType{Type: PositiveInteger} - -The library version uses an algorithm that is different from the four -above because it highly optimizes the recurrence relation definition of -{\bf factorial}. - -\spadcommand{factorial(50)} -$$ -30414093201713378043612608166064768844377641568960512000000000000 -$$ -\returnType{Type: PositiveInteger} - -You are not limited to one-line functions in Axiom. If you place your -function definitions in {\bf .input} files \index{file!input} (see -\ref{ugInOutIn} on page~\pageref{ugInOutIn}), you can have multi-line -functions that use indentation for grouping. - -Given $n$ elements, {\bf diagonalMatrix} creates an -$n$ by $n$ matrix with those elements down the diagonal. -This function uses a permutation matrix -that interchanges the $i$th and $j$th rows of a matrix -by which it is right-multiplied. - -This function definition shows a style of definition that can be used -in {\bf .input} files. Indentation is used to create {\sl blocks}: -sequences of expressions that are evaluated in sequence except as -modified by control statements such as {\tt if-then-else} and {\tt return}. - -\begin{verbatim} -permMat(n, i, j) == - m := diagonalMatrix - [(if i = k or j = k then 0 else 1) - for k in 1..n] - m(i,j) := 1 - m(j,i) := 1 - m -\end{verbatim} - -This creates a four by four matrix that interchanges the second and third -rows. -\spadcommand{p := permMat(4,2,3)} -\begin{verbatim} - Compiling function permMat with type (PositiveInteger, - PositiveInteger,PositiveInteger) -> Matrix Integer -\end{verbatim} -$$ -\left[ -\begin{array}{cccc} -1 & 0 & 0 & 0 \\ -0 & 0 & 1 & 0 \\ -0 & 1 & 0 & 0 \\ -0 & 0 & 0 & 1 -\end{array} -\right] -$$ -\returnType{Type: Matrix Integer} - -Create an example matrix to permute. -\spadcommand{m := matrix [ [4*i + j for j in 1..4] for i in 0..3]} -$$ -\left[ -\begin{array}{cccc} -1 & 2 & 3 & 4 \\ -5 & 6 & 7 & 8 \\ -9 & {10} & {11} & {12} \\ -{13} & {14} & {15} & {16} -\end{array} -\right] -$$ -\returnType{Type: Matrix Integer} - -Interchange the second and third rows of m. -\spadcommand{permMat(4,2,3) * m} -$$ -\left[ -\begin{array}{cccc} -1 & 2 & 3 & 4 \\ -9 & {10} & {11} & {12} \\ -5 & 6 & 7 & 8 \\ -{13} & {14} & {15} & {16} -\end{array} -\right] -$$ -\returnType{Type: Matrix Integer} - -A function can also be passed as an argument to another function, -which then applies the function or passes it off to some other -function that does. You often have to declare the type of a function -that has functional arguments. - -This declares {\bf t} to be a two-argument function that returns a -{\tt Float}. The first argument is a function that takes one -{\tt Float} argument and returns a {\tt Float}. - -\spadcommand{t : (Float -> Float, Float) -> Float} -\returnType{Type: Void} - -This is the definition of {\bf t}. - -\spadcommand{t(fun, x) == fun(x)**2 + sin(x)**2} -\returnType{Type: Void} - -We have not defined a {\bf cos} in the workspace. The one from the -Axiom library will do. - -\spadcommand{t(cos, 5.2058)} -$$ -1.0 -$$ -\returnType{Type: Float} - -Here we define our own (user-defined) function. -\spadcommand{cosinv(y) == cos(1/y)} -\returnType{Type: Void} - -Pass this function as an argument to {\bf t}. -\spadcommand{t(cosinv, 5.2058)} -$$ -1.7392237241\ 8005164925\ 4147684772\ 932520785 -$$ -\returnType{Type: Float} - -Axiom also has pattern matching capabilities for -\index{simplification} -simplification -\index{pattern matching} -of expressions and for defining new functions by rules. -For example, suppose that you want to apply regularly a transformation -that groups together products of radicals: -$$\sqrt{a}\sqrt{b} \mapsto \sqrt{ab}, \quad -(\forall a)(\forall b)$$ -Note that such a transformation is not generally correct. -Axiom never uses it automatically. - -Give this rule the name {\bf groupSqrt}. -\spadcommand{groupSqrt := rule(sqrt(a) * sqrt(b) == sqrt(a*b))} -$$ -{ \%C \ {\sqrt {a}} \ {\sqrt {b}}} \mbox{\rm == } { \%C \ {\sqrt {{a \ -b}}}} -$$ -\returnType{Type: RewriteRule(Integer,Integer,Expression Integer)} - -Here is a test expression. -\spadcommand{a := (sqrt(x) + sqrt(y) + sqrt(z))**4} -$$ -\begin{array}{@{}l} -\displaystyle -{{\left({{\left({4 \ z}+{4 \ y}+{{12}\ x}\right)}\ {\sqrt{y}}}+ -{{\left({4 \ z}+{{12}\ y}+{4 \ x}\right)}\ {\sqrt{x}}}\right)}\ {\sqrt{z}}}+ - -\\ -\\ -\displaystyle -{{\left({{12}\ z}+{4 \ y}+{4 \ x}\right)}\ {\sqrt{x}}\ {\sqrt{y}}}+ -{z^2}+{{\left({6 \ y}+{6 \ x}\right)}\ z}+{y^2}+{6 \ x \ -y}+{x^2} -\end{array} -$$ -\returnType{Type: Expression Integer} - -The rule -{\bf groupSqrt} successfully simplifies the expression. -\spadcommand{groupSqrt a} -$$ -\begin{array}{@{}l} -\displaystyle -{{\left({4 \ z}+{4 \ y}+{{12}\ x}\right)}\ {\sqrt{y \ z}}}+ -{{\left({4 \ z}+{{12}\ y}+{4 \ x}\right)}\ {\sqrt{x \ z}}}+ - -\\ -\\ -\displaystyle -{{\left({{12}\ z}+{4 \ y}+{4 \ x}\right)}\ {\sqrt{x \ y}}}+ -{z^2}+{{\left({6 \ y}+{6 \ x}\right)}\ z}+{y^2}+{6 \ x \ -y}+{x^2} -\end{array} -$$ -\returnType{Type: Expression Integer} - -\section{Polynomials} -\label{ugIntroVariables} -Polynomials are the commonly used algebraic types in symbolic -computation. \index{polynomial} Interactive users of Axiom -generally only see one type of polynomial: {\tt Polynomial(R)}. -This type represents polynomials in any number of unspecified -variables over a particular coefficient domain $R$. This type -represents its coefficients {\sl sparsely}: only terms with non-zero -coefficients are represented. - -In building applications, many other kinds of polynomial -representations are useful. Polynomials may have one variable or -multiple variables, the variables can be named or unnamed, the -coefficients can be stored sparsely or densely. So-called -``distributed multivariate polynomials'' store polynomials as -coefficients paired with vectors of exponents. This type is -particularly efficient for use in algorithms for solving systems of -non-linear polynomial equations. - -The polynomial constructor most familiar to the interactive user -is {\tt Polynomial}. -\spadcommand{(x**2 - x*y**3 +3*y)**2} -$$ -{{x \sp 2} \ {y \sp 6}} -{6 \ x \ {y \sp 4}} -{2 \ {x \sp 3} \ {y \sp -3}}+{9 \ {y \sp 2}}+{6 \ {x \sp 2} \ y}+{x \sp 4} -$$ -\returnType{Type: Polynomial Integer} - -If you wish to restrict the variables used, -{\tt UnivariatePolynomial} provides polynomials in one variable. - -\spadcommand{p: UP(x,INT) := (3*x-1)**2 * (2*x + 8)} -$$ -{{18} \ {x \sp 3}}+{{60} \ {x \sp 2}} -{{46} \ x}+8 -$$ -\returnType{Type: UnivariatePolynomial(x,Integer)} - -The constructor {\tt MultivariatePolynomial} provides polynomials -in one or more specified variables. - -\spadcommand{m: MPOLY([x,y],INT) := (x**2-x*y**3+3*y)**2} -$$ -{x \sp 4} -{2 \ {y \sp 3} \ {x \sp 3}}+{{\left( {y \sp 6}+{6 \ y} -\right)} -\ {x \sp 2}} -{6 \ {y \sp 4} \ x}+{9 \ {y \sp 2}} -$$ -\returnType{Type: MultivariatePolynomial([x,y],Integer)} - -You can change the way the polynomial appears by modifying the variable -ordering in the explicit list. -\spadcommand{m :: MPOLY([y,x],INT)} -$$ -{{x \sp 2} \ {y \sp 6}} -{6 \ x \ {y \sp 4}} -{2 \ {x \sp 3} \ {y \sp -3}}+{9 \ {y \sp 2}}+{6 \ {x \sp 2} \ y}+{x \sp 4} -$$ -\returnType{Type: MultivariatePolynomial([y,x],Integer)} - -The constructor {\tt DistributedMultivariatePolynomial} provides -polynomials in one or more specified variables with the monomials -ordered lexicographically. - -\spadcommand{m :: DMP([y,x],INT)} -$$ -{{y \sp 6} \ {x \sp 2}} -{6 \ {y \sp 4} \ x} -{2 \ {y \sp 3} \ {x \sp -3}}+{9 \ {y \sp 2}}+{6 \ y \ {x \sp 2}}+{x \sp 4} -$$ -\returnType{Type: DistributedMultivariatePolynomial([y,x],Integer)} - -The constructor -{\tt HomogeneousDistributedMultivariatePolynomial} is similar -except that the monomials are ordered by total order refined by -reverse lexicographic order. -\spadcommand{m :: HDMP([y,x],INT)} -$$ -{{y \sp 6} \ {x \sp 2}} -{2 \ {y \sp 3} \ {x \sp 3}} -{6 \ {y \sp 4} \ -x}+{x \sp 4}+{6 \ y \ {x \sp 2}}+{9 \ {y \sp 2}} -$$ -\returnType{Type: HomogeneousDistributedMultivariatePolynomial([y,x],Integer)} - -More generally, the domain constructor -{\tt GeneralDistributedMultivariatePolynomial} allows the user to -provide an arbitrary predicate to define his own term ordering. These -last three constructors are typically used in Gr\"{o}bner basis -applications and -when a flat (that is, non-recursive) display is wanted and the term -ordering is critical for controlling the computation. - -\section{Limits} -\label{ugIntroCalcLimits} - -Axiom's {\bf limit} function is usually used to evaluate -limits of quotients where the numerator and denominator \index{limit} -both tend to zero or both tend to infinity. To find the limit of an -expression $f$ as a real variable $x$ tends to a limit -value $a$, enter {\tt limit(f, x=a)}. Use -{\bf complexLimit} if the variable is complex. Additional -information and examples of limits are in -Section \ref{ugProblemLimits} on page~\pageref{ugProblemLimits}. - -You can take limits of functions with parameters. -\index{limit!of function with parameters} -\spadcommand{g := csc(a*x) / csch(b*x)} -$$ -{\csc -\left( -{{a \ x}} -\right)} -\over {\csch -\left( -{{b \ x}} -\right)} -$$ -\returnType{Type: Expression Integer} - -As you can see, the limit is expressed in terms of the parameters. -\spadcommand{limit(g,x=0)} -$$ -b \over a -$$ -\returnType{Type: Union(OrderedCompletion Expression Integer,...)} - -A variable may also approach plus or minus infinity: -\spadcommand{h := (1 + k/x)**x} -$$ -{{x+k} \over x} \sp x -$$ -\returnType{Type: Expression Integer} - -Use {\tt \%plusInfinity} and {\tt \%minusInfinity} to -denote $\infty$ and $-\infty$. -\spadcommand{limit(h,x=\%plusInfinity)} -$$ -e \sp k -$$ -\returnType{Type: Union(OrderedCompletion Expression Integer,...)} - -A function can be defined on both sides of a particular value, but -may tend to different limits as its variable approaches that value from the -left and from the right. - -\spadcommand{limit(sqrt(y**2)/y,y = 0)} -$$ -\left[ -{leftHandLimit=-1}, {rightHandLimit=1} -\right] -$$ -\returnType{Type: Union(Record(leftHandLimit: Union(OrderedCompletion Expression Integer,"failed"),rightHandLimit: Union(OrderedCompletion Expression Integer,"failed")),...)} - -As $x$ approaches $0$ along the real axis, {\tt exp(-1/x**2)} -tends to $0$. - -\spadcommand{limit(exp(-1/x**2),x = 0)} -$$ -0 -$$ -\returnType{Type: Union(OrderedCompletion Expression Integer,...)} - -However, if $x$ is allowed to approach $0$ along any path in the -complex plane, the limiting value of {\tt exp(-1/x**2)} depends on the -path taken because the function has an essential singularity at $x=0$. -This is reflected in the error message returned by the function. -\spadcommand{complexLimit(exp(-1/x**2),x = 0)} -$$ -\mbox{\tt "failed"} -$$ -\returnType{Type: Union("failed",...)} - -\section{Series} -\label{ugIntroSeries} - -Axiom also provides power series. \index{series!power} By default, -Axiom tries to compute and display the first ten elements of a series. -Use {\tt )set streams calculate} to change the default value to -something else. For the purposes of this document, we have used this -system command to display fewer than ten terms. For more information -about working with series, see \ref{ugProblemSeries} on -page~\pageref{ugProblemSeries}. - -You can convert a functional expression to a power series by using the -operation {\bf series}. In this example, {\tt sin(a*x)} is -expanded in powers of $(x - 0)$, that is, in powers of $x$. -\spadcommand{series(sin(a*x),x = 0)} -$$ -{a \ x} -{{{a \sp 3} \over 6} \ {x \sp 3}}+{{{a \sp 5} \over {120}} \ {x -\sp 5}} -{{{a \sp 7} \over {5040}} \ {x \sp 7}}+{{{a \sp 9} \over {362880}} -\ {x \sp 9}} -{{{a \sp {11}} \over {39916800}} \ {x \sp {11}}}+{O -\left( -{{x \sp {12}}} -\right)} -$$ -\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} - -This expression expands {\tt sin(a*x)} in powers of {\tt (x - \%pi/4)}. -\spadcommand{series(sin(a*x),x = \%pi/4)} -$$ -{\sin -\left({{{a \ \pi} \over 4}}\right)}+ -{a \ {\cos \left({{{a \ \pi} \over 4}} \right)} -\ {\left( x -{\pi \over 4} \right)}}- -\hbox{\hskip 2.0cm} -$$ -$$ -{{{{a \sp 2} \ {\sin \left({{{a \ \pi} \over 4}} \right)}}\over 2} -\ {{\left( x -{\pi \over 4} \right)}\sp 2}} - -{{{{a \sp 3} \ {\cos \left({{{a \ \pi} \over 4}} \right)}}\over 6} -\ {{\left( x -{\pi \over 4} \right)}\sp 3}} + -$$ -$$ -{{{{a \sp 4} \ {\sin \left({{{a \ \pi} \over 4}} \right)}}\over {24}} -\ {{\left( x -{\pi \over 4} \right)}\sp 4}} + -{{{{a \sp 5} \ {\cos \left({{{a \ \pi} \over 4}} \right)}}\over {120}} -\ {{\left( x -{\pi \over 4} \right)}\sp 5}} - -$$ -$$ -{{{{a \sp 6} \ {\sin \left({{{a \ \pi} \over 4}} \right)}}\over {720}} -\ {{\left( x -{\pi \over 4} \right)}\sp 6}} - -{{{{a \sp 7} \ {\cos \left({{{a \ \pi} \over 4}} \right)}}\over {5040}} -\ {{\left( x -{\pi \over 4} \right)}\sp 7}} + -$$ -$$ -{{{{a \sp 8} \ {\sin \left({{{a \ \pi} \over 4}} \right)}}\over {40320}} -\ {{\left( x -{\pi \over 4} \right)}\sp 8}} + -{{{{a \sp 9} \ {\cos \left({{{a \ \pi} \over 4}} \right)}}\over {362880}} -\ {{\left( x -{\pi \over 4} \right)}\sp 9}} - -$$ -$$ -{{{{a \sp {10}} \ {\sin \left({{{a \ \pi} \over 4}} \right)}} -\over {3628800}} -\ {{\left( x -{\pi \over 4} \right)}\sp {10}}} + -{O \left({{{\left( x -{\pi \over 4} \right)}\sp {11}}} \right)} -$$ -\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,pi/4)} - -Axiom provides \index{series!Puiseux} {\it Puiseux series:} -\index{Puiseux series} series with rational number exponents. The -first argument to {\bf series} is an in-place function that -computes the $n$-th coefficient. (Recall that the -``{\tt +->}'' is an infix operator meaning ``maps to.'') -\spadcommand{series(n +-> (-1)**((3*n - 4)/6)/factorial(n - 1/3),x=0,4/3..,2)} -%%NOTE: the paper book shows O(x^4) but Axiom computes O(x^5) -$$ -{x \sp {4 \over 3}} -{{1 \over 6} \ {x \sp {{10} \over 3}}}+{O -\left( -{{x \sp 5}} -\right)} -$$ -\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} - -Once you have created a power series, you can perform arithmetic -operations on that series. We compute the Taylor expansion of $1/(1-x)$. -\index{series!Taylor} -\spadcommand{f := series(1/(1-x),x = 0)} -$$ -1+x+{x \sp 2}+{x \sp 3}+{x \sp 4}+{x \sp 5}+{x \sp 6}+{x \sp 7}+{x \sp 8}+{x -\sp 9}+{x \sp {10}}+{O -\left( -{{x \sp {11}}} -\right)} -$$ -\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} - -Compute the square of the series. -\spadcommand{f ** 2} -$$ -1+{2 \ x}+{3 \ {x \sp 2}}+{4 \ {x \sp 3}}+{5 \ {x \sp 4}}+{6 \ {x \sp -5}}+{7 \ {x \sp 6}}+{8 \ {x \sp 7}}+{9 \ {x \sp 8}}+{{10} \ {x \sp -9}}+{{11} \ {x \sp {10}}}+{O -\left( -{{x \sp {11}}} -\right)} -$$ -\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} - -The usual elementary functions -({\bf log}, {\bf exp}, trigonometric functions, and so on) -are defined for power series. -\spadcommand{f := series(1/(1-x),x = 0)} -$$ -1+x+{x \sp 2}+{x \sp 3}+{x \sp 4}+{x \sp 5}+{x \sp 6}+{x \sp 7}+{x \sp 8}+{x -\sp 9}+{x \sp {10}}+{O -\left( -{{x \sp {11}}} -\right)} -$$ -\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} - -\spadcommand{g := log(f)} -$$ -\begin{array}{@{}l} -x+ -{{1 \over 2} \ {x \sp 2}}+ -{{1 \over 3} \ {x \sp 3}}+ -{{1 \over 4} \ {x \sp 4}}+ -{{1 \over 5} \ {x \sp 5}}+ -{{1 \over 6} \ {x \sp 6}}+ -{{1 \over 7} \ {x \sp 7}}+ -\\ -\\ -\displaystyle -{{1 \over 8} \ {x \sp 8}}+ -{{1 \over 9} \ {x \sp 9}}+ -{{1 \over {10}} \ {x \sp {10}}}+ -{{1 \over {11}} \ {x \sp {11}}}+ -{O \left({{x \sp {12}}} \right)} -\end{array} -$$ -\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} - -\spadcommand{exp(g)} -$$ -1+x+{x \sp 2}+{x \sp 3}+{x \sp 4}+{x \sp 5}+{x \sp 6}+{x \sp 7}+{x \sp 8}+{x -\sp 9}+{x \sp {10}}+{O -\left( -{{x \sp {11}}} -\right)} -$$ -\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} - -% Warning: currently there are (interpreter) problems with converting -% rational functions and polynomials to power series. - -Here is a way to obtain numerical approximations of -$e$ from the Taylor series expansion of {\bf exp}(x). -First create the desired Taylor expansion. -\spadcommand{f := taylor(exp(x))} -$$ -1+x+{{1 \over 2} \ {x \sp 2}}+{{1 \over 6} \ {x \sp 3}}+{{1 \over {24}} \ -{x \sp 4}}+{{1 \over {120}} \ {x \sp 5}}+{{1 \over {720}} \ {x \sp 6}} + -\hbox{\hskip 1.0cm} -$$ -$$ -{{1 -\over {5040}} \ {x \sp 7}} + -{{1 \over {40320}} \ {x \sp 8}}+{{1 \over -{362880}} \ {x \sp 9}}+{{1 \over {3628800}} \ {x \sp {10}}}+{O -\left( -{{x \sp {11}}} -\right)} -$$ -\returnType{Type: UnivariateTaylorSeries(Expression Integer,x,0)} - -Evaluate the series at the value $1.0$. -% Warning: syntax for evaluating power series may change. -As you see, you get a sequence of partial sums. -\spadcommand{eval(f,1.0)} -$$ -\left[ -{1.0}, {2.0}, {2.5}, {2.6666666666 666666667}, \hbox{\hskip 3.0cm} -\right. -$$ -$${2.7083333333 333333333}, {2.7166666666 666666667}, \hbox{\hskip 1.0cm} -$$ -$${2.7180555555 555555556}, {2.7182539682 53968254}, \hbox{\hskip 1.1cm} -$$ -$$\left. -{2.7182787698 412698413}, {2.7182815255 731922399}, \ldots -\hbox{\hskip 0.4cm} -\right] -$$ -\returnType{Type: Stream Expression Float} - -\section{Derivatives} -\label{ugIntroCalcDeriv} - -Use the Axiom function {\bf D} to differentiate an -\index{derivative} expression. \index{differentiation} - -To find the derivative of an expression $f$ with respect to a -variable $x$, enter {\bf D}(f, x). - -\spadcommand{f := exp exp x} -$$ -e \sp {e \sp x} -$$ -\returnType{Type: Expression Integer} - -\spadcommand{D(f, x)} -$$ -{e \sp x} \ {e \sp {e \sp x}} -$$ -\returnType{Type: Expression Integer} - -An optional third argument $n$ in {\bf D} asks Axiom for the $n$-th -derivative of $f$. This finds the fourth derivative of $f$ with -respect to $x$. - -\spadcommand{D(f, x, 4)} -$$ -{\left( {{e \sp x} \sp 4}+{6 \ {{e \sp x} \sp 3}}+{7 \ {{e \sp x} \sp -2}}+{e \sp x} -\right)} -\ {e \sp {e \sp x}} -$$ -\returnType{Type: Expression Integer} - -You can also compute partial derivatives by specifying the order of -\index{differentiation!partial} -differentiation. -\spadcommand{g := sin(x**2 + y)} -$$ -\sin -\left( -{{y+{x \sp 2}}} -\right) -$$ -\returnType{Type: Expression Integer} - -\spadcommand{D(g, y)} -$$ -\cos -\left( -{{y+{x \sp 2}}} -\right) -$$ -\returnType{Type: Expression Integer} - -\spadcommand{D(g, [y, y, x, x])} -$$ -{4 \ {x \sp 2} \ {\sin -\left( -{{y+{x \sp 2}}} -\right)}} --{2 \ {\cos -\left( -{{y+{x \sp 2}}} -\right)}} -$$ -\returnType{Type: Expression Integer} - -Axiom can manipulate the derivatives (partial and iterated) of -\index{differentiation!formal} expressions involving formal operators. -All the dependencies must be explicit. - -This returns $0$ since F (so far) does not explicitly depend on $x$. - -\spadcommand{D(F,x)} -$$ -0 -$$ -\returnType{Type: Polynomial Integer} - -Suppose that we have F a function of $x$, $y$, and $z$, -where $x$ and $y$ are themselves functions of $z$. - -Start by declaring that $F$, $x$, and $y$ are operators. -\index{operator} - -\spadcommand{F := operator 'F; x := operator 'x; y := operator 'y} -$$ -y -$$ -\returnType{Type: BasicOperator} - -You can use F, $x$, and $y$ in expressions. - -\spadcommand{a := F(x z, y z, z**2) + x y(z+1)} -$$ -{x -\left( -{{y -\left( -{{z+1}} -\right)}} -\right)}+{F -\left( -{{x -\left( -{z} -\right)}, - {y -\left( -{z} -\right)}, - {z \sp 2}} -\right)} -$$ -\returnType{Type: Expression Integer} - -Differentiate formally with respect to $z$. -The formal derivatives appearing in $dadz$ are not just formal symbols, -but do represent the derivatives of $x$, $y$, and F. - -\spadcommand{dadz := D(a, z)} -$$ -\begin{array}{@{}l} -\displaystyle -{2 \ z \ {{F_{, 3}}\left({{x \left({z}\right)}, {y \left({z}\right)}, - {z^2}}\right)}}+{{{y_{\ }^{,}}\left({z}\right)}\ {{F_{, 2}}\left({{x -\left({z}\right)}, {y \left({z}\right)}, {z^2}}\right)}}+ - -\\ -\\ -\displaystyle -{{{x_{\ }^{,}}\left({z}\right)}\ {{F_{, 1}}\left({{x \left({z}\right)}, - {y \left({z}\right)}, {z^2}}\right)}}+{{{x_{\ }^{,}}\left({y -\left({z + 1}\right)}\right)}\ {{y_{\ }^{,}}\left({z + 1}\right)}} -\end{array} -$$ -\returnType{Type: Expression Integer} - -You can evaluate the above for particular functional values of -F, $x$, and $y$. If $x(z)$ is {\bf exp}(z) and $y(z)$ is {\bf log}(z+1), -then evaluates {\tt dadz}. - -\spadcommand{eval(eval(dadz, 'x, z +-> exp z), 'y, z +-> log(z+1))} -$$ -{\left( -\begin{array}{@{}l} -\displaystyle -{{\left({2 \ {z^2}}+{2 \ z}\right)}\ {{F_{, 3}}\left({{e^z}, - {\log \left({z + 1}\right)}, {z^2}}\right)}}+ -\\ -\\ -\displaystyle -{{F_{, 2}}\left({{e^ -z}, {\log \left({z + 1}\right)}, {z^2}}\right)}+ -\\ -\\ -\displaystyle -{{\left(z + 1 \right)}\ {e^z}\ {{F_{, 1}}\left({{e^z}, {\log -\left({z + 1}\right)}, {z^2}}\right)}}+ z + 1 -\end{array} -\right)}\over{z + 1} -$$ -\returnType{Type: Expression Integer} - -You obtain the same result by first evaluating $a$ and -then differentiating. - -\spadcommand{eval(eval(a, 'x, z +-> exp z), 'y, z +-> log(z+1))} -$$ -{F -\left( -{{e \sp z}, {\log -\left( -{{z+1}} -\right)}, - {z \sp 2}} -\right)}+z+2 -$$ -\returnType{Type: Expression Integer} - -\spadcommand{D(\%, z)} -$$ -{\left( -\begin{array}{@{}l} -\displaystyle -{{\left({2 \ {z^2}}+{2 \ z}\right)}\ {{F_{, 3}}\left({{e^ -z}, {\log \left({z + 1}\right)}, {z^2}}\right)}}+ -\\ -\\ -\displaystyle -{{F_{, 2}}\left({{e^z}, {\log \left({z + 1}\right)}, {z^2}}\right)}+ -\\ -\\ -\displaystyle -{{\left(z -+ 1 \right)}\ {e^z}\ {{F_{, 1}}\left({{e^z}, {\log \left({z -+ 1}\right)}, {z^2}}\right)}}+ z + 1 -\end{array} -\right)} -\over{z + 1} -$$ -\returnType{Type: Expression Integer} - -\section{Integration} -\label{ugIntroIntegrate} - -Axiom has extensive library facilities for integration. -\index{integration} - -The first example is the integration of a fraction with denominator -that factors into a quadratic and a quartic irreducible polynomial. -The usual partial fraction approach used by most other computer -algebra systems either fails or introduces expensive unneeded -algebraic numbers. - -We use a factorization-free algorithm. -\spadcommand{integrate((x**2+2*x+1)/((x+1)**6+1),x)} -$$ -{\arctan -\left( -{{{x \sp 3}+{3 \ {x \sp 2}}+{3 \ x}+1}} -\right)} -\over 3 -$$ -\returnType{Type: Union(Expression Integer,...)} - -When real parameters are present, the form of the integral can depend on -the signs of some expressions. - -Rather than query the user or make sign assumptions, Axiom returns -all possible answers. -\spadcommand{integrate(1/(x**2 + a),x)} -$$ -\left[ -{{\log -\left( -{{{{{\left( {x \sp 2} -a -\right)} -\ {\sqrt {-a}}}+{2 \ a \ x}} \over {{x \sp 2}+a}}} -\right)} -\over {2 \ {\sqrt {-a}}}}, {{\arctan -\left( -{{{x \ {\sqrt {a}}} \over a}} -\right)} -\over {\sqrt {a}}} -\right] -$$ -\returnType{Type: Union(List Expression Integer,...)} - -The {\bf integrate} operation generally assumes that all -parameters are real. The only exception is when the integrand has -complex valued quantities. - -If the parameter is complex instead of real, then the notion of sign -is undefined and there is a unique answer. You can request this -answer by ``prepending'' the word ``complex'' to the command name: - -\spadcommand{complexIntegrate(1/(x**2 + a),x)} -%%NOTE: the expression in the book is different but they differentiate -%%to exactly the same answer. -$$ -{{\log -\left( -{{{{x \ {\sqrt {-a}}}+a} \over {\sqrt {-a}}}} -\right)} --{\log -\left( -{{{{x \ {\sqrt {-a}}} -a} \over {\sqrt {-a}}}} -\right)}} -\over {2 \ {\sqrt {-a}}} -$$ -\returnType{Type: Expression Integer} - -The following two examples illustrate the limitations of table-based -approaches. The two integrands are very similar, but the answer to -one of them requires the addition of two new algebraic numbers. - -This one is the easy one. -The next one looks very similar -but the answer is much more complicated. -\spadcommand{integrate(x**3 / (a+b*x)**(1/3),x)} -$$ -{{\left( {{120} \ {b \sp 3} \ {x \sp 3}} -{{135} \ a \ {b \sp 2} \ {x -\sp 2}}+{{162} \ {a \sp 2} \ b \ x} -{{243} \ {a \sp 3}} -\right)} -\ {{\root {3} \of {{{b \ x}+a}}} \sp 2}} \over {{440} \ {b \sp 4}} -$$ -\returnType{Type: Union(Expression Integer,...)} - -Only an algorithmic approach is guaranteed to find what new constants -must be added in order to find a solution. - -\spadcommand{integrate(1 / (x**3 * (a+b*x)**(1/3)),x)} -$$ -\left( -\begin{array}{@{}l} --{2 \ {b \sp 2} \ {x \sp 2} \ {\sqrt {3}} \ {\log -\left( -{{{{\root {3} \of {a}} \ {{\root {3} \of {{{b \ x}+a}}} \sp 2}}+{{{\root -{3} \of {a}} \sp 2} \ {\root {3} \of {{{b \ x}+a}}}}+a}} -\right)}}+ -\\ -\\ -\displaystyle -{4\ {b \sp 2} \ {x \sp 2} \ {\sqrt {3}} \ {\log -\left( -{{{{{\root {3} \of {a}} \sp 2} \ {\root {3} \of {{{b \ x}+a}}}} -a}} -\right)}}+ -\\ -\\ -\displaystyle -{{12}\ {b \sp 2} \ {x \sp 2} \ {\arctan -\left( -{{{{2 \ {\sqrt {3}} \ {{\root {3} \of {a}} \sp 2} \ {\root {3} \of {{{b \ -x}+a}}}}+{a \ {\sqrt {3}}}} \over {3 \ a}}} -\right)}}+ -\\ -\\ -\displaystyle -{{\left( -{{12} \ b \ x} -{9 \ a} -\right)} -\ {\sqrt {3}} \ {\root {3} \of {a}} \ {{\root {3} \of {{{b \ x}+a}}} \sp -2}} -\end{array} -\right) -\over {{18} \ {a \sp 2} \ {x \sp 2} \ {\sqrt {3}} \ {\root {3} \of -{a}}} -$$ -\returnType{Type: Union(Expression Integer,...)} - -Some computer algebra systems use heuristics or table-driven -approaches to integration. When these systems cannot determine the -answer to an integration problem, they reply ``I don't know.'' Axiom -uses an algorithm which is a {\sl decision procedure} for integration. -If Axiom returns the original integral that conclusively proves that -an integral cannot be expressed in terms of elementary functions. - -When Axiom returns an integral sign, it has proved that no answer -exists as an elementary function. - -\spadcommand{integrate(log(1 + sqrt(a*x + b)) / x,x)} -$$ -\int \sp{\displaystyle x} {{{\log -\left( -{{{\sqrt {{b+{ \%Q \ a}}}}+1}} -\right)} -\over \%Q} \ {d \%Q}} -$$ -\returnType{Type: Union(Expression Integer,...)} - -Axiom can handle complicated mixed functions much beyond what you -can find in tables. - -Whenever possible, Axiom tries to express the answer using the -functions present in the integrand. - -\spadcommand{integrate((sinh(1+sqrt(x+b))+2*sqrt(x+b)) / (sqrt(x+b) * (x + cosh(1+sqrt(x + b)))), x)} -%%NOTE: the book has the same answer with a trailing ``+4'' term. -%%This term is not generated by Axiom -$$ -{2 \ {\log -\left( -{{{-{2 \ {\cosh -\left( -{{{\sqrt {{x+b}}}+1}} -\right)}} --{2 \ x}} \over {{\sinh -\left( -{{{\sqrt {{x+b}}}+1}} -\right)} --{\cosh -\left( -{{{\sqrt {{x+b}}}+1}} -\right)}}}} -\right)}} --{2 \ {\sqrt {{x+b}}}} -$$ -\returnType{Type: Union(Expression Integer,...)} - -A strong structure-checking algorithm in Axiom finds hidden algebraic -relationships between functions. - -\spadcommand{integrate(tan(atan(x)/3),x)} -%%NOTE: the book has a trailing ``+16'' term in the numerator -%%This is not generated by Axiom -$$ -\left( -\begin{array}{@{}l} -{8 \ {\log -\left( -{{{3 \ {{\tan -\left( -{{{\arctan -\left( -{x} -\right)} -\over 3}} -\right)} -\sp 2}} -1}} -\right)}} --{3 \ {{\tan -\left( -{{{\arctan -\left( -{x} -\right)} -\over 3}} -\right)} -\sp 2}}+ -\\ -\\ -\displaystyle -{{18} \ x \ {\tan -\left( -{{{\arctan -\left( -{x} -\right)} -\over 3}} -\right)}} -\end{array} -\right) -\over {18} -$$ -\returnType{Type: Union(Expression Integer,...)} - -The discovery of this algebraic relationship is necessary for correct -integration of this function. -Here are the details: -\begin{enumerate} -\item -If $x=\tan t$ and $g=\tan (t/3)$ then the following -algebraic relation is true: $${g^3-3xg^2-3g+x=0}$$ -\item -Integrate $g$ using this algebraic relation; this produces: -$${{(24g^2 - 8)\log(3g^2 - 1) + (81x^2 + 24)g^2 + 72xg - 27x^2 - 16} -\over{54g^2 - 18}}$$ -\item -Rationalize the denominator, producing: -$${8\log(3g^2-1) - 3g^2 + 18xg + 16} \over {18}$$ -Replace $g$ by the initial definition -$g = \tan(\arctan(x)/3)$ -to produce the final result. -\end{enumerate} - -This is an example of a mixed function where -the algebraic layer is over the transcendental one. -\spadcommand{integrate((x + 1) / (x*(x + log x) ** (3/2)), x)} -$$ --{{2 \ {\sqrt {{{\log -\left( -{x} -\right)}+x}}}} -\over {{\log -\left( -{x} -\right)}+x}} -$$ -\returnType{Type: Union(Expression Integer,...)} - -While incomplete for non-elementary functions, Axiom can -handle some of them. -\spadcommand{integrate(exp(-x**2) * erf(x) / (erf(x)**3 - erf(x)**2 - erf(x) + 1),x)} -$$ -{{{\left( {\erf -\left( -{x} -\right)} --1 -\right)} -\ {\sqrt {\pi}} \ {\log -\left( -{{{{\erf -\left( -{x} -\right)} --1} \over {{\erf -\left( -{x} -\right)}+1}}} -\right)}} --{2 \ {\sqrt {\pi}}}} \over {{8 \ {\erf -\left( -{x} -\right)}} --8} -$$ -\returnType{Type: Union(Expression Integer,...)} - -More examples of Axiom's integration capabilities are discussed in -Section \ref{ugProblemIntegration} on page~\pageref{ugProblemIntegration}. - -\section{Differential Equations} -\label{ugIntroDiffEqns} -The general approach used in integration also carries over to the -solution of linear differential equations. - -Let's solve some differential equations. -Let $y$ be the unknown function in terms of $x$. -\spadcommand{y := operator 'y} -$$ -y -$$ -\returnType{Type: BasicOperator} - -Here we solve a third order equation with polynomial coefficients. -\spadcommand{deq := x**3 * D(y x, x, 3) + x**2 * D(y x, x, 2) - 2 * x * D(y x, x) + 2 * y x = 2 * x**4} -$$ -{{{x \sp 3} \ {{y \sb {{\ }} \sp {,,,}} -\left( -{x} -\right)}}+{{x -\sp 2} \ {{y \sb {{\ }} \sp {,,}} -\left( -{x} -\right)}} --{2 \ x \ {{y \sb {{\ }} \sp {,}} -\left( -{x} -\right)}}+{2 -\ {y -\left( -{x} -\right)}}}={2 -\ {x \sp 4}} -$$ -\returnType{Type: Equation Expression Integer} - -\spadcommand{solve(deq, y, x)} -%%NOTE: the book has a different solution and it appears to be -%%less complicated than this one. -$$ -\begin{array}{@{}l} -\left[ -{particular={{{x \sp 5} -{{10} \ {x \sp 3}}+{{20} \ {x \sp 2}}+4} \over -{{15} \ x}}}, -\right. -\\ -\\ -\displaystyle -\left. -{basis={\left[ {{{2 \ {x \sp 3}} -{3 \ {x \sp 2}}+1} -\over x}, {{{x \sp 3} -1} \over x}, {{{x \sp 3} -{3 \ {x \sp 2}} -1} -\over x} -\right]}} -\right] -\end{array} -$$ -\returnType{Type: Union(Record(particular: Expression Integer,basis: List Expression Integer),...)} - - -Here we find all the algebraic function solutions of the equation. -\spadcommand{deq := (x**2 + 1) * D(y x, x, 2) + 3 * x * D(y x, x) + y x = 0} -$$ -{{{\left( {x \sp 2}+1 -\right)} -\ {{y \sb {{\ }} \sp {,,}} -\left( -{x} -\right)}}+{3 -\ x \ {{y \sb {{\ }} \sp {,}} -\left( -{x} -\right)}}+{y -\left( -{x} -\right)}}=0 -$$ -\returnType{Type: Equation Expression Integer} - -\spadcommand{solve(deq, y, x)} -$$ -\left[ -{particular=0}, -{basis={\left[ {1 \over {\sqrt {{{x \sp 2}+1}}}}, -{{\log -\left( -{{{\sqrt {{{x \sp 2}+1}}} -x}} -\right)} -\over {\sqrt {{{x \sp 2}+1}}}} -\right]}} -\right] -$$ -\returnType{Type: Union(Record(particular: Expression Integer,basis: List Expression Integer),...)} - -Coefficients of differential equations can come from arbitrary -constant fields. For example, coefficients can contain algebraic -numbers. - -This example has solutions whose logarithmic derivative is an -algebraic function of degree two. - -\spadcommand{eq := 2*x**3 * D(y x,x,2) + 3*x**2 * D(y x,x) - 2 * y x} -$$ -{2 \ {x \sp 3} \ {{y \sb {{\ }} \sp {,,}} -\left( -{x} -\right)}}+{3 -\ {x \sp 2} \ {{y \sb {{\ }} \sp {,}} -\left( -{x} -\right)}} --{2 \ {y -\left( -{x} -\right)}} -$$ -\returnType{Type: Expression Integer} - -\spadcommand{solve(eq,y,x).basis} -$$ -\left[ -{e \sp {\left( -{2 \over {\sqrt {x}}} -\right)}}, - {e \sp {2 \over {\sqrt {x}}}} -\right] -$$ -\returnType{Type: List Expression Integer} - -Here's another differential equation to solve. -\spadcommand{deq := D(y x, x) = y(x) / (x + y(x) * log y x)} -$$ -{{y \sb {{\ }} \sp {,}} -\left( -{x} -\right)}={{y -\left( -{x} -\right)} -\over {{{y -\left( -{x} -\right)} -\ {\log -\left( -{{y -\left( -{x} -\right)}} -\right)}}+x}} -$$ -\returnType{Type: Equation Expression Integer} - -\spadcommand{solve(deq, y, x)} -$$ -{{{y -\left( -{x} -\right)} -\ {{\log -\left( -{{y -\left( -{x} -\right)}} -\right)} -\sp 2}} -{2 \ x}} \over {2 \ {y -\left( -{x} -\right)}} -$$ -\returnType{Type: Union(Expression Integer,...)} - -Rather than attempting to get a closed form solution of -a differential equation, you instead might want to find an -approximate solution in the form of a series. - -Let's solve a system of nonlinear first order equations and get a -solution in power series. Tell Axiom that $x$ is also an -operator. - -\spadcommand{x := operator 'x} -$$ -x -$$ -\returnType{Type: BasicOperator} - -Here are the two equations forming our system. -\spadcommand{eq1 := D(x(t), t) = 1 + x(t)**2} -$$ -{{x \sb {{\ }} \sp {,}} -\left( -{t} -\right)}={{{x -\left( -{t} -\right)} -\sp 2}+1} -$$ -\returnType{Type: Equation Expression Integer} - -\spadcommand{eq2 := D(y(t), t) = x(t) * y(t)} -$$ -{{y \sb {{\ }} \sp {,}} -\left( -{t} -\right)}={{x -\left( -{t} -\right)} -\ {y -\left( -{t} -\right)}} -$$ -\returnType{Type: Equation Expression Integer} - -We can solve the system around $t = 0$ with the initial -conditions $x(0) = 0$ and $y(0) = 1$. Notice that since -we give the unknowns in the order $[x, y]$, the answer is a list -of two series in the order -$[{\rm series\ for\ }x(t), {\rm series\ for\ }y(t)]$. - -\spadcommand{seriesSolve([eq2, eq1], [x, y], t = 0, [y(0) = 1, x(0) = 0])} -$$ -\left[ -{\ t+ -{{1 \over 3} \ {t \sp 3}}+ -{{2 \over {15}} \ {t \sp 5}}+ -{{{17} \over {315}} \ {t \sp 7}}+ -{{{62} \over {2835}} \ {t \sp 9}}+ -{O \left({{t \sp {11}}} \right)}}, -\right. -\hbox{\hskip 2.0cm} -$$ -$$ -\hbox{\hskip 0.4cm} -\left. -{1+ -{{1 \over 2} \ {t \sp 2}}+ -{{5 \over {24}} \ {t \sp 4}}+ -{{{61} \over {720}} \ {t \sp 6}}+ -{{{277} \over {8064}} \ {t \sp 8}}+ -{{{50521} \over {3628800}} \ {t \sp {10}}}+ -{O \left({{t \sp {11}}}\right)}} -\right] -$$ -\returnType{Type: List UnivariateTaylorSeries(Expression Integer,t,0)} - -\section{Solution of Equations} -\label{ugIntroSolution} -Axiom also has state-of-the-art algorithms for the solution of -systems of polynomial equations. When the number of equations and -unknowns is the same, and you have no symbolic coefficients, you can -use {\bf solve} for real roots and {\bf complexSolve} for -complex roots. In each case, you tell Axiom how accurate you -want your result to be. All operations in the {\it solve} family -return answers in the form of a list of solution sets, where each -solution set is a list of equations. - -A system of two equations involving a symbolic parameter $t$. -\spadcommand{S(t) == [x**2-2*y**2 - t,x*y-y-5*x + 5]} -\returnType{Type: Void} - -Find the real roots of $S(19)$ with -rational arithmetic, correct to within $1/10^{20}$. -\spadcommand{solve(S(19),1/10**20)} -$$ -\left[ -{\left[ {y=5}, {x=-{{2451682632253093442511} \over -{295147905179352825856}}} -\right]}, -\right. -$$ -$$ -\left. -{\left[ {y=5}, {x={{2451682632253093442511} \over -{295147905179352825856}}} -\right]} -\right] -$$ -\returnType{Type: List List Equation Polynomial Fraction Integer} - -Find the complex roots of $S(19)$ with floating -point coefficients to $20$ digits accuracy in the mantissa. - -\spadcommand{complexSolve(S(19),10.e-20)} -$$ -\left[ -{\left[ {y={5.0}}, {x={8.3066238629 180748526}} \right]}, -\right. -$$ -$$ -{\left[ {y={5.0}}, {x=-{8.3066238629 180748526}} \right]}, -$$ -$$ -\left. -{\left[ {y=-{{3.0} \ i}}, {x={1.0}} \right]}, -{\left[ {y={{3.0} \ i}}, {x={1.0}} \right]} -\right] -$$ -\returnType{Type: List List Equation Polynomial Complex Float} - -If a system of equations has symbolic coefficients and you want -a solution in radicals, try {\bf radicalSolve}. -\spadcommand{radicalSolve(S(a),[x,y])} -$$ -\left[ -{\left[ {x=-{\sqrt {{a+{50}}}}}, {y=5} \right]}, -{\left[ {x={\sqrt {{a+{50}}}}}, {y=5} \right]}, -\right. -$$ -$$ -\hbox{\hskip 0.7cm} -\left. -{\left[ {x=1}, {y={\sqrt {{{-a+1} \over 2}}}} \right]}, -{\left[ {x=1}, {y=-{\sqrt {{{-a+1} \over 2}}}} \right]} -\right] -$$ -\returnType{Type: List List Equation Expression Integer} - -For systems of equations with symbolic coefficients, you can apply -{\bf solve}, listing the variables that you want Axiom to -solve for. For polynomial equations, a solution cannot usually be -expressed solely in terms of the other variables. Instead, the -solution is presented as a ``triangular'' system of equations, where -each polynomial has coefficients involving only the succeeding -variables. This is analogous to converting a linear system of -equations to ``triangular form''. - -A system of three equations in five variables. -\spadcommand{eqns := [x**2 - y + z,x**2*z + x**4 - b*y, y**2 *z - a - b*x]} -$$ -\left[ -{z -y+{x \sp 2}}, {{{x \sp 2} \ z} -{b \ y}+{x \sp 4}}, {{{y \sp 2} \ -z} -{b \ x} -a} -\right] -$$ -\returnType{Type: List Polynomial Integer} - -Solve the system for unknowns $[x,y,z]$, -reducing the solution to triangular form. -\spadcommand{solve(eqns,[x,y,z])} -$$ -\left[ -{\left[ {x=-{a \over b}}, {y=0}, {z=-{{a \sp 2} \over {b \sp 2}}} -\right]}, -\right. -\hbox{\hskip 10.0cm} -$$ -$$ -\left. -\begin{array}{@{}l} -\left[ -{x={{{z \sp 3}+{2 \ b \ {z \sp 2}}+{{b \sp 2} \ z} -a} \over b}}, -{y={z+b}}, -\right. -\hbox{\hskip 10.0cm} -\\ -\\ -\displaystyle -{z \sp 6}+{4 \ b \ {z \sp 5}}+ -{6 \ {b \sp 2} \ {z \sp 4}}+ -{{\left( {4 \ {b \sp 3}} -{2 \ a} \right)}\ {z \sp 3}}+ -{{\left( {b \sp 4} -{4 \ a \ b} \right)}\ {z \sp 2}}- -\hbox{\hskip 4.0cm} -\\ -\\ -\displaystyle -\left. -{2 \ a \ {b \sp 2} \ z} -{b \sp 3}+{a \sp 2}=0 -\right] -\end{array} -\right] -\hbox{\hskip 7.0cm} -$$ -\returnType{Type: List List Equation Fraction Polynomial Integer} - -\section{System Commands} -\label{ugIntroSysCmmands} -We conclude our tour of Axiom with a brief discussion of -{\it system commands}. System commands are special statements -that start with a closing parenthesis ({\tt )}). They are used -to control or display your Axiom environment, start the -HyperDoc system, issue operating system commands and leave -Axiom. For example, {\tt )system} is used to issue commands -to the operating system from Axiom. Here -is a brief description of some of these commands. For more -information on specific commands, see Appendix A -on page~\pageref{ugSysCmd}. - -Perhaps the most important user command is the {\tt )clear all} -command that initializes your environment. Every section and -subsection in this document has an invisible {\tt )clear all} that is -read prior to the examples given in the section. {\tt )clear all} -gives you a fresh, empty environment with no user variables defined -and the step number reset to $1$. The {\tt )clear} command -can also be used to selectively clear values and properties of system -variables. - -Another useful system command is {\tt )read}. A preferred way to -develop an application in Axiom is to put your interactive -commands into a file, say {\bf my.input} file. To get Axiom to -read this file, you use the system command {\tt )read my.input}. -If you need to make changes to your approach or definitions, go into -your favorite editor, change {\bf my.input}, then {\tt )read -my.input} again. - -Other system commands include: {\tt )history}, to display -previous input and/or output lines; {\tt )display}, to display -properties and values of workspace variables; and {\tt )what}. - -Issue {\tt )what} to get a list of Axiom objects that -contain a given substring in their name. -\spadcommand{)what operations integrate} -\begin{verbatim} - -Operations whose names satisfy the above pattern(s): - -HermiteIntegrate algintegrate complexIntegrate -expintegrate extendedIntegrate fintegrate -infieldIntegrate integrate internalIntegrate -internalIntegrate0 lazyGintegrate lazyIntegrate -lfintegrate limitedIntegrate monomialIntegrate -nagPolygonIntegrate palgintegrate pmComplexintegrate -pmintegrate primintegrate tanintegrate - -To get more information about an operation such as -limitedIntegrate , issue the command )display op limitedIntegrate - -\end{verbatim} - -\subsection{Undo} -\label{ugIntroUndo} -A useful system command is {\tt )undo}. Sometimes while computing -interactively with Axiom, you make a mistake and enter an -incorrect definition or assignment. Or perhaps you need to try one of -several alternative approaches, one after another, to find the best -way to approach an application. For this, you will find the -{\it undo} facility of Axiom helpful. - -System command {\tt )undo n} means ``undo back to step -$n$''; it restores the values of user variables to those that -existed immediately after input expression $n$ was evaluated. -Similarly, {\tt )undo -n} undoes changes caused by the last -$n$ input expressions. Once you have done an {\tt )undo}, -you can continue on from there, or make a change and {\bf redo} all -your input expressions from the point of the {\tt )undo} forward. -The {\tt )undo} is completely general: it changes the environment -like any user expression. Thus you can {\tt )undo} any previous -undo. - -Here is a sample dialogue between user and Axiom. - -``Let me define -two mutually dependent functions $f$ and $g$ piece-wise.'' -\spadcommand{f(0) == 1; g(0) == 1} -\returnType{Type: Void} - -``Here is the general term for $f$.'' -\spadcommand{f(n) == e/2*f(n-1) - x*g(n-1)} -\returnType{Type: Void} - -``And here is the general term for $g$.'' -\spadcommand{g(n) == -x*f(n-1) + d/3*g(n-1)} -\returnType{Type: Void} - -``What is value of $f(3)$?'' -\spadcommand{f(3)} -$$ --{x \sp 3}+{{\left( e+{{1 \over 3} \ d} -\right)} -\ {x \sp 2}}+{{\left( -{{1 \over 4} \ {e \sp 2}} -{{1 \over 6} \ d \ e} --{{1 \over 9} \ {d \sp 2}} -\right)} -\ x}+{{1 \over 8} \ {e \sp 3}} -$$ -\returnType{Type: Polynomial Fraction Integer} - -``Hmm, I think I want to define $f$ differently. -Undo to the environment right after I defined $f$.'' -\spadcommand{)undo 2} - -``Here is how I think I want $f$ to be defined instead.'' -\spadcommand{f(n) == d/3*f(n-1) - x*g(n-1)} -\begin{verbatim} - 1 old definition(s) deleted for function or rule f -\end{verbatim} -\returnType{Type: Void} - -Redo the computation from expression $3$ forward. -\spadcommand{)undo )redo} -\begin{verbatim} -g(n) == -x*f(n-1) + d/3*g(n-1) - - Type: Void -f(3) - - Compiling function g with type Integer -> Polynomial Fraction - Integer - Compiling function g as a recurrence relation. - -+++ |*1;g;1;G82322;AUX| redefined - -+++ |*1;g;1;G82322| redefined - Compiling function g with type Integer -> Polynomial Fraction - Integer - Compiling function g as a recurrence relation. - -+++ |*1;g;1;G82322;AUX| redefined - -+++ |*1;g;1;G82322| redefined - Compiling function f with type Integer -> Polynomial Fraction - Integer - Compiling function f as a recurrence relation. - -+++ |*1;f;1;G82322;AUX| redefined - -+++ |*1;f;1;G82322| redefined -\end{verbatim} -$$ --{x \sp 3}+{d \ {x \sp 2}} -{{1 \over 3} \ {d \sp 2} \ x}+{{1 \over {27}} -\ {d \sp 3}} -$$ -\returnType{Type: Polynomial Fraction Integer} - -``I want my old definition of -$f$ after all. Undo the undo and restore -the environment to that immediately after $(4)$.'' -\spadcommand{)undo 4} - -``Check that the value of $f(3)$ is restored.'' -\spadcommand{f(3)} -\begin{verbatim} - Compiling function g with type Integer -> Polynomial Fraction - Integer - Compiling function g as a recurrence relation. - -+++ |*1;g;1;G82322;AUX| redefined - -+++ |*1;g;1;G82322| redefined - Compiling function g with type Integer -> Polynomial Fraction - Integer - Compiling function g as a recurrence relation. - -+++ |*1;g;1;G82322;AUX| redefined - -+++ |*1;g;1;G82322| redefined - Compiling function f with type Integer -> Polynomial Fraction - Integer - Compiling function f as a recurrence relation. - -+++ |*1;f;1;G82322;AUX| redefined - -+++ |*1;f;1;G82322| redefined -\end{verbatim} -$$ --{x \sp 3}+{{\left( e+{{1 \over 3} \ d} -\right)} -\ {x \sp 2}}+{{\left( -{{1 \over 4} \ {e \sp 2}} -{{1 \over 6} \ d \ e} --{{1 \over 9} \ {d \sp 2}} -\right)} -\ x}+{{1 \over 8} \ {e \sp 3}} -$$ -\returnType{Type: Polynomial Fraction Integer} - -After you have gone off on several tangents, then backtracked to -previous points in your conversation using {\tt )undo}, you might -want to save all the ``correct'' input commands you issued, -disregarding those undone. The system command {\tt )history -)write mynew.input} writes a clean straight-line program onto the file -{\bf mynew.input} on your disk. - -\section{Graphics} -\label{ugIntroGraphics} -Axiom has a two- and three-dimensional drawing and rendering -\index{graphics} package that allows you to draw, shade, color, -rotate, translate, map, clip, scale and combine graphic output of -Axiom computations. The graphics interface is capable of -plotting functions of one or more variables and plotting parametric -surfaces. Once the graphics figure appears in a window, move your -mouse to the window and click. A control panel appears immediately -and allows you to interactively transform the object. - -This is an example of Axiom's two-dimensional plotting. -From the 2D Control Panel you can rescale the plot, turn axes and units -on and off and save the image, among other things. -This PostScript image was produced by clicking on the -{\bf PS} 2D Control Panel button. -\spadgraph{draw(cos(5*t/8), t=0..16*\%pi, coordinates==polar)} -% window was 256 x 256 -%\epsffile[72 72 300 300]{ps/rose-1.ps} -\begin{figure}[htbp] -\includegraphics[bbllx=14, bblly=14, bburx=176, bbury=186]{ps/p28a.eps} -\caption{$J_0(\sqrt{x^2+y^2})$ for $-20 \leq x,y \leq 20$} -\label{tpdhere} -\end{figure} - -This is an example of Axiom's three-dimensional plotting. -It is a monochrome graph of the complex arctangent -function. -The image displayed was rotated and had the ``shade'' and ``outline'' -display options set from the 3D Control Panel. -The PostScript output was produced by clicking on the -{\bf save} 3D Control Panel button and then -clicking on the {\bf PS} button. -See Section \ref{ugProblemNumeric} on page~\pageref{ugProblemNumeric} -for more details and examples of Axiom's numeric and graphics capabilities. - -\spadgraph{draw((x,y) +-> real atan complex(x,y), -\%pi..\%pi, -\%pi..\%pi, colorFunction == (x,y) +-> argument atan complex(x,y))} -% window was 256 x 256 -%\epsffile[72 72 285 285]{ps/atan-1.ps} -\begin{figure}[htbp] -\includegraphics[bbllx=14, bblly=14, bburx=175, bbury=185]{ps/p28b.eps} -\caption{atan} -\label{tpdhere1} -\end{figure} - -An exhibit of Axiom images is given later. For a description of the -commands and programs that produced these figures, see -\ref{ugAppGraphics} on page~\pageref{ugAppGraphics}. PostScript -\index{PostScript} output is available so that Axiom images can be -printed.\footnote{PostScript is a trademark of Adobe Systems -Incorporated, registered in the United States.} See \ref{ugGraph} on -page~\pageref{ugGraph} for more examples and details about using -Axiom's graphics facilities. - -This concludes your tour of Axiom. -To disembark, issue the system command {\tt )quit} to leave Axiom -and return to the operating system. - -\setcounter{chapter}{1} - -\chapter{Using Types and Modes} -\begin{quote} -Only recently have I begun to realize that the problem is not merely -one of technical mastery or the competent application of the rules -\ldots -but that there is actually something else which is guiding these -rules. It actually involves a different level of mastery. It's quite -a different process to do it right; and every single act that you -do can be done in that sense well or badly. But even assuming that -you have got the technical part clear, the creation of this quality -is a much more complicated process of the most utterly absorbing and -fascinating dimensions. It is in fact a major creative or artistic -act -- every single little thing you do -- \ldots - --- Christopher Alexander - -(from Patterns of Software by Richard Gabriel) - -\end{quote} -\label{ugTypes} - -In this chapter we look at the key notion of {\it type} and its -generalization {\it mode}. We show that every Axiom object has a type -that determines what you can do with the object. In particular, we -explain how to use types to call specific functions from particular -parts of the library and how types and modes can be used to create new -objects from old. We also look at {\tt Record} and {\tt Union} types -and the special type {\tt Any}. Finally, we give you an idea of how -Axiom manipulates types and modes internally to resolve ambiguities. - -\section{The Basic Idea} -\label{ugTypesBasic} - -The Axiom world deals with many kinds of objects. There are -mathematical objects such as numbers and polynomials, data structure -objects such as lists and arrays, and graphics objects such as points -and graphic images. Functions are objects too. - -Axiom organizes objects using the notion of domain of computation, or -simply {\it domain}. Each domain denotes a class of objects. The -class of objects it denotes is usually given by the name of the -domain: {\tt Integer} for the integers, {\tt Float} for floating-point -numbers, and so on. The convention is that the first letter of a -domain name is capitalized. Similarly, the domain -{\tt Polynomial(Integer)} denotes ``polynomials with integer -coefficients.'' Also, {\tt Matrix(Float)} denotes ``matrices with -floating-point entries.'' - -Every basic Axiom object belongs to a unique domain. The integer $3$ -belongs to the domain {\tt Integer} and the polynomial $x + 3$ belongs -to the domain {\tt Polynomial(Integer)}. The domain of an object is -also called its {\it type}. Thus we speak of ``the type -{\tt Integer}'' and ``the type {\tt Polynomial(Integer)}.'' - -After an Axiom computation, the type is displayed toward the -right-hand side of the page (or screen). -\spadcommand{-3} -$$ --3 -$$ -\returnType{Type: Integer} - -Here we create a rational number but it looks like the last result. -The type however tells you it is different. You cannot identify the -type of an object by how Axiom displays the object. -\spadcommand{-3/1} -$$ --3 -$$ -\returnType{Type: Fraction Integer} - -When a computation produces a result of a simpler type, Axiom leaves -the type unsimplified. Thus no information is lost. -\spadcommand{x + 3 - x} -$$ -3 -$$ -\returnType{Type: Polynomial Integer} - -This seldom matters since Axiom retracts the answer to the -simpler type if it is necessary. -\spadcommand{factorial(\%)} -$$ -6 -$$ -\returnType{Type: Expression Integer} - -When you issue a positive number, the type {\tt PositiveInteger} is -printed. Surely, $3$ also has type {\tt Integer}! The curious reader -may now have two questions. First, is the type of an object not -unique? Second, how is {\tt PositiveInteger} related to {\tt -Integer}? -\spadcommand{3} -$$ -3 -$$ -\returnType{Type: PositiveInteger} - -Any domain can be refined to a {\it subdomain} by a membership -{\tt predicate}. A {\tt predicate} is a function that, when applied to an -object of the domain, returns either {\tt true} or {\tt false}. For -example, the domain {\tt Integer} can be refined to the subdomain -{\tt PositiveInteger}, the set of integers $x$ such that $x > 0$, by giving -the Axiom predicate $x +-> x > 0$. Similarly, Axiom can define -subdomains such as ``the subdomain of diagonal matrices,'' ``the -subdomain of lists of length two,'' ``the subdomain of monic -irreducible polynomials in $x$,'' and so on. Trivially, any domain is -a subdomain of itself. - -While an object belongs to a unique domain, it can belong to any -number of subdomains. Any subdomain of the domain of an object can be -used as the {\it type} of that object. The type of $3$ is indeed both -{\tt Integer} and {\tt PositiveInteger} as well as any other subdomain -of integer whose predicate is satisfied, such as ``the prime -integers,'' ``the odd positive integers between 3 and 17,'' and so on. - -\subsection{Domain Constructors} -\label{ugTypesBasicDomainCons} - -In Axiom, domains are objects. You can create them, pass them to -functions, and, as we'll see later, test them for certain properties. - -In Axiom, you ask for a value of a function by applying its name -to a set of arguments. - -To ask for ``the factorial of $7$'' you enter this expression to -Axiom. This applies the function {\tt factorial} to the value $7$ to -compute the result. -\spadcommand{factorial(7)} -$$ -5040 -$$ -\returnType{Type: PositiveInteger} - -Enter the type {\tt Polynomial (Integer)} as an expression to Axiom. -This looks much like a function call as well. It is! The result is -appropriately stated to be of type {\tt Domain}, which according to -our usual convention, denotes the class of all domains. -\spadcommand{Polynomial(Integer)} -$$ -\mbox{\rm Polynomial Integer} -$$ -\returnType{Type: Domain} - -The most basic operation involving domains is that of building a new -domain from a given one. To create the domain of ``polynomials over -the integers,'' Axiom applies the function {\tt Polynomial} to the -domain {\tt Integer}. A function like {\tt Polynomial} is called a -{\it domain constructor} or, \index{constructor!domain} more simply, a -{\it constructor}. A domain constructor is a function that creates a -domain. An argument to a domain constructor can be another domain or, -in general, an arbitrary kind of object. {\tt Polynomial} takes a -single domain argument while {\tt SquareMatrix} takes a positive -integer as an argument to give its dimension and a domain argument to -give the type of its components. - -What kinds of domains can you use as the argument to {\tt Polynomial} -or {\tt SquareMatrix} or {\tt List}? Well, the first two are -mathematical in nature. You want to be able to perform algebraic -operations like ``{\tt +}'' and ``{\tt *}'' on polynomials and square -matrices, and operations such as {\bf determinant} on square -matrices. So you want to allow polynomials of integers {\it and} -polynomials of square matrices with complex number coefficients and, -in general, anything that ``makes sense.'' At the same time, you don't -want Axiom to be able to build nonsense domains such as ``polynomials -of strings!'' - -In contrast to algebraic structures, data structures can hold any kind -of object. Operations on lists such as \spadfunFrom{insert}{List}, -\spadfunFrom{delete}{List}, and \spadfunFrom{concat}{List} just -manipulate the list itself without changing or operating on its -elements. Thus you can build {\tt List} over almost any datatype, -including itself. - -Create a complicated algebraic domain. -\spadcommand{List (List (Matrix (Polynomial (Complex (Fraction (Integer))))))} -$$ -\mbox{\rm List List Matrix Polynomial Complex Fraction Integer} -$$ -\returnType{Type: Domain} - -Try to create a meaningless domain. -\spadcommand{Polynomial(String)} -\begin{verbatim} - Polynomial String is not a valid type. -\end{verbatim} - -Evidently from our last example, Axiom has some mechanism that tells -what a constructor can use as an argument. This brings us to the -notion of {\it category}. As domains are objects, they too have a -domain. The domain of a domain is a category. A category is simply a -type whose members are domains. - -A common algebraic category is {\tt Ring}, the class of all domains -that are ``rings.'' A ring is an algebraic structure with constants -$0$ and $1$ and operations \spadopFrom{+}{Ring}, \spadopFrom{-}{Ring}, -and \spadopFrom{*}{Ring}. These operations are assumed ``closed'' -with respect to the domain, meaning that they take two objects of the -domain and produce a result object also in the domain. The operations -are understood to satisfy certain ``axioms,'' certain mathematical -principles providing the algebraic foundation for rings. For example, -the {\it additive inverse axiom} for rings states: \begin{center} -Every element $x$ has an additive inverse $y$ such that $x + y = 0$. -\end{center} The prototypical example of a domain that is a ring is -the integers. Keep them in mind whenever we mention {\tt Ring}. - -Many algebraic domain constructors such as {\tt Complex}, -{\tt Polynomial}, {\tt Fraction}, take rings as arguments and return rings -as values. You can use the infix operator ``$has$'' to ask a domain -if it belongs to a particular category. - -All numerical types are rings. Domain constructor {\tt Polynomial} -builds ``the ring of polynomials over any other ring.'' -\spadcommand{Polynomial(Integer) has Ring} -$$ -{\rm true} -$$ -\returnType{Type: Boolean} - -Constructor {\tt List} never produces a ring. -\spadcommand{List(Integer) has Ring} -$$ -{\rm false} -$$ -\returnType{Type: Boolean} - -The constructor {\tt Matrix(R)} builds ``the domain of all matrices -over the ring $R$.'' This domain is never a ring since the operations -``{\tt +}'', ``{\tt -}'', and ``{\tt *}'' on matrices of arbitrary -shapes are undefined. -\spadcommand{Matrix(Integer) has Ring} -$$ -{\rm false} -$$ -\returnType{Type: Boolean} - -Thus you can never build polynomials over matrices. -\spadcommand{Polynomial(Matrix(Integer))} -\begin{verbatim} - Polynomial Matrix Integer is not a valid type. -\end{verbatim} - -Use {\tt SquareMatrix(n,R)} instead. For any positive integer $n$, it -builds ``the ring of $n$ by $n$ matrices over $R$.'' -\spadcommand{Polynomial(SquareMatrix(7,Complex(Integer)))} -$$ -\mbox{\rm Polynomial SquareMatrix(7,Complex Integer)} -$$ -\returnType{Type: Domain} - -Another common category is {\tt Field}, the class of all fields. -\index{field} A field is a ring with additional operations. For -example, a field has commutative multiplication and a closed operation -\spadopFrom{/}{Field} for the division of two elements. {\tt Integer} -is not a field since, for example, $3/2$ does not have an integer -result. The prototypical example of a field is the rational numbers, -that is, the domain {\tt Fraction(Integer)}. In general, the -constructor {\tt Fraction} takes an IntegralDomain, which is a ring -with additional properties, as an argument and returns a field. -\footnote{Actually, the argument domain must have some additional -so as to belong to the category {\tt IntegralDomain}} -Other domain constructors, such as {\tt Complex}, build fields only if their -argument domain is a field. - -The complex integers (often called the ``Gaussian integers'') do not form -a field. -\spadcommand{Complex(Integer) has Field} -$$ -{\tt false} -$$ -\returnType{Type: Boolean} - -But fractions of complex integers do. -\spadcommand{Fraction(Complex(Integer)) has Field} -$$ -{\tt true} -$$ -\returnType{Type: Boolean} - -The algebraically equivalent domain of complex rational numbers is a field -since domain constructor {\tt Complex} produces a field whenever its -argument is a field. -\spadcommand{Complex(Fraction(Integer)) has Field} -$$ -{\tt true} -$$ -\returnType{Type: Boolean} - -The most basic category is {\tt Type}. \index{Type} It denotes the -class of all domains and subdomains. Note carefully that {\tt Type} -does not denote the class of all types. The type of all categories is -{\tt Category}. The type of {\tt Type} itself is undefined. Domain -constructor {\tt List} is able to build ``lists of elements from -domain $D$'' for arbitrary $D$ simply by requiring that $D$ belong to -category {\tt Type}. - -Now, you may ask, what exactly is a category? \index{category} Like -domains, categories can be defined in the Axiom language. A category -is defined by three components: -% -\begin{enumerate} -\item a name (for example, {\tt Ring}), -used to refer to the class of domains that the category represents; -\item a set of operations, used to refer to the operations that -the domains of this class support -(for example, ``{\tt +}'', ``{\tt -}'', and ``{\tt *}'' for rings); and -\item an optional list of other categories that this category extends. -\end{enumerate} -% -This last component is a new idea. And it is key to the design of -Axiom! Because categories can extend one another, they form -hierarchies. Detailed charts showing the category hierarchies -in Axiom are displayed in Appendix (TPDHERE). There you see -that all categories are extensions of {\tt Type} and that {\tt Field} -is an extension of {\tt Ring}. - -The operations supported by the domains of a category are called the -{\sl exports} of that category because these are the operations made -available for system-wide use. The exports of a domain of a given -category are not only the ones explicitly mentioned by the category. -Since a category extends other categories, the operations of these -other categories---and all categories these other categories -extend---are also exported by the domains. - -For example, polynomial domains belong to {\tt PolynomialCategory}. -This category explicitly mentions some twenty-nine operations on -polynomials, but it extends eleven other categories (including -{\tt Ring}). As a result, the current system has over one hundred -operations on polynomials. - -If a domain belongs to a category that extends, say, {\tt Ring}, it is -convenient to say that the domain exports {\tt Ring}. The name of the -category thus provides a convenient shorthand for the list of -operations exported by the category. Rather than listing operations -such as \spadopFrom{+}{Ring} and \spadopFrom{*}{Ring} of {\tt Ring} -each time they are needed, the definition of a type simply asserts -that it exports category {\tt Ring}. - -The category name, however, is more than a shorthand. The name -{\tt Ring}, in fact, implies that the operations exported by rings are -required to satisfy a set of ``axioms'' associated with the name -{\tt Ring}. This subtle but important feature distinguishes Axiom from -other abstract datatype designs. - -Why is it not correct to assume that some type is a ring if it exports -all of the operations of {\tt Ring}? Here is why. Some languages -such as {\bf APL} \index{APL} denote the {\tt Boolean} constants -{\tt true} and {\tt false} by the integers $1$ and $0$ respectively, then use -``{\tt +}'' and ``{\tt *}'' to denote the logical operators {\bf or} and -{\bf and}. But with these definitions {\tt Boolean} is not a -ring since the additive inverse axiom is violated. That is, there is -no inverse element $a$ such that $1 + a = 0$, or, in the usual terms: -{\tt true or a = false}. This alternative definition of {\tt Boolean} -can be easily and correctly implemented in Axiom, since {\tt Boolean} -simply does not assert that it is of category {\tt Ring}. This -prevents the system from building meaningless domains such as -{\tt Polynomial(Boolean)} and then wrongfully applying algorithms that -presume that the ring axioms hold. - -Enough on categories. To learn more about them, see Chapter -\ref{ugCategories} on page~\pageref{ugCategories}. -We now return to our discussion of domains. - -Domains {\it export} a set of operations to make them available for -system-wide use. {\tt Integer}, for example, exports the operations -\spadopFrom{+}{Integer} and \spadopFrom{=}{Integer} given by the -signatures \spadopFrom{+}{Integer}: -\spadsig{(Integer,Integer)}{Integer} and \spadopFrom{=}{Integer}: -\spadsig{(Integer,Integer)}{Boolean}, respectively. Each of these -operations takes two {\tt Integer} arguments. The -\spadopFrom{+}{Integer} operation also returns an {\tt Integer} but -\spadopFrom{=}{Integer} returns a {\tt Boolean}: {\tt true} or {\tt false}. -The operations exported by a domain usually manipulate objects of the -domain---but not always. - -The operations of a domain may actually take as arguments, and return -as values, objects from any domain. For example, {\tt Fraction -(Integer)} exports the operations \spadopFrom{/}{Fraction}: -\spadsig{(Integer,Integer)}{Fraction(Integer)} and -\spadfunFrom{characteristic}{Fraction}: -\spadsig{}{NonNegativeInteger}. - -Suppose all operations of a domain take as arguments and return as -values, only objects from {\it other} domains. \index{package} This -kind of domain \index{constructor!package} is what Axiom calls a {\it -package}. - -A package does not designate a class of objects at all. Rather, a -package is just a collection of operations. Actually the bulk of the -Axiom library of algorithms consists of packages. The facilities for -factorization; integration; solution of linear, polynomial, and -differential equations; computation of limits; and so on, are all -defined in packages. Domains needed by algorithms can be passed to a -package as arguments or used by name if they are not ``variable.'' -Packages are useful for defining operations that convert objects of -one type to another, particularly when these types have different -parameterizations. As an example, the package {\tt PolynomialFunction2(R,S)} -defines operations that convert polynomials -over a domain $R$ to polynomials over $S$. To convert an object from -{\tt Polynomial(Integer)} to {\tt Polynomial(Float)}, Axiom builds the -package {\tt PolynomialFunctions2(Integer,Float)} in order to create -the required conversion function. (This happens ``behind the scenes'' -for you: see \ref{ugTypesConvert} on page~\pageref{ugTypesConvert} -for details on how to convert objects.) - -Axiom categories, domains and packages and all their contained -functions are written in the Axiom programming language and have been -compiled into machine code. This is what comprises the Axiom -{\it library}. We will show you how to use these -domains and their functions and how to write your own functions. - -\section{Writing Types and Modes} -\label{ugTypesWriting} - -We have already seen in the last section \ref{ugTypesBasic} on -page~\pageref{ugTypesBasic} several examples of types. Most of these -examples had either no arguments (for example, {\tt Integer}) or one -argument (for example, {\tt Polynomial (Integer)}). In this section -we give details about writing arbitrary types. We then define modes -and discuss how to write them. We conclude the section with a -discussion on constructor abbreviations. - -When might you need to write a type or mode? You need to do so when -you declare variables. -\spadcommand{a : PositiveInteger} -\returnType{Type: Void} - -You need to do so when you declare functions -(See Section \ref{ugTypesDeclare} on page~\pageref{ugTypesDeclare}), -\spadcommand{f : Integer -> String} -\returnType{Type: Void} - -You need to do so when you convert an object from one type to another -(See Section \ref{ugTypesConvert} on page~\pageref{ugTypesConvert}). -\spadcommand{factor(2 :: Complex(Integer))} -$$ --{i \ {{\left( 1+i -\right)} -\sp 2}} -$$ -\returnType{Type: Factored Complex Integer} - -\spadcommand{(2 = 3)\$Integer} -$$ -{\tt false} -$$ -\returnType{Type: Boolean} - -You need to do so when you give computation target type information -(See Section \ref{ugTypesPkgCall} on page~\pageref{ugTypesPkgCall}). -\spadcommand{(2 = 3)@Boolean} -$$ -{\tt false} -$$ -\returnType{Type: Boolean} - -\subsection{Types with No Arguments} -\label{ugTypesWritingZero} - -A constructor with no arguments can be written either -\index{type!using parentheses} with or without -\index{parentheses!using with types} trailing opening and closing -parentheses ``{\tt ()}''. - -\begin{center} -{\tt Boolean()} is the same as {\tt Boolean} \\ -{\tt Integer()} is the same as {\tt Integer} \\ -{\tt String()} is the same as {\tt String} \\ -{\tt Void()} is the same as {\tt Void} -\end{center} - -It is customary to omit the parentheses. - -\subsection{Types with One Argument} -\label{ugTypesWritingOne} - -A constructor with one argument can frequently be -\index{type!using parentheses} written with no -\index{parentheses!using with types} parentheses. Types nest from -right to left so that {\tt Complex Fraction Polynomial Integer} -is the same as {\tt Complex (Fraction (Polynomial (Integer)))}. -You need to use parentheses to force the application of a constructor -to the correct argument, but you need not use any more than is necessary -to remove ambiguities. - -Here are some guidelines for using parentheses (they are possibly slightly -more restrictive than they need to be). - -If the argument is an expression like $2 + 3$ -then you must enclose the argument in parentheses. -\spadcommand{e : PrimeField(2 + 3)} -\returnType{Type: Void} - -If the type is to be used with package calling -then you must enclose the argument in parentheses. -\spadcommand{content(2)\$Polynomial(Integer)} -$$ -2 -$$ -\returnType{Type: Integer} - -Alternatively, you can write the type without parentheses -then enclose the whole type expression with parentheses. -\spadcommand{content(2)\$(Polynomial Complex Fraction Integer)} -$$ -2 -$$ -\returnType{Type: Complex Fraction Integer} - -If you supply computation target type information -(See Section \ref{ugTypesPkgCall} on page~\pageref{ugTypesPkgCall}) -then you should enclose the argument in parentheses. -\spadcommand{(2/3)@Fraction(Polynomial(Integer))} -$$ -2 \over 3 -$$ -\returnType{Type: Fraction Polynomial Integer} - -If the type itself has parentheses around it and we are not in the -case of the first example above, then the parentheses can usually be -omitted. -\spadcommand{(2/3)@Fraction(Polynomial Integer)} -$$ -2 \over 3 -$$ -\returnType{Type: Fraction Polynomial Integer} - -If the type is used in a declaration and the argument is a single-word -type, integer or symbol, then the parentheses can usually be omitted. -\spadcommand{(d,f,g) : Complex Polynomial Integer} -\returnType{Type: Void} - -\subsection{Types with More Than One Argument} -\label{ugTypesWritingMore} - -If a constructor \index{type!using parentheses} has more than -\index{parentheses!using with types} one argument, you must use -parentheses. Some examples are \\ - -{\tt UnivariatePolynomial(x, Float)} \\ -{\tt MultivariatePolynomial([z,w,r], Complex Float)} \\ -{\tt SquareMatrix(3, Integer)} \\ -{\tt FactoredFunctions2(Integer,Fraction Integer)} - -\subsection{Modes} -\label{ugTypesWritingModes} - -A {\it mode} is a type that possibly is a question mark ({\tt ?}) or -contains one in an argument position. For example, the following are -all modes.\\ - -{\tt ?} \\ -{\tt Polynomial ?} \\ -{\tt Matrix Polynomial ?} \\ -{\tt SquareMatrix(3,?)} \\ -{\tt Integer} \\ -{\tt OneDimensionalArray(Float)} - -As is evident from these examples, a mode is a type with a part that -is not specified (indicated by a question mark). Only one ``{\tt ?}'' is -allowed per mode and it must appear in the most deeply nested argument -that is a type. Thus {\tt ?(Integer)}, {\tt Matrix(? (Polynomial))}, -{\tt SquareMatrix(?, Integer)} (it requires a numeric argument) -and {\tt SquareMatrix(?, ?)} are all -invalid. The question mark must take the place of a domain, not data. -This rules out, for example, the two {\tt SquareMatrix} expressions. - -Modes can be used for declarations (See Section \ref{ugTypesDeclare} -on page~\pageref{ugTypesDeclare}) and conversions (Section -\ref{ugTypesConvert} on page~\pageref{ugTypesConvert}). However, you -cannot use a mode for package calling or giving target type information. - -\subsection{Abbreviations} -\label{ugTypesWritingAbbr} - -Every constructor has an abbreviation that -\index{abbreviation!constructor} you can freely -\index{constructor!abbreviation} substitute for the constructor name. -In some cases, the abbreviation is nothing more than the capitalized -version of the constructor name. - -\boxed{4.6in}{ - -\vskip 0.1cm -Aside from allowing types to be written more concisely, abbreviations -are used by Axiom to name various system files for constructors (such -as library filenames, test input files and example files). Here are -some common abbreviations. - -\begin{center} -\begin{tabular}{ll} -\small{\tt COMPLEX} abbreviates {\tt Complex} & -\small{\tt DFLOAT} abbreviates {\tt DoubleFloat} \\ -\small{\tt EXPR} abbreviates {\tt Expression} & -\small{\tt FLOAT} abbreviates {\tt Float} \\ -\small{\tt FRAC} abbreviates {\tt Fraction} & -\small{\tt INT} abbreviates {\tt Integer} \\ -\small{\tt MATRIX} abbreviates {\tt Matrix} & -\small{\tt NNI} abbreviates {\tt NonNegativeInteger} \\ -\small{\tt PI} abbreviates {\tt PositiveInteger} & -\small{\tt POLY} abbreviates {\tt Polynomial} \\ -\small{\tt STRING} abbreviates {\tt String} & -\small{\tt UP} abbreviates {\tt UnivariatePolynomial}\\ -\end{tabular} -\end{center} -\vskip 0.1cm -} - -You can combine both full constructor names and abbreviations in a -type expression. Here are some types using abbreviations. - -\begin{center} -\begin{tabular}{rcl} -{\tt POLY INT} & is the same as & {\tt Polynomial(INT)} \\ -{\tt POLY(Integer)} & is the same as & {\tt Polynomial(Integer)} \\ -{\tt POLY(Integer)} & is the same as & {\tt Polynomial(INT)} \\ -{\tt FRAC(COMPLEX(INT))} & is the same as & {\tt Fraction Complex Integer} \\ -{\tt FRAC(COMPLEX(INT))} & is the same as & {\tt FRAC(Complex Integer)} -\end{tabular} -\end{center} - -There are several ways of finding the names of constructors and their -abbreviations. For a specific constructor, use {\tt )abbreviation -query}. \index{abbreviation} You can also use the {\tt )what} system -command to see the names and abbreviations of constructors. -\index{what} For more information about {\tt )what}, see -\ref{ugSysCmdwhat} on page~\pageref{ugSysCmdwhat}. - -{\tt )abbreviation query} can be abbreviated (no pun intended) to -{\tt )abb q}. -\spadcommand{)abb q Integer} -\begin{verbatim} - INT abbreviates domain Integer -\end{verbatim} - -The {\tt )abbreviation query} command lists the constructor name if -you give the abbreviation. Issue {\tt )abb q} if you want to see the -names and abbreviations of all Axiom constructors. -\spadcommand{)abb q DMP} -\begin{verbatim} - DMP abbreviates domain DistributedMultivariatePolynomial -\end{verbatim} - -Issue this to see all packages whose -names contain the string ``ode''. \index{what packages} -\spadcommand{)what packages ode} -\begin{verbatim} ----------------------- Packages ----------------------- - -Packages with names matching patterns: - ode - - EXPRODE ExpressionSpaceODESolver - FCPAK1 FortranCodePackage1 - GRAY GrayCode - LODEEF ElementaryFunctionLODESolver - NODE1 NonLinearFirstOrderODESolver - ODECONST ConstantLODE - ODEEF ElementaryFunctionODESolver - ODEINT ODEIntegration - ODEPAL PureAlgebraicLODE - ODERAT RationalLODE - ODERED ReduceLODE - ODESYS SystemODESolver - ODETOOLS ODETools - UTSODE UnivariateTaylorSeriesODESolver - UTSODETL UTSodetools -\end{verbatim} - -\section{Declarations} -\label{ugTypesDeclare} - -A {\it declaration} is an expression used to restrict the type of -values that can be assigned to variables. A colon ``{\tt :}'' is always -used after a variable or list of variables to be declared. - -\boxed{4.6in}{ -\vskip 0.1cm -For a single variable, the syntax for declaration is -\begin{center} -{\it variableName $:$ typeOrMode} -\end{center} - -For multiple variables, the syntax is -\begin{center} -{\tt ($\hbox{\it variableName}_{1}$, $\hbox{\it variableName}_{2}$, -\ldots $\hbox{\it variableName}_{N}$): {\it typeOrMode}} -\end{center} -\vskip 0.1cm -} - -You can always combine a declaration with an assignment. When you do, -it is equivalent to first giving a declaration statement, then giving -an assignment. For more information on assignment, see -Section \ref{ugIntroAssign} on page~\pageref{ugIntroAssign} and -Section \ref{ugLangAssign} on page~\pageref{ugLangAssign}. -To see how to declare your own functions, -see \ref{ugUserDeclare} on page~\pageref{ugUserDeclare}. - -This declares one variable to have a type. -\spadcommand{a : Integer} -\returnType{Type: Void} - -This declares several variables to have a type. -\spadcommand{(b,c) : Integer} -\returnType{Type: Void} - -$a$, $b$ and $c$ can only hold integer values. -\spadcommand{a := 45} -$$ -45 -$$ -\returnType{Type: Integer} - -If a value cannot be converted to a declared type, -an error message is displayed. -\spadcommand{b := 4/5} -\begin{verbatim} - - Cannot convert right-hand side of assignment - 4 - - - 5 - - to an object of the type Integer of the left-hand side. -\end{verbatim} - -This declares a variable with a mode. -\spadcommand{n : Complex ?} -\returnType{Type: Void} - -This declares several variables with a mode. -\spadcommand{(p,q,r) : Matrix Polynomial ?} -\returnType{Type: Void} - -This complex object has integer real and imaginary parts. -\spadcommand{n := -36 + 9 * \%i} -$$ --{36}+{9 \ i} -$$ -\returnType{Type: Complex Integer} - -This complex object has fractional symbolic real and imaginary parts. -\spadcommand{n := complex(4/(x + y),y/x)} -$$ -{4 \over {y+x}}+{{y \over x} \ i} -$$ -\returnType{Type: Complex Fraction Polynomial Integer} - -This matrix has entries that are polynomials with integer -coefficients. -\spadcommand{p := [ [1,2],[3,4],[5,6] ]} -$$ -\left[ -\begin{array}{cc} -1 & 2 \\ -3 & 4 \\ -5 & 6 -\end{array} -\right] -$$ -\returnType{Type: Matrix Polynomial Integer} - -This matrix has a single entry that is a polynomial with -rational number coefficients. -\spadcommand{q := [ [x - 2/3] ]} -$$ -\left[ -\begin{array}{c} -{x -{2 \over 3}} -\end{array} -\right] -$$ -\returnType{Type: Matrix Polynomial Fraction Integer} - -This matrix has entries that are polynomials with complex integer -coefficients. - -\spadcommand{r := [ [1-\%i*x,7*y+4*\%i] ]} -$$ -\left[ -\begin{array}{cc} -{-{i \ x}+1} & {{7 \ y}+{4 \ i}} -\end{array} -\right] -$$ -\returnType{Type: Matrix Polynomial Complex Integer} - -Note the difference between this and the next example. -This is a complex object with polynomial real and imaginary parts. - -\spadcommand{f : COMPLEX POLY ? := (x + y*\%i)**2} -$$ --{y \sp 2}+{x \sp 2}+{2 \ x \ y \ i} -$$ -\returnType{Type: Complex Polynomial Integer} - -This is a polynomial with complex integer coefficients. The objects -are convertible from one to the other. See \ref{ugTypesConvert} on -page~\pageref{ugTypesConvert} for more information. - -\spadcommand{g : POLY COMPLEX ? := (x + y*\%i)**2} -$$ --{y \sp 2}+{2 \ i \ x \ y}+{x \sp 2} -$$ -\returnType{Type: Polynomial Complex Integer} - -\section{Records} -\label{ugTypesRecords} - -A {\tt Record} is an object composed of one or more other objects, -\index{Record} each of which is referenced \index{selector!record} -with \index{record!selector} a {\it selector}. Components can all -belong to the same type or each can have a different type. - -\boxed{4.6in}{ -\vskip 0.1cm -The syntax for writing a {\tt Record} type is \begin{center} {\tt -Record($\hbox{\it selector}_{1}$:$\hbox{\it type}_{1}$, -$\hbox{\it selector}_{2}$:$\hbox{\it type}_{2}$, \ldots, -$\hbox{\it selector}_{N}$:$\hbox{\it type}_{N}$)} \end{center} You must be -careful if a selector has the same name as a variable in the -workspace. If this occurs, precede the selector name by a single -\index{quote} quote.\\ -} - -Record components are implicitly ordered. All the components of a -record can be set at once by assigning the record a bracketed {\it -tuple} of values of the proper length. For example: -\spadcommand{r : Record(a:Integer, b: String) := [1, "two"]} -$$ -\left[ -{a=1}, {b= \mbox{\tt "two"} } -\right] -$$ -\returnType{Type: Record(a: Integer,b: String)} -To access a component of a record $r$, write the name $r$, followed by -a period, followed by a selector. - -The object returned by this computation is a record with two components: a -$quotient$ part and a $remainder$ part. -\spadcommand{u := divide(5,2)} -$$ -\left[ -{quotient=2}, {remainder=1} -\right] -$$ -\returnType{Type: Record(quotient: Integer,remainder: Integer)} - -This is the quotient part. -\spadcommand{u.quotient} -$$ -2 -$$ -\returnType{Type: PositiveInteger} - -This is the remainder part. -\spadcommand{u.remainder} -$$ -1 -$$ -\returnType{Type: PositiveInteger} - -You can use selector expressions on the left-hand side of an assignment -to change destructively the components of a record. -\spadcommand{u.quotient := 8978} -$$ -8978 -$$ -\returnType{Type: PositiveInteger} - -The selected component $quotient$ has the value $8978$, which is what -is returned by the assignment. Check that the value of $u$ was -modified. -\spadcommand{u} -$$ -\left[ -{quotient={8978}}, {remainder=1} -\right] -$$ -\returnType{Type: Record(quotient: Integer,remainder: Integer)} - -Selectors are evaluated. Thus you can use variables that evaluate to -selectors instead of the selectors themselves. -\spadcommand{s := 'quotient} -$$ -quotient -$$ -\returnType{Type: Variable quotient} - -Be careful! A selector could have the same name as a variable in the -workspace. If this occurs, precede the selector name by a single -quote, as in $u.'quotient$. \index{selector!quoting} -\spadcommand{divide(5,2).s} -$$ -2 -$$ -\returnType{Type: PositiveInteger} - -Here we declare that the value of $bd$ has two components: a string, -to be accessed via {\tt name}, and an integer, to be accessed via -{\tt birthdayMonth}. -\spadcommand{bd : Record(name : String, birthdayMonth : Integer)} -\returnType{Type: Void} - -You must initially set the value of the entire {\tt Record} at once. -\spadcommand{bd := ["Judith", 3]} -$$ -\left[ -{name= \mbox{\tt "Judith"} }, {birthdayMonth=3} -\right] -$$ -\returnType{Type: Record(name: String,birthdayMonth: Integer)} - -Once set, you can change any of the individual components. -\spadcommand{bd.name := "Katie"} -$$ -\mbox{\tt "Katie"} -$$ -\returnType{Type: String} - -Records may be nested and the selector names can be shared at -different levels. -\spadcommand{r : Record(a : Record(b: Integer, c: Integer), b: Integer)} -\returnType{Type: Void} - -The record $r$ has a $b$ selector at two different levels. -Here is an initial value for $r$. -\spadcommand{r := [ [1,2], 3 ]} -$$ -\left[ -{a={\left[ {b=1}, {c=2} -\right]}}, - {b=3} -\right] -$$ -\returnType{Type: Record(a: Record(b: Integer,c: Integer),b: Integer)} - -This extracts the $b$ component from the $a$ component of $r$. -\spadcommand{r.a.b} -$$ -1 -$$ -\returnType{Type: PositiveInteger} - -This extracts the $b$ component from $r$. -\spadcommand{r.b} -$$ -3 -$$ -\returnType{Type: PositiveInteger} - -You can also use spaces or parentheses to refer to {\tt Record} -components. This is the same as $r.a$. -\spadcommand{r(a)} -$$ -\left[ -{b=1}, {c=2} -\right] -$$ -\returnType{Type: Record(b: Integer,c: Integer)} -This is the same as $r.b$. -\spadcommand{r b} -$$ -3 -$$ -\returnType{Type: PositiveInteger} - -This is the same as $r.b := 10$. -\spadcommand{r(b) := 10} -$$ -10 -$$ -\returnType{Type: PositiveInteger} - -Look at $r$ to make sure it was modified. -\spadcommand{r} -$$ -\left[ -{a={\left[ {b=1}, {c=2} -\right]}}, - {b={10}} -\right] -$$ -\returnType{Type: Record(a: Record(b: Integer,c: Integer),b: Integer)} - -\section{Unions} -\label{ugTypesUnions} - -Type {\tt Union} is used for objects that can be of any of a specific -finite set of types. \index{Union} Two versions of unions are -available, one with selectors (like records) and one without. -\index{union} - -\subsection{Unions Without Selectors} -\label{ugTypesUnionsWOSel} - -The declaration $x : Union(Integer, String, Float)$ states that $x$ -can have values that are integers, strings or ``big'' floats. If, for -example, the {\tt Union} object is an integer, the object is said to -belong to the {\tt Integer} {\it branch} of the {\tt Union}. Note -that we are being a bit careless with the language here. Technically, -the type of $x$ is always {\tt Union(Integer, String, Float)}. If it -belongs to the {\tt Integer} branch, $x$ may be converted to an object -of type {\tt Integer}. - -\boxed{4.6in}{ -\vskip 0.1cm -The syntax for writing a {\tt Union} type without selectors is -\begin{center} -{\tt Union($\hbox{\it type}_{1}$, $\hbox{\it type}_{2}$, -\ldots, $\hbox{\it type}+{N}$)} -\end{center} -The types in a union without selectors must be distinct.\\ -} - -It is possible to create unions like {\tt Union(Integer, PositiveInteger)} -but they are difficult to work with because of the overlap in the branch -types. See below for the rules Axiom uses for converting something into -a union object. - -The {\tt case} infix \index{case} operator returns a {\tt Boolean} and can -be used to determine the branch in which an object lies. - -This function displays a message stating in which branch of the -{\tt Union} the object (defined as $x$ above) lies. - -\begin{verbatim} -sayBranch(x : Union(Integer,String,Float)) : Void == - output - x case Integer => "Integer branch" - x case String => "String branch" - "Float branch" -\end{verbatim} - -This tries {\bf sayBranch} with an integer. -\spadcommand{sayBranch 1} -\begin{verbatim} -Compiling function sayBranch with type Union(Integer,String,Float) - -> Void - Integer branch -\end{verbatim} -\returnType{Type: Void} - -This tries {\bf sayBranch} with a string. -\spadcommand{sayBranch "hello"} -\begin{verbatim} - String branch -\end{verbatim} -\returnType{Type: Void} - -This tries {\bf sayBranch} with a floating-point number. -\spadcommand{sayBranch 2.718281828} -\begin{verbatim} - Float branch -\end{verbatim} -\returnType{Type: Void} - -There are two things of interest about this particular -example to which we would like to draw your attention. -\begin{enumerate} -\item Axiom normally converts a result to the target value -before passing it to the function. -If we left the declaration information out of this function definition -then the {\bf sayBranch} call would have been attempted with an -{\tt Integer} rather than a {\tt Union}, and an error would have -resulted. -\item The types in a {\tt Union} are searched in the order given. -So if the type were given as - -%\noindent -{\tt sayBranch(x: Union(String,Integer,Float,Any)): Void} - -then the result would have been ``String branch'' because there -is a conversion from {\tt Integer} to {\tt String}. -\end{enumerate} - -Sometimes {\tt Union} types can have extremely long names. Axiom -therefore abbreviates the names of unions by printing the type of the -branch first within the {\tt Union} and then eliding the remaining -types with an ellipsis ({\tt ...}). - -Here the {\tt Integer} branch is displayed first. Use ``{\tt ::}'' to -create a {\tt Union} object from an object. -\spadcommand{78 :: Union(Integer,String)} -$$ -78 -$$ -\returnType{Type: Union(Integer,...)} - -Here the {\tt String} branch is displayed first. -\spadcommand{s := "string" :: Union(Integer,String)} -$$ -\mbox{\tt "string"} -$$ -\returnType{Type: Union(String,...)} - -Use {\tt typeOf} to see the full and actual {\tt Union} type. \index{typeOf} -\spadcommand{typeOf s} -$$ -Union(Integer,String) -$$ -\returnType{Type: Domain} - -A common operation that returns a union is \spadfunFrom{exquo}{Integer} -which returns the ``exact quotient'' if the quotient is exact, -\spadcommand{three := exquo(6,2)} -$$ -3 -$$ -\returnType{Type: Union(Integer,...)} - -and {\tt "failed"} if the quotient is not exact. -\spadcommand{exquo(5,2)} -$$ -\mbox{\tt "failed"} -$$ -\returnType{Type: Union("failed",...)} - -A union with a {\tt "failed"} is frequently used to indicate the failure -or lack of applicability of an object. As another example, assign an -integer a variable $r$ declared to be a rational number. -\spadcommand{r: FRAC INT := 3} -$$ -3 -$$ -\returnType{Type: Fraction Integer} - -The operation \spadfunFrom{retractIfCan}{Fraction} tries to retract -the fraction to the underlying domain {\tt Integer}. It produces a -union object. Here it succeeds. -\spadcommand{retractIfCan(r)} -$$ -3 -$$ -\returnType{Type: Union(Integer,...)} - -Assign it a rational number. -\spadcommand{r := 3/2} -$$ -3 \over 2 -$$ -\returnType{Type: Fraction Integer} - -Here the retraction fails. -\spadcommand{retractIfCan(r)} -$$ -\mbox{\tt "failed"} -$$ -\returnType{Type: Union("failed",...)} - -\subsection{Unions With Selectors} -\label{ugTypesUnionsWSel} - -Like records (\ref{ugTypesRecords} on page~\pageref{ugTypesRecords}), -you can write {\tt Union} types \index{selector!union} with selectors. -\index{union!selector} - -\boxed{4.6in}{ -\vskip 0.1cm -The syntax for writing a {\tt Union} type with selectors is -\begin{center} -{\tt Union($\hbox{\it selector}_{1}$:$\hbox{\it type}_{1}$, -$\hbox{\it selector}_{2}$:$\hbox{\it type}_{2}$, \ldots, -$\hbox{\it selector}_{N}$:$\hbox{\it type}_{N}$)} -\end{center} -You must be careful if a selector has the same name as a variable in -the workspace. If this occurs, precede the selector name by a single -\index{quote} quote. \index{selector!quoting} It is an error to use a -selector that does not correspond to the branch of the {\tt Union} in -which the element actually lies. \\ -} - -Be sure to understand the difference between records and unions with -selectors. \index{union!difference from record} Records can have more -than one component and the selectors are used to refer to the -components. \index{record!difference from union} Unions always have -one component but the type of that one component can vary. An object -of type {\tt Record(a: Integer, b: Float, c: String)} contains an -integer {\it and} a float {\it and} a string. An object of type -{\tt Union(a: Integer, b: Float, c: String)} contains an integer -{\it or} a float {\it or} a string. - -Here is a version of the {\bf sayBranch} function (cf. -\ref{ugTypesUnionsWOSel} on page~\pageref{ugTypesUnionsWOSel}) that -works with a union with selectors. It displays a message stating in -which branch of the {\tt Union} the object lies. - -\begin{verbatim} -sayBranch(x:Union(i:Integer,s:String,f:Float)):Void== - output - x case i => "Integer branch" - x case s => "String branch" - "Float branch" -\end{verbatim} - -Note that {\tt case} uses the selector name as its right-hand argument. -\index{case} If you accidentally use the branch type on the right-hand -side of {\tt case}, {\tt false} will be returned. - -Declare variable $u$ to have a union type with selectors. -\spadcommand{u : Union(i : Integer, s : String)} -\returnType{Type: Void} - -Give an initial value to $u$. -\spadcommand{u := "good morning"} -$$ -\mbox{\tt "good morning"} -$$ -\returnType{Type: Union(s: String,...)} - -Use $case$ to determine in which branch of a {\tt Union} an object lies. -\spadcommand{u case i} -$$ -{\tt false} -$$ -\returnType{Type: Boolean} - -\spadcommand{u case s} -$$ -{\tt true} -$$ -\returnType{Type: Boolean} - -To access the element in a particular branch, use the selector. -\spadcommand{u.s} -$$ -\mbox{\tt "good morning"} -$$ -\returnType{Type: String} - -\section{The ``Any'' Domain} -\label{ugTypesAnyNone} - -With the exception of objects of type {\tt Record}, all Axiom data -structures are homogenous, that is, they hold objects all of the same -type. \index{Any} If you need to get around this, you can use type -{\tt Any}. Using {\tt Any}, for example, you can create lists whose -elements are integers, rational numbers, strings, and even other -lists. - -Declare $u$ to have type {\tt Any}. -\spadcommand{u: Any} -\returnType{Type: Void} - -Assign a list of mixed type values to $u$ -\spadcommand{u := [1, 7.2, 3/2, x**2, "wally"]} -$$ -\left[ -1, {7.2}, {3 \over 2}, {x \sp 2}, \mbox{\tt "wally"} -\right] -$$ -\returnType{Type: List Any} - -When we ask for the elements, Axiom displays these types. -\spadcommand{u.1} -$$ -1 -$$ -\returnType{Type: PositiveInteger} - -Actually, these objects belong to {\tt Any} but Axiom -automatically converts them to their natural types for you. -\spadcommand{u.3} -$$ -3 \over 2 -$$ -\returnType{Type: Fraction Integer} - -Since type {\tt Any} can be anything, it can only belong to type -{\tt Type}. Therefore it cannot be used in algebraic domains. -\spadcommand{v : Matrix(Any)} -\begin{verbatim} - Matrix Any is not a valid type. -\end{verbatim} - -Perhaps you are wondering how Axiom internally represents objects of -type {\tt Any}. An object of type {\tt Any} consists not only a data -part representing its normal value, but also a type part (a -{\it badge}) giving \index{badge} its type. For example, the value $1$ of -type {\tt PositiveInteger} as an object of type {\tt Any} internally -looks like $[1,{\tt PositiveInteger()}]$. - -When should you use {\tt Any} instead of a {\tt Union} type? For a -{\tt Union}, you must know in advance exactly which types you are -going to -allow. For {\tt Any}, anything that comes along can be accommodated. - -\section{Conversion} -\label{ugTypesConvert} - -\boxed{4.6in}{ -\vskip 0.1cm -Conversion is the process of changing an object of one type into an -object of another type. The syntax for conversion is: -$$ -{\it object} {\tt ::} {\it newType} -$$ -} - -By default, $3$ has the type {\tt PositiveInteger}. -\spadcommand{3} -$$ -3 -$$ -\returnType{Type: PositiveInteger} - -We can change this into an object of type {\tt Fraction Integer} -by using ``{\tt ::}''. -\spadcommand{3 :: Fraction Integer} -$$ -3 -$$ -\returnType{Type: Fraction Integer} - -A {\it coercion} is a special kind of conversion that Axiom is allowed -to do automatically when you enter an expression. Coercions are -usually somewhat safer than more general conversions. The Axiom -library contains operations called {\bf coerce} and {\bf convert}. -Only the {\bf coerce} operations can be used by the interpreter to -change an object into an object of another type unless you explicitly -use a {\tt ::}. - -By now you will be quite familiar with what types and modes look like. -It is useful to think of a type or mode as a pattern for what you want -the result to be. - -Let's start with a square matrix of polynomials with complex rational -number coefficients. \index{SquareMatrix} -\spadcommand{m : SquareMatrix(2,POLY COMPLEX FRAC INT)} -\returnType{Type: Void} - -\spadcommand{m := matrix [ [x-3/4*\%i,z*y**2+1/2],[3/7*\%i*y**4 - x,12-\%i*9/5] ]} -$$ -\left[ -\begin{array}{cc} -{x -{{3 \over 4} \ i}} & {{{y \sp 2} \ z}+{1 \over 2}} \\ -{{{3 \over 7} \ i \ {y \sp 4}} -x} & {{12} -{{9 \over 5} \ i}} -\end{array} -\right] -$$ -\returnType{Type: SquareMatrix(2,Polynomial Complex Fraction Integer)} - -We first want to interchange the {\tt Complex} and {\tt Fraction} -layers. We do the conversion by doing the interchange in the type -expression. -\spadcommand{m1 := m :: SquareMatrix(2,POLY FRAC COMPLEX INT)} -$$ -\left[ -\begin{array}{cc} -{x -{{3 \ i} \over 4}} & {{{y \sp 2} \ z}+{1 \over 2}} \\ -{{{{3 \ i} \over 7} \ {y \sp 4}} -x} & {{{60} -{9 \ i}} \over 5} -\end{array} -\right] -$$ -\returnType{Type: SquareMatrix(2,Polynomial Fraction Complex Integer)} - -Interchange the {\tt Polynomial} and the {\tt Fraction} levels. -\spadcommand{m2 := m1 :: SquareMatrix(2,FRAC POLY COMPLEX INT)} -$$ -\left[ -\begin{array}{cc} -{{{4 \ x} -{3 \ i}} \over 4} & {{{2 \ {y \sp 2} \ z}+1} \over 2} \\ -{{{3 \ i \ {y \sp 4}} -{7 \ x}} \over 7} & {{{60} -{9 \ i}} \over 5} -\end{array} -\right] -$$ -\returnType{Type: SquareMatrix(2,Fraction Polynomial Complex Integer)} - -Interchange the {\tt Polynomial} and the {\tt Complex} levels. -\spadcommand{m3 := m2 :: SquareMatrix(2,FRAC COMPLEX POLY INT)} -$$ -\left[ -\begin{array}{cc} -{{{4 \ x} -{3 \ i}} \over 4} & {{{2 \ {y \sp 2} \ z}+1} \over 2} \\ -{{-{7 \ x}+{3 \ {y \sp 4} \ i}} \over 7} & {{{60} -{9 \ i}} \over 5} -\end{array} -\right] -$$ -\returnType{Type: SquareMatrix(2,Fraction Complex Polynomial Integer)} - -All the entries have changed types, although in comparing the -last two results only the entry in the lower left corner looks different. -We did all the intermediate steps to show you what Axiom can do. - -In fact, we could have combined all these into one conversion. -\spadcommand{m :: SquareMatrix(2,FRAC COMPLEX POLY INT)} -$$ -\left[ -\begin{array}{cc} -{{{4 \ x} -{3 \ i}} \over 4} & {{{2 \ {y \sp 2} \ z}+1} \over 2} \\ -{{-{7 \ x}+{3 \ {y \sp 4} \ i}} \over 7} & {{{60} -{9 \ i}} \over 5} -\end{array} -\right] -$$ -\returnType{Type: SquareMatrix(2,Fraction Complex Polynomial Integer)} - -There are times when Axiom is not be able to do the conversion in one -step. You may need to break up the transformation into several -conversions in order to get an object of the desired type. - -We cannot move either {\tt Fraction} or {\tt Complex} above (or to the -left of, depending on how you look at it) {\tt SquareMatrix} because -each of these levels requires that its argument type have commutative -multiplication, whereas {\tt SquareMatrix} does not. That is because -{\tt Fraction} requires that its argument belong to the category -{\tt IntegralDomain} and \index{category} {\tt Complex} requires that its -argument belong to {\tt CommutativeRing}. -See \ref{ugTypesBasic} on page~\pageref{ugTypesBasic} for a -brief discussion of categories. The {\tt Integer} level did not move -anywhere because it does not allow any arguments. We also did not -move the {\tt SquareMatrix} part anywhere, but we could have. - -Recall that $m$ looks like this. - -\spadcommand{m} -$$ -\left[ -\begin{array}{cc} -{x -{{3 \over 4} \ i}} & {{{y \sp 2} \ z}+{1 \over 2}} \\ -{{{3 \over 7} \ i \ {y \sp 4}} -x} & {{12} -{{9 \over 5} \ i}} -\end{array} -\right] -$$ -\returnType{Type: SquareMatrix(2,Polynomial Complex Fraction Integer)} - -If we want a polynomial with matrix coefficients rather than a matrix -with polynomial entries, we can just do the conversion. - -\spadcommand{m :: POLY SquareMatrix(2,COMPLEX FRAC INT)} -$$ -{{\left[ -\begin{array}{cc} -0 & 1 \\ -0 & 0 -\end{array} -\right]} -\ {y \sp 2} \ z}+{{\left[ -\begin{array}{cc} -0 & 0 \\ -{{3 \over 7} \ i} & 0 -\end{array} -\right]} -\ {y \sp 4}}+{{\left[ -\begin{array}{cc} -1 & 0 \\ --1 & 0 -\end{array} -\right]} -\ x}+{\left[ -\begin{array}{cc} --{{3 \over 4} \ i} & {1 \over 2} \\ -0 & {{12} -{{9 \over 5} \ i}} -\end{array} -\right]} -$$ -\returnType{Type: Polynomial SquareMatrix(2,Complex Fraction Integer)} - -We have not yet used modes for any conversions. Modes are a great -shorthand for indicating the type of the object you want. Instead of -using the long type expression in the last example, we could have -simply said this. - -\spadcommand{m :: POLY ?} -$$ -{{\left[ -\begin{array}{cc} -0 & 1 \\ -0 & 0 -\end{array} -\right]} -\ {y \sp 2} \ z}+{{\left[ -\begin{array}{cc} -0 & 0 \\ -{{3 \over 7} \ i} & 0 -\end{array} -\right]} -\ {y \sp 4}}+{{\left[ -\begin{array}{cc} -1 & 0 \\ --1 & 0 -\end{array} -\right]} -\ x}+{\left[ -\begin{array}{cc} --{{3 \over 4} \ i} & {1 \over 2} \\ -0 & {{12} -{{9 \over 5} \ i}} -\end{array} -\right]} -$$ -\returnType{Type: Polynomial SquareMatrix(2,Complex Fraction Integer)} - -We can also indicate more structure if we want the entries of the -matrices to be fractions. - -\spadcommand{m :: POLY SquareMatrix(2,FRAC ?)} -$$ -{{\left[ -\begin{array}{cc} -0 & 1 \\ -0 & 0 -\end{array} -\right]} -\ {y \sp 2} \ z}+{{\left[ -\begin{array}{cc} -0 & 0 \\ -{{3 \ i} \over 7} & 0 -\end{array} -\right]} -\ {y \sp 4}}+{{\left[ -\begin{array}{cc} -1 & 0 \\ --1 & 0 -\end{array} -\right]} -\ x}+{\left[ -\begin{array}{cc} --{{3 \ i} \over 4} & {1 \over 2} \\ -0 & {{{60} -{9 \ i}} \over 5} -\end{array} -\right]} -$$ -\returnType{Type: Polynomial SquareMatrix(2,Fraction Complex Integer)} - -\section{Subdomains Again} -\label{ugTypesSubdomains} - -A {\it subdomain} {\rm S} of a domain {\rm D} is a domain consisting of -\begin{enumerate} -\item those elements of {\rm D} that satisfy some -{\it predicate} (that is, a test that returns {\tt true} or {\tt false}) and -\item a subset of the operations of {\rm D}. -\end{enumerate} -Every domain is a subdomain of itself, trivially satisfying the -membership test: {\tt true}. - -Currently, there are only two system-defined subdomains in Axiom that -receive substantial use. {\tt PositiveInteger} and -{\tt NonNegativeInteger} are subdomains of {\tt Integer}. An element $x$ -of {\tt NonNegativeInteger} is an integer that is greater than or -equal to zero, that is, satisfies $x >= 0$. An element $x$ of -{\tt PositiveInteger} is a nonnegative integer that is, in fact, greater -than zero, that is, satisfies $x > 0$. Not all operations from -{\tt Integer} are available for these subdomains. For example, negation -and subtraction are not provided since the subdomains are not closed -under those operations. When you use an integer in an expression, -Axiom assigns to it the type that is the most specific subdomain whose -predicate is satisfied. - -This is a positive integer. -\spadcommand{5} -$$ -5 -$$ -\returnType{Type: PositiveInteger} - -This is a nonnegative integer. -\spadcommand{0} -$$ -0 -$$ -\returnType{Type: NonNegativeInteger} - -This is neither of the above. -\spadcommand{-5} -$$ --5 -$$ -\returnType{Type: Integer} - -Furthermore, unless you are assigning an integer to a declared variable -or using a conversion, any integer result has as type the most -specific subdomain. -\spadcommand{(-2) - (-3)} -$$ -1 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{0 :: Integer} -$$ -0 -$$ -\returnType{Type: Integer} - -\spadcommand{x : NonNegativeInteger := 5} -$$ -5 -$$ -\returnType{Type: NonNegativeInteger} - -When necessary, Axiom converts an integer object into one belonging to -a less specific subdomain. For example, in $3-2$, the arguments to -\spadopFrom{-}{Integer} are both elements of {\tt PositiveInteger}, -but this type does not provide a subtraction operation. Neither does -{\tt NonNegativeInteger}, so $3$ and $2$ are viewed as elements of -{\tt Integer}, where their difference can be calculated. The result -is $1$, which Axiom then automatically assigns the type -{\tt PositiveInteger}. - -Certain operations are very sensitive to the subdomains to which their -arguments belong. This is an element of {\tt PositiveInteger}. -\spadcommand{2 ** 2} -$$ -4 -$$ -\returnType{Type: PositiveInteger} - -This is an element of {\tt Fraction Integer}. -\spadcommand{2 ** (-2)} -$$ -1 \over 4 -$$ -\returnType{Type: Fraction Integer} - -It makes sense then that this is a list of elements of {\tt -PositiveInteger}. -\spadcommand{[10**i for i in 2..5]} -$$ -\left[ -{100}, {1000}, {10000}, {100000} -\right] -$$ -\returnType{Type: List PositiveInteger} - -What should the type of {\tt [10**(i-1) for i in 2..5]} be? On one hand, -$i-1$ is always an integer greater than zero as $i$ ranges from $2$ to -$5$ and so $10**i$ is also always a positive integer. On the other, -$i-1$ is a very simple function of $i$. Axiom does not try to analyze -every such function over the index's range of values to determine -whether it is always positive or nowhere negative. For an arbitrary -Axiom function, this analysis is not possible. - -So, to be consistent no such analysis is done and we get this. -\spadcommand{[10**(i-1) for i in 2..5]} -$$ -\left[ -{10}, {100}, {1000}, {10000} -\right] -$$ -\returnType{Type: List Fraction Integer} - -To get a list of elements of {\tt PositiveInteger} instead, you have -two choices. You can use a conversion. - -\spadcommand{[10**((i-1) :: PI) for i in 2..5]} -\begin{verbatim} -Compiling function G82696 with type Integer -> Boolean -Compiling function G82708 with type NonNegativeInteger -> Boolean -\end{verbatim} -$$ -\left[ -{10}, {100}, {1000}, {10000} -\right] -$$ -\returnType{Type: List PositiveInteger} - -Or you can use {\tt pretend}. \index{pretend} -\spadcommand{[10**((i-1) pretend PI) for i in 2..5]} -$$ -\left[ -{10}, {100}, {1000}, {10000} -\right] -$$ -\returnType{Type: List PositiveInteger} - -The operation {\tt pretend} is used to defeat the Axiom type system. -The expression {\tt object pretend D} means ``make a new object -(without copying) of type {\tt D} from {\tt object}.'' If -{\tt object} were an integer and you told Axiom to pretend it was a list, -you would probably see a message about a fatal error being caught and -memory possibly being damaged. Lists do not have the same internal -representation as integers! - -You use {\tt pretend} at your peril. \index{peril} - -Use $pretend$ with great care! Axiom trusts you that the value is of -the specified type. - -\spadcommand{(2/3) pretend Complex Integer} -$$ -2+{3 \ i} -$$ -\returnType{Type: Complex Integer} - -\section{Package Calling and Target Types} -\label{ugTypesPkgCall} - -Axiom works hard to figure out what you mean by an expression without -your having to qualify it with type information. Nevertheless, there -are times when you need to help it along by providing hints (or even -orders!) to get Axiom to do what you want. - -We saw in \ref{ugTypesDeclare} on page~\pageref{ugTypesDeclare} that -declarations using types and modes control the type of the results -produced. For example, we can either produce a complex object with -polynomial real and imaginary parts or a polynomial with complex -integer coefficients, depending on the declaration. - -Package calling is how you tell Axiom to use a particular function -from a particular part of the library. - -Use the \spadopFrom{/}{Fraction} from {\tt Fraction Integer} to create -a fraction of two integers. -\spadcommand{2/3} -$$ -2 \over 3 -$$ -\returnType{Type: Fraction Integer} - -If we wanted a floating point number, we can say ``use the -\spadopFrom{/}{Float} in {\tt Float}.'' -\spadcommand{(2/3)\$Float} -$$ -0.6666666666 6666666667 -$$ -\returnType{Type: Float} - -Perhaps we actually wanted a fraction of complex integers. -\spadcommand{(2/3)\$Fraction(Complex Integer)} -$$ -2 \over 3 -$$ -\returnType{Type: Float} - -In each case, AXIOM used the indicated operations, sometimes first -needing to convert the two integers into objects of the appropriate type. -In these examples, ``/'' is written as an infix operator. - -\boxed{4.6in}{ -\vskip 0.1cm -To use package calling with an infix operator, use the following syntax: -$$(\ arg_1{\rm \ op\ }arg_2\ )\$type$$ -} - -We used, for example, $(2/3)\${\rm Float}$. The expression $2+3+4$ -is equivalent to $(2+3)+4$. Therefore in the expression -$(2+3+4)\${\rm Float}$ the second ``+'' comes from the {\rm Float} -domain. The first ``+'' comes from {\rm Float} because the package -call causes AXIOM to convert $(2+3)$ and $4$ to type -{\rm Float}. Before the sum is converted, it is given a target type -of {\rm Float} by AXIOM and then evaluated. The target type causes the -``+'' from {\tt Float} to be used. - -\boxed{4.6in}{ -\vskip 0.1cm -For an operator written before its arguments, you must use parentheses -around the arguments (even if there is only one), and follow the closing -parenthesis by a ``\$'' and then the type. -$$ fun\ (\ arg_1, arg_2, \ldots, arg_N\ )\$type$$ -} - -For example, to call the ``minimum'' function from {\rm SmallFloat} on two -integers, you could write {\bf min}(4,89){\tt SmallFloat}. Another use of -package calling is to tell AXIOM to use a library function rather than a -function you defined. We discuss this in -Section \ref{ugUserUse} on page~\pageref{ugUserUse}. - -Sometimes rather than specifying where an operation comes from, you -just want to say what type the result should be. We say that you provide a -{\sl target type} for the expression. Instead of using a ``\$'', use a ``@'' -to specify the requested target type. Otherwise, the syntax is the same. -Note that giving a target type is not the same as explicitly doing a -conversion. The first says ``try to pick operations so that the result has -such-and-such a type.'' The second says ``compute the result and then convert -to an object of such-and-such a type.'' - -Sometimes it makes sense, as in this expression, to say ``choose the -operations in this expression so that the final result is {\rm Float}. -\spadcommand{(2/3)@Float} -$$ -0.6666666666 6666666667 -$$ -\returnType{Type: Float} - -Here we used ``{\tt @}'' to say that the target type of the left-hand side -was {\tt Float}. In this simple case, there was no real difference -between using ``{\tt \$}'' and ``{\tt @}''. -You can see the difference if you try the following. - -This says to try to choose ``{\tt +}'' so that the result is a string. -Axiom cannot do this. -\spadcommand{(2 + 3)@String} -\begin{verbatim} -An expression involving @ String actually evaluated to one of - type PositiveInteger . Perhaps you should use :: String . -\end{verbatim} - -This says to get the {\tt +} from {\tt String} and apply it to the two -integers. Axiom also cannot do this because there is no {\tt +} -exported by {\tt String}. -\spadcommand{(2 + 3)\$String} -\begin{verbatim} - The function + is not implemented in String . -\end{verbatim} - -(By the way, the operation \spadfunFrom{concat}{String} or juxtaposition -is used to concatenate two strings.) -\index{String} - -When we have more than one operation in an expression, the difference -is even more evident. The following two expressions show that Axiom -uses the target type to create different objects. -The ``{\tt +}'', ``{\tt *}'' and ``{\tt **}'' operations are all -chosen so that an object of the correct final type is created. - -This says that the operations should be chosen so that the result is a -{\tt Complex} object. -\spadcommand{((x + y * \%i)**2)@(Complex Polynomial Integer)} -$$ --{y \sp 2}+{x \sp 2}+{2 \ x \ y \ i} -$$ -\returnType{Type: Complex Polynomial Integer} - -This says that the operations should be chosen so that the result is a -{\tt Polynomial} object. -\spadcommand{((x + y * \%i)**2)@(Polynomial Complex Integer)} -$$ --{y \sp 2}+{2 \ i \ x \ y}+{x \sp 2} -$$ -\returnType{Type: Polynomial Complex Integer} - -What do you think might happen if we left off all target type and -package call information in this last example? -\spadcommand{(x + y * \%i)**2} -$$ --{y \sp 2}+{2 \ i \ x \ y}+{x \sp 2} -$$ -\returnType{Type: Polynomial Complex Integer} - -We can convert it to {\tt Complex} as an afterthought. But this is -more work than just saying making what we want in the first place. -\spadcommand{\% :: Complex ?} -$$ --{y \sp 2}+{x \sp 2}+{2 \ x \ y \ i} -$$ -\returnType{Type: Complex Polynomial Integer} - -Finally, another use of package calling is to qualify fully an -operation that is passed as an argument to a function. - -Start with a small matrix of integers. -\spadcommand{h := matrix [ [8,6],[-4,9] ]} -$$ -\left[ -\begin{array}{cc} -8 & 6 \\ --4 & 9 -\end{array} -\right] -$$ -\returnType{Type: Matrix Integer} - -We want to produce a new matrix that has for entries the -multiplicative inverses of the entries of $h$. One way to do this is -by calling \spadfunFrom{map}{MatrixCategoryFunctions2} with the -\spadfunFrom{inv}{Fraction} function from {\tt Fraction (Integer)}. - -\spadcommand{map(inv\$Fraction(Integer),h)} -$$ -\left[ -\begin{array}{cc} -{1 \over 8} & {1 \over 6} \\ --{1 \over 4} & {1 \over 9} -\end{array} -\right] -$$ -\returnType{Type: Matrix Fraction Integer} - -We could have been a bit less verbose and used abbreviations. -\spadcommand{map(inv\$FRAC(INT),h)} -$$ -\left[ -\begin{array}{cc} -{1 \over 8} & {1 \over 6} \\ --{1 \over 4} & {1 \over 9} -\end{array} -\right] -$$ -\returnType{Type: Matrix Fraction Integer} - -As it turns out, Axiom is smart enough to know what we mean anyway. -We can just say this. -\spadcommand{map(inv,h)} -$$ -\left[ -\begin{array}{cc} -{1 \over 8} & {1 \over 6} \\ --{1 \over 4} & {1 \over 9} -\end{array} -\right] -$$ -\returnType{Type: Matrix Fraction Integer} - -\section{Resolving Types} -\label{ugTypesResolve} - -In this section we briefly describe an internal process by which -\index{resolve} Axiom determines a type to which two objects of -possibly different types can be converted. We do this to give you -further insight into how Axiom takes your input, analyzes it, and -produces a result. - -What happens when you enter $x + 1$ to Axiom? Let's look at what you -get from the two terms of this expression. - -This is a symbolic object whose type indicates the name. -\spadcommand{x} -$$ -x -$$ -\returnType{Type: Variable x} - -This is a positive integer. -\spadcommand{1} -$$ -1 -$$ -\returnType{Type: PositiveInteger} - -There are no operations in {\tt PositiveInteger} that add positive -integers to objects of type {\tt Variable(x)} nor are there any in -{\tt Variable(x)}. Before it can add the two parts, Axiom must come -up with a common type to which both $x$ and $1$ can be converted. We -say that Axiom must {\it resolve} the two types into a common type. -In this example, the common type is {\tt Polynomial(Integer)}. - -Once this is determined, both parts are converted into polynomials, -and the addition operation from {\tt Polynomial(Integer)} is used to -get the answer. -\spadcommand{x + 1} -$$ -x+1 -$$ -\returnType{Type: Polynomial Integer} - -Axiom can always resolve two types: if nothing resembling the original -types can be found, then {\tt Any} is be used. \index{Any} This is -fine and useful in some cases. - -\spadcommand{["string",3.14159]} -$$ -\left[ -\mbox{\tt "string"} , {3.14159} -\right] -$$ -\returnType{Type: List Any} - -In other cases objects of type {\tt Any} can't be used by the -operations you specified. -\spadcommand{"string" + 3.14159} -\begin{verbatim} -There are 11 exposed and 5 unexposed library operations named + - having 2 argument(s) but none was determined to be applicable. - Use HyperDoc Browse, or issue - )display op + - to learn more about the available operations. Perhaps - package-calling the operation or using coercions on the - arguments will allow you to apply the operation. - -Cannot find a definition or applicable library operation named + - with argument type(s) - String - Float - - Perhaps you should use "@" to indicate the required return type, - or "$" to specify which version of the function you need. -\end{verbatim} - -Although this example was contrived, your expressions may need to be -qualified slightly to help Axiom resolve the types involved. You may -need to declare a few variables, do some package calling, provide some -target type information or do some explicit conversions. - -We suggest that you just enter the expression you want evaluated and -see what Axiom does. We think you will be impressed with its ability -to ``do what I mean.'' If Axiom is still being obtuse, give it some -hints. As you work with Axiom, you will learn where it needs a little -help to analyze quickly and perform your computations. - -\section{Exposing Domains and Packages} -\label{ugTypesExpose} - -In this section we discuss how Axiom makes some operations available -to you while hiding others that are meant to be used by developers or -only in rare cases. If you are a new user of Axiom, it is likely that -everything you need is available by default and you may want to skip -over this section on first reading. - -Every \index{constructor!exposed} domain and package in the Axiom -library \index{constructor!hidden} is \index{exposed!constructor} -either exposed (meaning that you can use its operations without doing -anything special) or it is {\it hidden} (meaning you have to either -package call (see \ref{ugTypesPkgCall} on -page~\pageref{ugTypesPkgCall}) the operations it contains or -explicitly expose it to use the operations). The initial exposure -status for a constructor is set in the file {\bf exposed.lsp} (see the -{\it Installer's Note} \index{exposed.lsp @{\bf exposed.lsp}} for -Axiom \index{file!exposed.lsp @{\bf exposed.lsp}} if you need to know -the location of this file). Constructors are collected together in -\index{group!exposure} {\it exposure groups}. \index{exposure!group} -Categories are all in the exposure group ``categories'' and the bulk -of the basic set of packages and domains that are exposed are in the -exposure group ``basic.'' Here is an abbreviated sample of the file -(without the Lisp parentheses): - -\begin{verbatim} -basic - AlgebraicNumber AN - AlgebraGivenByStructuralConstants ALGSC - Any ANY - AnyFunctions1 ANY1 - BinaryExpansion BINARY - Boolean BOOLEAN - CardinalNumber CARD - CartesianTensor CARTEN - Character CHAR - CharacterClass CCLASS - CliffordAlgebra CLIF - Color COLOR - Complex COMPLEX - ContinuedFraction CONTFRAC - DecimalExpansion DECIMAL - ... -\end{verbatim} -\begin{verbatim} -categories - AbelianGroup ABELGRP - AbelianMonoid ABELMON - AbelianMonoidRing AMR - AbelianSemiGroup ABELSG - Aggregate AGG - Algebra ALGEBRA - AlgebraicallyClosedField ACF - AlgebraicallyClosedFunctionSpace ACFS - ArcHyperbolicFunctionCategory AHYP - ... -\end{verbatim} - -For each constructor in a group, the full name and the abbreviation is -given. There are other groups in {\bf exposed.lsp} but initially only -the constructors in exposure groups ``basic'' ``categories'' -``naglink'' and ``anna'' are exposed. - -As an interactive user of Axiom, you do not need to modify this file. -Instead, use {\tt )set expose} to expose, hide or query the exposure -status of an individual constructor or exposure group. \index{set expose} -The reason for having exposure groups is to be able to expose -or hide multiple constructors with a single command. For example, you -might group together into exposure group ``quantum'' a number of -domains and packages useful for quantum mechanical computations. -These probably should not be available to every user, but you want an -easy way to make the whole collection visible to Axiom when it is -looking for operations to apply. - -If you wanted to hide all the basic constructors available by default, -you would issue {\tt )set expose drop group basic}. -\index{set expose drop group} We do not recommend that you do this. -If, however, you discover that you have hidden all the basic constructors, -you should issue {\tt )set expose add group basic} to restore your default -environment. \index{set expose add group} - -It is more likely that you would want to expose or hide individual -constructors. In \ref{ugUserTriangle} on -page~\pageref{ugUserTriangle} we use several operations from -{\tt OutputForm}, a domain usually hidden. To avoid package calling every -operation from {\tt OutputForm}, we expose the domain and let Axiom -conclude that those operations should be used. Use {\tt )set expose -add constructor} and {\tt )set expose drop constructor} to expose and -hide a constructor, respectively. \index{set expose drop constructor} -You should use the constructor name, not the abbreviation. The -{\tt )set expose} command guides you through these options. -\index{set expose add constructor} - -If you expose a previously hidden constructor, Axiom exhibits new -behavior (that was your intention) though you might not expect the -results that you get. {\tt OutputForm} is, in fact, one of the worst -offenders in this regard. \index{OutputForm} This domain is meant to -be used by other domains for creating a structure that Axiom knows how -to display. It has functions like \spadopFrom{+}{OutputForm} that -form output representations rather than do mathematical calculations. -Because of the order in which Axiom looks at constructors when it is -deciding what operation to apply, {\tt OutputForm} might be used -instead of what you expect. - -This is a polynomial. -\spadcommand{x + x} -$$ -2 \ x -$$ -\returnType{Type: Polynomial Integer} - -Expose {\tt OutputForm}. -\spadcommand{)set expose add constructor OutputForm } -\begin{verbatim} - OutputForm is now explicitly exposed in frame G82322 -\end{verbatim} - -This is what we get when {\tt OutputForm} is automatically available. -\spadcommand{x + x} -$$ -x+x -$$ -\returnType{Type: OutputForm} - -Hide {\tt OutputForm} so we don't run into problems with any later examples! -\spadcommand{)set expose drop constructor OutputForm } -\begin{verbatim} - OutputForm is now explicitly hidden in frame G82322 -\end{verbatim} - -Finally, exposure is done on a frame-by-frame basis. A {\it frame} -(see \ref{ugSysCmdframe} on page~\pageref{ugSysCmdframe}) -\index{frame!exposure and} is one of possibly several logical Axiom -workspaces within a physical one, each having its own environment (for -example, variables and function definitions). If you have several -Axiom workspace windows on your screen, they are all different frames, -automatically created for you by HyperDoc. Frames can be manually -created, made active and destroyed by the {\tt )frame} system command. -\index{frame} They do not share exposure information, so you need to -use {\tt )set expose} in each one to add or drop constructors from -view. - -\section{Commands for Snooping} -\label{ugAvailSnoop} - -To conclude this chapter, we introduce you to some system commands -that you can use for getting more information about domains, packages, -categories, and operations. The most powerful Axiom facility for -getting information about constructors and operations is the Browse -component of HyperDoc. This is discussed in Chapter -\ref{ugBrowse} on page~\pageref{ugBrowse}. - -Use the {\tt )what} system command to see lists of system objects -whose name contain a particular substring (uppercase or lowercase is -not significant). \index{what} - -Issue this to see a list of all operations with ``{\tt complex}'' in -their names. \index{what operation} -\spadcommand{)what operation complex} -\begin{verbatim} - -Operations whose names satisfy the above pattern(s): - -complex complex? -complexEigenvalues complexEigenvectors -complexElementary complexExpand -complexForm complexIntegrate -complexLimit complexNormalize -complexNumeric complexNumericIfCan -complexRoots complexSolve -complexZeros createLowComplexityNormalBasis -createLowComplexityTable doubleComplex? -drawComplex drawComplexVectorField -fortranComplex fortranDoubleComplex -pmComplexintegrate - -To get more information about an operation such as -complexZeros, issue the command )display op complexZeros -\end{verbatim} - -If you want to see all domains with ``{\tt matrix}'' in their names, -issue this. \index{what domain} -\spadcommand{)what domain matrix} -\begin{verbatim} ------------------------ Domains ----------------------- - -Domains with names matching patterns: - matrix - - DHMATRIX DenavitHartenbergMatrix - DPMM DirectProductMatrixModule - IMATRIX IndexedMatrix - LSQM LieSquareMatrix - M3D ThreeDimensionalMatrix - MATCAT- MatrixCategory& - MATRIX Matrix - RMATCAT- RectangularMatrixCategory& - RMATRIX RectangularMatrix - SMATCAT- SquareMatrixCategory& - SQMATRIX SquareMatrix -\end{verbatim} - -Similarly, if you wish to see all packages whose names contain ``{\tt -gauss}'', enter this. \index{what packages} -\spadcommand{)what package gauss} -\begin{verbatim} ----------------------- Packages ----------------------- - -Packages with names matching patterns: - gauss - - GAUSSFAC GaussianFactorizationPackage -\end{verbatim} - -This command shows all the operations that {\tt Any} provides. -Wherever {\tt \$} appears, it means ``{\tt Any}''. \index{show} -\spadcommand{)show Any} -\begin{verbatim} - Any is a domain constructor - Abbreviation for Any is ANY - This constructor is exposed in this frame. - Issue )edit /usr/local/axiom/mnt/algebra/any.spad - to see algebra source code for ANY - ---------------------- Operations ---------------------- - ?=? : (%,%) -> Boolean - any : (SExpression,None) -> % - coerce : % -> OutputForm - dom : % -> SExpression - domainOf : % -> OutputForm - hash : % -> SingleInteger - latex : % -> String - obj : % -> None - objectOf : % -> OutputForm - ?~=? : (%,%) -> Boolean - showTypeInOutput : Boolean -> String - -\end{verbatim} - -This displays all operations with the name {\tt complex}. -\index{display operation} -\spadcommand{)display operation complex} -\begin{verbatim} -There is one exposed function called complex : - [1] (D1,D1) -> D from D if D has COMPCAT D1 and D1 has COMRING -\end{verbatim} - -Let's analyze this output. - -First we find out what some of the abbreviations mean. -\spadcommand{)abbreviation query COMPCAT} -\begin{verbatim} - COMPCAT abbreviates category ComplexCategory -\end{verbatim} - -\spadcommand{)abbreviation query COMRING} -\begin{verbatim} - COMRING abbreviates category CommutativeRing -\end{verbatim} - -So if {\tt D1} is a commutative ring (such as the integers or floats) and -{\tt D} belongs to {\tt ComplexCategory D1}, then there is an operation -called {\bf complex} that takes two elements of {\tt D1} and creates an -element of {\tt D}. The primary example of a constructor implementing -domains belonging to {\tt ComplexCategory} is {\tt Complex}. See -\ref{Complex} on page~\pageref{Complex} for more information on that and see -\ref{ugUserDeclare} on page~\pageref{ugUserDeclare} -for more information on function types. - -\setcounter{chapter}{2} - -\chapter{Using HyperDoc} -\label{ugHyper} - -\begin{figure}[htbp] -\begin{picture}(324,260)%(-54,0) -\special{psfile=ps/h-root.ps} -\end{picture} -\caption{The HyperDoc root window page.} -\end{figure} - -HyperDoc is the gateway to Axiom. \index{HyperDoc} It's both an -on-line tutorial and an on-line reference manual. It also enables you -to use Axiom simply by using the mouse and filling in templates. -HyperDoc is available to you if you are running Axiom under the X -Window System. - -Pages usually have active areas, marked in {\bf this font} (bold -face). As you move the mouse pointer to an active area, the pointer -changes from a filled dot to an open circle. The active areas are -usually linked to other pages. When you click on an active area, you -move to the linked page. - -\section{Headings} -\label{ugHyperHeadings} -Most pages have a standard set of buttons at the top of the page. -This is what they mean: - -\begin{description} - -\item[\HelpBitmap] Click on this to get help. The button only appears -if there is specific help for the page you are viewing. You can get -{\it general} help for HyperDoc by clicking the help button on the -home page. - -\item[\UpBitmap] Click here to go back one page. -By clicking on this button repeatedly, you can go back several pages and -then take off in a new direction. - -\item[\ReturnBitmap] Go back to the home page, that is, the page on -which you started. Use HyperDoc to explore, to make forays into new -topics. Don't worry about how to get back. HyperDoc remembers where -you came from. Just click on this button to return. - -\item[\ExitBitmap] From the root window (the one that is displayed -when you start the system) this button leaves the HyperDoc program, -and it must be restarted if you want to use it again. From any other -HyperDoc window, it just makes that one window go away. You {\it must} -use this button to get rid of a window. If you use the window -manager ``Close'' button, then all of HyperDoc goes away. - -\end{description} - -The buttons are not displayed if they are not applicable to the page -you are viewing. For example, there is no \ReturnBitmap button on the -top-level menu. - -\section{Key Definitions} -\label{ugHyperKeys} - -The following keyboard definitions are in effect throughout HyperDoc. -See \ref{ugHyperScroll} on page~\pageref{ugHyperScroll} and -\ref{ugHyperInput} on page~\pageref{ugHyperInput} for some contextual key -definitions. - -\begin{description} -\item[F1] Display the main help page. -\item[F3] Same as \ExitBitmap{}, makes the window go away if you are not at the top-level window or quits the HyperDoc facility if you are at the top-level. -\item[F5] Rereads the HyperDoc database, if necessary (for system developers). -\item[F9] Displays this information about key definitions. -\item[F12] Same as {\bf F3}. -\item[Up Arrow] Scroll up one line. -\item[Down Arrow] Scroll down one line. -\item[Page Up] Scroll up one page. -\item[Page Down] Scroll down one page. -\end{description} - -\section{Scroll Bars} -\label{ugHyperScroll} - -Whenever there is too much text to fit on a page, a -{\it scroll \index{scroll bar} bar} -automatically appears along the right side. - -With a scroll bar, your page becomes an aperture, that is, a window -into a larger amount of text than can be displayed at one time. The -scroll bar lets you move up and down in the text to see different -parts. It also shows where the aperture is relative to the whole -text. The aperture is indicated by a strip on the scroll bar. - -Move the cursor with the mouse to the ``down-arrow'' at the bottom of -the scroll bar and click. See that the aperture moves down one line. -Do it several times. Each time you click, the aperture moves down one -line. Move the mouse to the ``up-arrow'' at the top of the scroll bar -and click. The aperture moves up one line each time you click. - -Next move the mouse to any position along the middle of the scroll bar -and click. HyperDoc attempts to move the top of the aperture to this -point in the text. - -You cannot make the aperture go off the bottom edge. When the -aperture is about half the size of text, the lowest you can move the -aperture is halfway down. - -To move up or down one screen at a time, use the \fbox{\bf PageUp} and -\fbox{\bf PageDown} keys on your keyboard. They move the visible part of the -region up and down one page each time you press them. - -If the HyperDoc page does not contain an input area (see -\ref{ugHyperInput} on page~\pageref{ugHyperInput}), you can also use -the \fbox{\bf Home} and \fbox{$\uparrow$} and \fbox{$\downarrow$} -arrow keys to navigate. When you press the \fbox{\bf Home} key, the -screen is positioned at the very top of the page. Use the -\fbox{$\uparrow$} and \fbox{$\downarrow$} arrow keys to move the -screen up and down one line at a time, respectively. - -\section{Input Areas} -\label{ugHyperInput} - -Input areas are boxes where you can put data. - -To enter characters, first move your mouse cursor to somewhere within -the HyperDoc page. Characters that you type are inserted in front of -the underscore. This means that when you type characters at your -keyboard, they go into this first input area. - -The input area grows to accommodate as many characters as you type. -Use the \fbox{\bf Backspace} key to erase characters to the left. To -modify what you type, use the right-arrow \fbox{$\rightarrow$} and -left-arrow keys \fbox{$\leftarrow$} and the keys \fbox{\bf Insert}, -\fbox{\bf Delete}, \fbox{\bf Home} and \fbox{\bf End}. These keys are -found immediately on the right of the standard IBM keyboard. - -If you press the \fbox{\bf Home} key, the cursor moves to the -beginning of the line and if you press the \fbox{\bf End} key, the -cursor moves to the end of the line. Pressing -\fbox{\bf Ctrl}--\fbox{\bf End} deletes all the text from the -cursor to the end of the line. - -A page may have more than one input area. Only one input area has an -underscore cursor. When you first see apage, the top-most input area -contains the cursor. To type information into another input area, use -the \fbox{\bf Enter} or \fbox{\bf Tab} key to move from one input area to -xanother. To move in the reverse order, use \fbox{\bf Shift}--\fbox{\bf Tab}. - -You can also move from one input area to another using your mouse. -Notice that each input area is active. Click on one of the areas. -As you can see, the underscore cursor moves to that window. - -\section{Radio Buttons and Toggles} -\label{ugHyperButtons} - -Some pages have {\it radio buttons} and {\it toggles}. -Radio buttons are a group of buttons like those on car radios: you can -select only one at a time. - -Once you have selected a button, it appears to be inverted and -contains a checkmark. To change the selection, move the cursor with -the mouse to a different radio button and click. - -A toggle is an independent button that displays some on/off state. -When ``on'', the button appears to be inverted and contains a -checkmark. When ``off'', the button is raised. - -Unlike radio buttons, you can set a group of them any way you like. -To change toggle the selection, move the cursor with the mouse to the -button and click. - -\section{Search Strings} -\label{ugHyperSearch} - -A {\it search string} is used for searching some database. To learn -about search strings, we suggest that you bring up the HyperDoc -glossary. To do this from the top-level page of HyperDoc: -\begin{enumerate} -\item Click on Reference, bringing up the Axiom Reference page. -\item Click on Glossary, bringing up the glossary. -\end{enumerate} - -The glossary has an input area at its bottom. We review the various -kinds of search strings you can enter to search the glossary. - -The simplest search string is a word, for example, {\tt operation}. A -word only matches an entry having exactly that spelling. Enter the -word {\tt operation} into the input area above then click on -{\bf Search}. As you can see, {\tt operation} matches only one entry, -namely with {\tt operation} itself. - -Normally matching is insensitive to whether the alphabetic characters -of your search string are in uppercase or lowercase. Thus -{\tt operation} and {\tt OperAtion} both have the same effect. -%If you prefer that matching be case-sensitive, issue the command -%{\tt set HHyperName mixedCase} command to the interpreter. - -You will very often want to use the wildcard ``{\tt *}'' in your search -string so as to match multiple entries in the list. The search key -``{\tt *}'' matches every entry in the list. You can also use ``{\tt *}'' -anywhere within a search string to match an arbitrary substring. Try -``{\tt cat*}'' for example: enter ``{\tt cat*}'' into the input area and click -on {\bf Search}. This matches several entries. - -You use any number of wildcards in a search string as long as they are -not adjacent. Try search strings such as ``{\tt *dom*}''. As you see, -this search string matches ``{\tt domain}'', ``{\tt domain constructor}'', -``{\tt subdomain}'', and so on. - -\subsection{Logical Searches} -\label{ugLogicalSearches} - -For more complicated searches, you can use ``{\tt and}'', ``{\tt or}'', and -``{\tt not}'' with basic search strings; write logical expressions using -these three operators just as in the Axiom language. For example, -{\tt domain or package} matches the two entries {\tt domain} and -{\tt package}. Similarly, ``{\tt dom* and *con*}'' matches -``{\tt domain constructor}'' and others. Also ``{\tt not *a*}'' matches -every entry that does not contain the letter ``{\tt a}'' somewhere. - -Use parentheses for grouping. For example, ``{\tt dom* and (not *con*)}'' -matches ``{\tt domain}'' but not ``{\tt domain constructor}''. - -There is no limit to how complex your logical expression can be. -For example, -\begin{center} -{\tt a* or b* or c* or d* or e* and (not *a*)} -\end{center} -is a valid expression. - -\section{Example Pages} -\label{ugHyperExample} - -Many pages have Axiom example commands. - -Each command has an active ``button'' along the left margin. When you -click on this button, the output for the command is ``pasted-in.'' -Click again on the button and you see that the pasted-in output -disappears. - -Maybe you would like to run an example? To do so, just click on any -part of its text! When you do, the example line is copied into a new -interactive Axiom buffer for this HyperDoc page. - -Sometimes one example line cannot be run before you run an earlier one. -Don't worry---HyperDoc automatically runs all the necessary -lines in the right order! - -The new interactive Axiom buffer disappears when you leave HyperDoc. -If you want to get rid of it beforehand, use the {\bf Cancel} button -of the X Window manager or issue the Axiom system command -{\tt )close.} \index{close} - -\section{X Window Resources for HyperDoc} -\label{ugHyperResources} - -You can control the appearance of HyperDoc while running under Version -11 \index{HyperDoc X Window System defaults} of the X Window System by -placing the following resources \index{X Window System} in the file -{\bf .Xdefaults} in your home directory. \index{file!.Xdefaults} -In what follows, {\it font} is any valid X11 font name -\index{font} (for example, {\tt Rom14}) and {\it color} is any valid -X11 color \index{color} specification (for example, {\tt NavyBlue}). -For more information about fonts and colors, refer to the X Window -documentation for your system. - -\begin{description} -\item[{\tt Axiom.hyperdoc.RmFont:} {\it font}] \ \newline -This is the standard text font. -The default value is {\tt Rom14} -\item[{\tt Axiom.hyperdoc.RmColor:} {\it color}] \ \newline -This is the standard text color. -The default value is {\tt black} -\item[{\tt Axiom.hyperdoc.ActiveFont:} {\it font}] \ \newline -This is the font used for HyperDoc link buttons. -The default value is {\tt Bld14} -\item[{\tt Axiom.hyperdoc.ActiveColor:} {\it color}] \ \newline -This is the color used for HyperDoc link buttons. -The default value is {\tt black} -\item[{\tt Axiom.hyperdoc.AxiomFont:} {\it font}] \ \newline -This is the font used for active Axiom commands. -The default value is {\tt Bld14} -\item[{\tt Axiom.hyperdoc.AxiomColor:} {\it color}] \ \newline -This is the color used for active Axiom commands. -The default value is {\tt black} -\item[{\tt Axiom.hyperdoc.BoldFont:} {\it font}] \ \newline -This is the font used for bold face. -The default value is {\tt Bld14} -\item[{\tt Axiom.hyperdoc.BoldColor:} {\it color}] \ \newline -This is the color used for bold face. -The default value is {\tt black} -\item[{\tt Axiom.hyperdoc.TtFont:} {\it font}] \ \newline -This is the font used for Axiom output in HyperDoc. -This font must be fixed-width. -The default value is {\tt Rom14} -\item[{\tt Axiom.hyperdoc.TtColor:} {\it color}] \ \newline -This is the color used for Axiom output in HyperDoc. -The default value is {\tt black} -\item[{\tt Axiom.hyperdoc.EmphasizeFont:} {\it font}] \ \newline -This is the font used for italics. -The default value is {\tt Itl14} -\item[{\tt Axiom.hyperdoc.EmphasizeColor:} {\it color}] \ \newline -This is the color used for italics. -The default value is {\tt black} -\item[{\tt Axiom.hyperdoc.InputBackground:} {\it color}] \ \newline -This is the color used as the background for input areas. -The default value is {\tt black} -\item[{\tt Axiom.hyperdoc.InputForeground:} {\it color}] \ \newline -This is the color used as the foreground for input areas. -The default value is {\tt white} -\item[{\tt Axiom.hyperdoc.BorderColor:} {\it color}] \ \newline -This is the color used for drawing border lines. -The default value is {\tt black} -\item[{\tt Axiom.hyperdoc.Background:} {\it color}] \ \newline -This is the color used for the background of all windows. -The default value is {\tt white} -\end{description} -\vfill -\eject - -\setcounter{chapter}{3} - -\chapter{Input Files and Output Styles} -\label{ugInOut} - -In this chapter we discuss how to collect Axiom statements -and commands into files and then read the contents into the -workspace. -We also show how to display the results of your computations in -several different styles including \TeX{}, FORTRAN and -monospace two-dimensional format.\footnote{\TeX{} is a -trademark of the American Mathematical Society.} - -The printed version of this book uses the Axiom \TeX{} output formatter. -When we demonstrate a particular output style, we will need to turn -\TeX{} formatting off and the output style on so that the correct output -is shown in the text. - -\section{Input Files} -\label{ugInOutIn} - -In this section we explain what an {\it input file} is and -\index{file!input} why you would want to know about it. We discuss -where Axiom looks for input files and how you can direct it to look -elsewhere. We also show how to read the contents of an input file -into the {\it workspace} and how to use the {\it history} facility to -generate an input file from the statements you have entered directly -into the workspace. - -An {\it input} file contains Axiom expressions and system commands. -Anything that you can enter directly to Axiom can be put into an input -file. This is how you save input functions and expressions that you -wish to read into Axiom more than one time. - -To read an input file into Axiom, use the {\tt )read} system command. -\index{read} For example, you can read a file in a particular -directory by issuing -\begin{verbatim} -)read /spad/src/input/matrix.input -\end{verbatim} - -The ``{\bf .input}'' is optional; this also works: -\begin{verbatim} -)read /spad/src/input/matrix -\end{verbatim} - -What happens if you just enter {\tt )read matrix.input} or even {\tt -)read matrix}? Axiom looks in your current working directory for -input files that are not qualified by a directory name. Typically, -this directory is the directory from which you invoked Axiom. - -To change the current working directory, use the {\tt )cd} system -command. The command {\tt {)cd}} by itself shows the current working -\index{directory!default for searching} directory. \index{cd} To -change it to \index{file!input!where found} the {\tt {src/input}} -subdirectory for user ``babar'', issue -\begin{verbatim} -)cd /u/babar/src/input -\end{verbatim} -Axiom looks first in this directory for an input file. If it is not -found, it looks in the system's directories, assuming you meant some -input file that was provided with Axiom. - -\boxed{4.6in}{ -\vskip 0.1cm -If you have the Axiom history facility turned on (which it is -by default), you can save all the lines you have entered into the -workspace by entering - -)history )write - -\index{history )write} - -Axiom tells you what input file to edit to see your statements. The -file is in your home directory or in the directory you specified with -\index{cd} {\tt {)cd}}.\\ -} - -In \ref{ugLangBlocks} on page~\pageref{ugLangBlocks} -we discuss using indentation in input files to -group statements into {\it blocks.} - -\section{The .axiom.input File} -\label{ugInOutSpadprof} - -When Axiom starts up, it tries to read the input file {\bf -.axiom.input}\footnote{{\bf.axiom.input} used to be called -{\bf axiom.input} in the NAG version} -from your home \index{start-up profile file} -directory. \index{file!start-up profile} It -there is no {\bf .axiom.input} in your home directory, it reads the -copy located in its own {\bf src/input} directory. -\index{file!.axiom.input @{\bf .axiom.input}} The file usually -contains system commands to personalize your Axiom environment. In -the remainder of this section we mention a few things that users -frequently place in their {\bf .axiom.input} files. - -In order to have FORTRAN output always produced from your -computations, place the system command {\tt )set output fortran on} in -{\bf .axiom.input}. \index{quit} If you do not want to be prompted -for confirmation when you issue the {\tt )quit} system command, place -{\tt )set quit unprotected} in {\bf .axiom.input}. -\index{set quit unprotected} -If you then decide that you do want to be prompted, issue -{\tt )set quit protected}. \index{set quit protected} This is the -default setting so that new users do not leave Axiom -inadvertently.\footnote{The system command {\tt )pquit} always -prompts you for confirmation.} - -To see the other system variables you can set, issue {\tt {)set}} -or use the HyperDoc {\bf Settings} facility to view and change -Axiom system variables. - -\section{Common Features of Using Output Formats} -\label{ugInOutOut} - -In this section we discuss how to start and stop the display -\index{output formats!common features} of the different output formats -and how to send the output to the screen or to a file. -\index{file!sending output to} To fix ideas, we use FORTRAN output -format for most of the examples. - -You can use the {\tt )set output} system \index{output -formats!starting} command to \index{output formats!stopping} toggle or -redirect the different kinds of output. \index{set output} The name -of the kind of output follows ``output'' in the command. The names are - -\begin{tabular}{@{}ll} -{\bf fortran} & for FORTRAN output. \\ -{\bf algebra} & for monospace two-dimensional mathematical output. \\ -{\bf tex} & for \TeX{} output. \\ -{\bf script} & for IBM Script Formula Format output. -\end{tabular} - -For example, issue {\tt {)set output fortran on}} to turn on FORTRAN -format and issue {\tt {)set output fortran off}} to turn it off. By -default, {\tt algebra} is {\tt on} and all others are {\tt off}. -\index{set output fortran} When output is started, it is sent to the -screen. To send the output to a file, give the file name without -\index{output formats!sending to file} directory or extension. Axiom -appends a file extension depending on the kind of output being -produced. - -Issue this to redirect FORTRAN output to, for example, the file -{\bf linalg.sfort}. -\spadcommand{)set output fortran linalg} -\begin{verbatim} - FORTRAN output will be written to file linalg.sfort . -\end{verbatim} - -You must {\it also} turn on the creation of FORTRAN output. -The above just says where it goes if it is created. -\spadcommand{)set output fortran on} - -In what directory is this output placed? It goes into the directory -from which you started Axiom, or if you have used the {\tt {)cd}} -system command, the one that you specified with {\tt {)cd}}. -\index{cd} You should use {\tt )cd} before you send the output to the file. - -You can always direct output back to the screen by issuing this. -\index{output formats!sending to screen} -\spadcommand{)set output fortran console} - -Let's make sure FORTRAN formatting is off so that nothing we -do from now on produces FORTRAN output. -\spadcommand{)set output fortran off} - -We also delete the demonstrated output file we created. -\spadcommand{)system rm linalg.sfort} - -You can abbreviate the words ``{\tt on},'' ``{\tt off},'' and -``{\tt console}'' to the minimal number of characters needed to distinguish -them. Because of this, you cannot send output to files called -{\bf on.sfort, off.sfort, of.sfort, console.sfort, consol.sfort} and so on. - -The width of the output on the page is set by \index{output -formats!line length} {\tt )set output length} for all formats except -FORTRAN. \index{set output length} Use {\tt )set fortran fortlength} -to change the FORTRAN line length from its default value of $72$. - -\section{Monospace Two-Dimensional Mathematical Format} -\label{ugInOutAlgebra} - -This is the default output format for Axiom. -It is usually on when you start the system. -\index{set output algebra} -\index{output formats!monospace 2D} -\index{monospace 2D output format} - -If it is not, issue this. -\spadcommand{)set output algebra on} - -Since the printed version of this book (as opposed to the HyperDoc -version) shows output produced by the \TeX{} output formatter, let us -temporarily turn off \TeX{} output. -\spadcommand{)set output tex off} - -Here is an example of what it looks like. -\spadcommand{matrix [ [i*x**i + j*\%i*y**j for i in 1..2] for j in 3..4]} -\begin{verbatim} - - + 3 3 2+ - |3%i y + x 3%i y + 2x | - (1) | | - | 4 4 2| - +4%i y + x 4%i y + 2x + -\end{verbatim} -\returnType{Type: Matrix Polynomial Complex Integer} - -Issue this to turn off this kind of formatting. -\spadcommand{)set output algebra off} - -Turn \TeX{} output on again. -\spadcommand{)set output tex on} - -The characters used for the matrix brackets above are rather ugly. -You get this character set when you issue \index{character set} -{\tt )set output characters plain}. \index{set output characters} This -character set should be used when you are running on a machine that -does not support the IBM extended ASCII character set. If you are -running on an IBM workstation, for example, issue -{\tt )set output characters default} to get better looking output. - -\section{TeX Format} -\label{ugInOutTeX} - -Axiom can produce \TeX{} output for your \index{output formats!TeX -@{\TeX{}}} expressions. \index{TeX output format @{\TeX{}} output format} -The output is produced using macros from the \LaTeX{} document -preparation system by Leslie Lamport\cite{1}. The printed version -of this book was produced using this formatter. - -To turn on \TeX{} output formatting, issue this. -\index{set output tex} -\spadcommand{)set output tex on} - -Here is an example of its output. -\begin{verbatim} -matrix [ [i*x**i + j*\%i*y**j for i in 1..2] for j in 3..4] - -$$ -\left[ -\begin{array}{cc} -{{3 \ i \ {y \sp 3}}+x} & -{{3 \ i \ {y \sp 3}}+{2 \ {x \sp 2}}} \\ -{{4 \ i \ {y \sp 4}}+x} & -{{4 \ i \ {y \sp 4}}+{2 \ {x \sp 2}}} -\end{array} -\right] -$$ - -\end{verbatim} -This formats as -$$ -\left[ -\begin{array}{cc} -{{3 \ i \ {y \sp 3}}+x} & -{{3 \ i \ {y \sp 3}}+{2 \ {x \sp 2}}} \\ -{{4 \ i \ {y \sp 4}}+x} & -{{4 \ i \ {y \sp 4}}+{2 \ {x \sp 2}}} -\end{array} -\right] -$$ - -To turn \TeX{} output formatting off, issue -{\tt {)set output tex off}}. -The \LaTeX macros in the output generated by Axiom -are all standard except for the following definitions: -\begin{verbatim} -\def\csch{\mathop{\rm csch}\nolimits} - -\def\erf{\mathop{\rm erf}\nolimits} - -\def\zag#1#2{ - {{\hfill \left. {#1} \right|} - \over - {\left| {#2} \right. \hfill} - } -} -\end{verbatim} - -\section{IBM Script Formula Format} -\label{ugInOutScript} - -Axiom can \index{output formats!IBM Script Formula Format} produce IBM -Script Formula Format output for your -\index{IBM Script Formula Format} expressions. - -To turn IBM Script Formula Format on, issue this. -\index{set output script} -\spadcommand{)set output script on} - -Here is an example of its output. -\begin{verbatim} -matrix [ [i*x**i + j*%i*y**j for i in 1..2] for j in 3..4] - -.eq set blank @ -:df. - >+x> here < <3 @@ %i @@ - >+<2 @@ > > > habove < < <4 @@ %i @@ - >+x> here < <4 @@ %i @@ >+<2 @@ - > > > > right rb> -:edf. -\end{verbatim} - -To turn IBM Script Formula Format output formatting off, issue this. -\spadcommand{)set output script off} - -\section{FORTRAN Format} -\label{ugInOutFortran} - -In addition to turning FORTRAN output on and off and stating where the -\index{output formats!FORTRAN} output should be placed, there are many -options that control the \index{FORTRAN output format} appearance of -the generated code. In this section we describe some of the basic -options. Issue {\tt )set fortran} to see a full list with their -current settings. - -The output FORTRAN expression usually begins in column 7. If the -expression needs more than one line, the ampersand character {\tt \&} -is used in column 6. Since some versions of FORTRAN have restrictions -on the number of lines per statement, Axiom breaks long expressions -into segments with a maximum of 1320 characters (20 lines of 66 -characters) per segment. \index{set fortran} If you want to change -this, say, to 660 characters, issue the system command -\index{set fortran explength} {\tt )set fortran explength 660}. -\index{FORTRAN output format!breaking into multiple statements} -You can turn off the line breaking by issuing {\tt )set fortran segment off}. -\index{set fortran segment} Various code optimization levels are available. - -FORTRAN output is produced after you issue this. -\index{set output fortran} -\spadcommand{)set output fortran on} - -For the initial examples, we set the optimization level to 0, which is the -lowest level. -\index{set fortran optlevel} -\spadcommand{)set fortran optlevel 0} - -The output is usually in columns 7 through 72, although fewer columns -are used in the following examples so that the output -\index{FORTRAN output format!line length} -fits nicely on the page. -\spadcommand{)set fortran fortlength 60} - -By default, the output goes to the screen and is displayed before the -standard Axiom two-dimensional output. In this example, an assignment -to the variable $R1$ was generated because this is the result of step 1. -\spadcommand{(x+y)**3} -\begin{verbatim} - R1=y**3+3*x*y*y+3*x*x*y+x**3 -\end{verbatim} -$$ -{y \sp 3}+{3 \ x \ {y \sp 2}}+{3 \ {x \sp 2} \ y}+{x \sp 3} -$$ -\returnType{Type: Polynomial Integer} - -Here is an example that illustrates the line breaking. -\spadcommand{(x+y+z)**3} -\begin{verbatim} - R2=z**3+(3*y+3*x)*z*z+(3*y*y+6*x*y+3*x*x)*z+y**3+3*x*y - &*y+3*x*x*y+x**3 -\end{verbatim} -$$ -{z \sp 3}+{{\left( {3 \ y}+{3 \ x} -\right)} -\ {z \sp 2}}+{{\left( {3 \ {y \sp 2}}+{6 \ x \ y}+{3 \ {x \sp 2}} -\right)} -\ z}+{y \sp 3}+{3 \ x \ {y \sp 2}}+{3 \ {x \sp 2} \ y}+{x \sp 3} -$$ -\returnType{Type: Polynomial Integer} - -Note in the above examples that integers are generally converted to -\index{FORTRAN output format!integers vs. floats} floating point -numbers, except in exponents. This is the default behavior but can be -turned off by issuing {\tt )set fortran ints2floats off}. -\index{set fortran ints2floats} The rules governing when the conversion -is done are: -\begin{enumerate} -\item If an integer is an exponent, convert it to a floating point -number if it is greater than 32767 in absolute value, otherwise leave it -as an integer. -\item Convert all other integers in an expression to floating point numbers. -\end{enumerate} -These rules only govern integers in expressions. Numbers generated by -Axiom for $DIMENSION$ statements are also integers. - -To set the type of generated FORTRAN data, -\index{FORTRAN output format!data types} -use one of the following: -\begin{verbatim} -)set fortran defaulttype REAL -)set fortran defaulttype INTEGER -)set fortran defaulttype COMPLEX -)set fortran defaulttype LOGICAL -)set fortran defaulttype CHARACTER -\end{verbatim} - -When temporaries are created, they are given a default type of {\tt REAL.} -Also, the {\tt REAL} versions of functions are used by default. -\spadcommand{sin(x)} -\begin{verbatim} - R3=DSIN(x) -\end{verbatim} -$$ -\sin -\left( -{x} -\right) -$$ -\returnType{Type: Expression Integer} - -At optimization level 1, Axiom removes common subexpressions. -\index{FORTRAN output format!optimization level} -\index{set fortran optlevel} -\spadcommand{)set fortran optlevel 1} - -\spadcommand{(x+y+z)**3} -\begin{verbatim} - T2=y*y - T3=x*x - R4=z**3+(3*y+3*x)*z*z+(3*T2+6*x*y+3*T3)*z+y**3+3*x*T2+ - &3*T3*y+x**3 -\end{verbatim} -$$ -{z \sp 3}+{{\left( {3 \ y}+{3 \ x} -\right)} -\ {z \sp 2}}+{{\left( {3 \ {y \sp 2}}+{6 \ x \ y}+{3 \ {x \sp 2}} -\right)} -\ z}+{y \sp 3}+{3 \ x \ {y \sp 2}}+{3 \ {x \sp 2} \ y}+{x \sp 3} -$$ -\returnType{Type: Polynomial Integer} - -This changes the precision to {\tt DOUBLE}. \index{set fortran -precision double} Substitute {\tt single} for {\tt double} -\index{FORTRAN output format!precision} to return to single precision. -\index{set fortran precision single} - -\spadcommand{)set fortran precision double} - -Complex constants display the precision. -\spadcommand{2.3 + 5.6*\%i } -\begin{verbatim} - R5=(2.3D0,5.6D0) -\end{verbatim} -$$ -{2.3}+{{5.6} \ i} -$$ -\returnType{Type: Complex Float} - -The function names that Axiom generates depend on the chosen precision. -\spadcommand{sin \%e} -%%NOTE: the book shows DSIN(DEXP(1.0D0)) -\begin{verbatim} - R6=DSIN(DEXP(1)) -\end{verbatim} -$$ -\sin -\left( -{e} -\right) -$$ -\returnType{Type: Expression Integer} - -Reset the precision to {\tt single} and look at these two examples again. -\spadcommand{)set fortran precision single} - -\spadcommand{2.3 + 5.6*\%i} -\begin{verbatim} - R7=(2.3,5.6) -\end{verbatim} -$$ -{2.3}+{{5.6} \ i} -$$ -\returnType{Type: Complex Float} - -\spadcommand{sin \%e} -%%NOTE: the book shows SIN(EXP(1.)) -\begin{verbatim} - R8=SIN(EXP(1)) -\end{verbatim} -$$ -\sin -\left( -{e} -\right) -$$ -\returnType{Type: Expression Integer} -Expressions that look like lists, streams, sets or matrices cause -array code to be generated. -\spadcommand{[x+1,y+1,z+1]} -\begin{verbatim} - T1(1)=x+1 - T1(2)=y+1 - T1(3)=z+1 - R9=T1 -\end{verbatim} -$$ -\left[ -{x+1}, {y+1}, {z+1} -\right] -$$ -\returnType{Type: List Polynomial Integer} - - -A temporary variable is generated to be the name of the array. -\index{FORTRAN output format!arrays} This may have to be changed in -your particular application. -\spadcommand{set[2,3,4,3,5]} -\begin{verbatim} - T1(1)=2 - T1(2)=3 - T1(3)=4 - T1(4)=5 - R10=T1 -\end{verbatim} -$$ -\left\{ -2, 3, 4, 5 -\right\} -$$ -\returnType{Type: Set PositiveInteger} - -By default, the starting index for generated FORTRAN arrays is $0$. -\spadcommand{matrix [ [2.3,9.7],[0.0,18.778] ]} -\begin{verbatim} - T1(0,0)=2.3 - T1(0,1)=9.7 - T1(1,0)=0.0 - T1(1,1)=18.778 - T1 -\end{verbatim} -$$ -\left[ -\begin{array}{cc} -{2.3} & {9.7} \\ -{0.0} & {18.778} -\end{array} -\right] -$$ -\returnType{Type: Matrix Float} - -To change the starting index for generated FORTRAN arrays to be $1$, -\index{set fortran startindex} issue this. This value can only be $0$ -or $1$. -\spadcommand{)set fortran startindex 1} - -Look at the code generated for the matrix again. -\spadcommand{matrix [ [2.3,9.7],[0.0,18.778] ]} -\begin{verbatim} - T1(1,1)=2.3 - T1(1,2)=9.7 - T1(2,1)=0.0 - T1(2,2)=18.778 - T1 -\end{verbatim} -$$ -\left[ -\begin{array}{cc} -{2.3} & {9.7} \\ -{0.0} & {18.778} -\end{array} -\right] -$$ -\returnType{Type: Matrix Float} - - -\setcounter{chapter}{4} - -\chapter{Overview of Interactive Language} -\label{ugLang} - -In this chapter we look at some of the basic components of the Axiom -language that you can use interactively. We show how to create a {\it -block} of expressions, how to form loops and list iterations, how to -modify the sequential evaluation of a block and how to use -{\tt if-then-else} to evaluate parts of your program conditionally. We -suggest you first read the boxed material in each section and then -proceed to a more thorough reading of the chapter. - -\section{Immediate and Delayed Assignments} -\label{ugLangAssign} - -A {\it variable} in Axiom refers to a value. A variable has a name -beginning with an uppercase or lowercase alphabetic character, -``{\tt \%}'', or ``{\tt !}''. Successive characters (if any) can be any of -the above, digits, or ``{\tt ?}''. Case is distinguished. The -following are all examples of valid, distinct variable names: - -\begin{verbatim} -a tooBig? a1B2c3%!? -A %j numberOfPoints -beta6 %J numberofpoints -\end{verbatim} - -The ``{\tt :=}'' operator is the immediate {\it assignment} operator. -\index{assignment!immediate} Use it to associate a value with a -variable. \index{immediate assignment} - -\boxed{4.6in}{ -\vskip 0.1cm -The syntax for immediate assignment for a single variable is -\begin{center} -{\it variable} $:=$ {\it expression} -\end{center} -The value returned by an immediate assignment is the value of -{\it expression}.\\ -} - -The right-hand side of the expression is evaluated, yielding $1$. -This value is then assigned to $a$. -\spadcommand{a := 1} -$$ -1 -$$ -\returnType{Type: PositiveInteger} - -The right-hand side of the expression is evaluated, yielding $1$. -This value is then assigned to $b$. Thus $a$ and $b$ both have the -value $1$ after the sequence of assignments. -\spadcommand{b := a} -$$ -1 -$$ -\returnType{Type: PositiveInteger} - -What is the value of $b$ if $a$ is assigned the value $2$? -\spadcommand{a := 2} -$$ -2 -$$ -\returnType{Type: PositiveInteger} - -As you see, the value of $b$ is left unchanged. -\spadcommand{b} -$$ -1 -$$ -\returnType{Type: PositiveInteger} - -This is what we mean when we say this kind of assignment is {\it -immediate}; $b$ has no dependency on $a$ after the initial assignment. -This is the usual notion of assignment found in programming languages -such as C, \index{C language!assignment} PASCAL -\index{PASCAL!assignment} and FORTRAN. \index{FORTRAN!assignment} - -Axiom provides delayed assignment with ``{\tt ==}''. -\index{assignment!delayed} This implements a \index{delayed -assignment} delayed evaluation of the right-hand side and dependency -checking. - -\boxed{4.6in}{ -\vskip 0.1cm -The syntax for delayed assignment is -\begin{center} -{\it variable} $==$ {\it expression} -\end{center} -The value returned by a delayed assignment is the unique value of {\tt Void}.\\ -} - -Using $a$ and $b$ as above, these are the corresponding delayed assignments. -\spadcommand{a == 1} -\returnType{Type: Void} - -\spadcommand{b == a} -\returnType{Type: Void} - -The right-hand side of each delayed assignment is left unevaluated -until the variables on the left-hand sides are evaluated. Therefore -this evaluation and \ldots -\spadcommand{a} -\begin{verbatim} -Compiling body of rule a to compute value of type PositiveInteger -\end{verbatim} -$$ -1 -$$ -\returnType{Type: PositiveInteger} - -this evaluation seem the same as before. -\spadcommand{b} -\begin{verbatim} -Compiling body of rule b to compute value of type PositiveInteger -\end{verbatim} -$$ -1 -$$ -\returnType{Type: PositiveInteger} - -If we change $a$ to $2$ -\spadcommand{a == 2} -\begin{verbatim} - Compiled code for a has been cleared. - Compiled code for b has been cleared. - 1 old definition(s) deleted for function or rule a -\end{verbatim} -\returnType{Type: Void} - -then $a$ evaluates to $2$, as expected, but -\spadcommand{a} -\begin{verbatim} -Compiling body of rule a to compute value of type PositiveInteger - -+++ |*0;a;1;G82322| redefined -\end{verbatim} -$$ -2 -$$ -\returnType{Type: PositiveInteger} - -the value of $b$ reflects the change to $a$. -\spadcommand{b} -\begin{verbatim} -Compiling body of rule b to compute value of type PositiveInteger - -+++ |*0;b;1;G82322| redefined -\end{verbatim} -$$ -2 -$$ -\returnType{Type: PositiveInteger} - -It is possible to set several variables at the same time -\index{assignment!multiple immediate} by using \index{multiple -immediate assignment} a {\it tuple} of variables and a tuple of -expressions. Note that a {\it tuple} is a collection of things -separated by commas, often surrounded by parentheses. - -\boxed{4.6in}{ -\vskip 0.1cm -The syntax for multiple immediate assignments is -\begin{center} -{\tt ( $\hbox{\it var}_{1}$, $\hbox{\it var}_{2}$, \ldots, -$\hbox{\it var}_{N}$ ) := ( $\hbox{\it expr}_{1}$, $\hbox{\it expr}_{2}$, -\ldots, $\hbox{\it expr}_{N}$ ) } -\end{center} -The value returned by an immediate assignment is the value of -$\hbox{\it expr}_{N}$.\\ -} - -This sets $x$ to $1$ and $y$ to $2$. -\spadcommand{(x,y) := (1,2)} -$$ -2 -$$ -\returnType{Type: PositiveInteger} - -Multiple immediate assigments are parallel in the sense that the -expressions on the right are all evaluated before any assignments on -the left are made. However, the order of evaluation of these -expressions is undefined. - -You can use multiple immediate assignment to swap the values held by -variables. -\spadcommand{(x,y) := (y,x)} -$$ -1 -$$ -\returnType{Type: PositiveInteger} - -$x$ has the previous value of $y$. -\spadcommand{x} -$$ -2 -$$ -\returnType{Type: PositiveInteger} - -$y$ has the previous value of $x$. -\spadcommand{y} -$$ -1 -$$ -\returnType{Type: PositiveInteger} - -There is no syntactic form for multiple delayed assignments. See the -discussion in section \ref{ugUserDelay} on page~\pageref{ugUserDelay} -about how Axiom differentiates between delayed assignments and user -functions of no arguments. - -\section{Blocks} -\label{ugLangBlocks} - -A {\it block} is a sequence of expressions evaluated in the order that -they appear, except as modified by control expressions such as -{\tt break}, \index{break} {\tt return}, \index{return} {\tt iterate} and -\index{iterate} {\tt if-then-else} constructions. The value of a block is -the value of the expression last evaluated in the block. - -To leave a block early, use ``{\tt =>}''. For example, $i < 0 => x$. The -expression before the ``{\tt =>}'' must evaluate to {\tt true} or {\tt false}. -The expression following the ``{\tt =>}'' is the return value for the block. - -A block can be constructed in two ways: -\begin{enumerate} -\item the expressions can be separated by semicolons -and the resulting expression surrounded by parentheses, and -\item the expressions can be written on succeeding lines with each line -indented the same number of spaces (which must be greater than zero). -\index{indentation} -A block entered in this form is -called a {\it pile}. -\end{enumerate} -Only the first form is available if you are entering expressions -directly to Axiom. Both forms are available in {\bf .input} files. - -\boxed{4.6in}{ -\vskip 0.1cm -The syntax for a simple block of expressions entered interactively is -\begin{center} -{\tt ( $\hbox{\it expression}_{1}$; $\hbox{\it expression}_{2}$; \ldots; -$\hbox{\it expression}_{N}$ )} -\end{center} -The value returned by a block is the value of an {\tt =>} expression, -or $\hbox{\it expression}_{N}$ if no {\tt =>} is encountered.\\ -} - -In {\bf .input} files, blocks can also be written using piles. The -examples throughout this book are assumed to come from {\bf .input} files. - -In this example, we assign a rational number to $a$ using a block -consisting of three expressions. This block is written as a pile. -Each expression in the pile has the same indentation, in this case two -spaces to the right of the first line. -\begin{verbatim} -a := - i := gcd(234,672) - i := 3*i**5 - i + 1 - 1 / i -\end{verbatim} -$$ -1 \over {23323} -$$ -\returnType{Type: Fraction Integer} - -Here is the same block written on one line. This is how you are -required to enter it at the input prompt. -\spadcommand{a := (i := gcd(234,672); i := 3*i**5 - i + 1; 1 / i)} -$$ -1 \over {23323} -$$ -\returnType{Type: Fraction Integer} - -Blocks can be used to put several expressions on one line. The value -returned is that of the last expression. -\spadcommand{(a := 1; b := 2; c := 3; [a,b,c])} -$$ -\left[ -1, 2, 3 -\right] -$$ -\returnType{Type: List PositiveInteger} - -Axiom gives you two ways of writing a block and the preferred way in -an {\bf .input} file is to use a pile. \index{file!input} Roughly -speaking, a pile is a block whose constituent expressions are indented -the same amount. You begin a pile by starting a new line for the -first expression, indenting it to the right of the previous line. You -then enter the second expression on a new line, vertically aligning it -with the first line. And so on. If you need to enter an inner pile, -further indent its lines to the right of the outer pile. Axiom knows -where a pile ends. It ends when a subsequent line is indented to the -left of the pile or the end of the file. - -Blocks can be used to perform several steps before an assignment -(immediate or delayed) is made. -\begin{verbatim} -d := - c := a**2 + b**2 - sqrt(c * 1.3) -\end{verbatim} -$$ -2.5495097567 96392415 -$$ -\returnType{Type: Float} - -Blocks can be used in the arguments to functions. (Here $h$ is -assigned $2.1 + 3.5$.) -\begin{verbatim} -h := 2.1 + - 1.0 - 3.5 -\end{verbatim} -$$ -5.6 -$$ -\returnType{Type: Float} - -Here the second argument to {\bf eval} is $x = z$, where the value of -$z$ is computed in the first line of the block starting on the second -line. -\begin{verbatim} -eval(x**2 - x*y**2, - z := %pi/2.0 - exp(4.1) - x = z - ) -\end{verbatim} -$$ -{{58.7694912705 67072878} \ {y \sp 2}}+{3453.8531042012 59382} -$$ -\returnType{Type: Polynomial Float} - -Blocks can be used in the clauses of {\tt if-then-else} expressions -(see \ref{ugLangIf} on page~\pageref{ugLangIf}). - -\spadcommand{if h > 3.1 then 1.0 else (z := cos(h); max(z,0.5))} -$$ -1.0 -$$ -\returnType{Type: Float} - -This is the pile version of the last block. -\begin{verbatim} -if h > 3.1 then - 1.0 - else - z := cos(h) - max(z,0.5) -\end{verbatim} -$$ -1.0 -$$ -\returnType{Type: Float} - -Blocks can be nested. -\spadcommand{a := (b := factorial(12); c := (d := eulerPhi(22); factorial(d));b+c)} -$$ -482630400 -$$ -\returnType{Type: PositiveInteger} - -This is the pile version of the last block. -\begin{verbatim} -a := - b := factorial(12) - c := - d := eulerPhi(22) - factorial(d) - b+c -\end{verbatim} -$$ -482630400 -$$ -\returnType{Type: PositiveInteger} - -Since $c + d$ does equal $3628855$, $a$ has the value of $c$ and the -last line is never evaluated. -\begin{verbatim} -a := - c := factorial 10 - d := fibonacci 10 - c + d = 3628855 => c - d -\end{verbatim} -$$ -3628800 -$$ -\returnType{Type: PositiveInteger} - -\section{if-then-else} -\label{ugLangIf} - -Like many other programming languages, Axiom uses the three keywords -\index{if} {\tt if}, {\tt then} \index{then} and {\tt else} -\index{else} to form \index{conditional} conditional expressions. The -{\tt else} part of the conditional is optional. The expression -between the {\tt if} and {\tt then} keywords is a {\it predicate}: an -expression that evaluates to or is convertible to either {\tt true} or -{\tt false}, that is, a {\tt Boolean}. \index{Boolean} - -\boxed{4.6in}{ -\vskip 0.1cm -The syntax for conditional expressions is -\begin{center} -{\tt if\ }{\it predicate} -{\tt then\ }$\hbox{\it expression}_{1}$ -{\tt else\ }$\hbox{\it expression}_{2}$ -\end{center} -where the {\tt else} $\hbox{\it expression}_{2}$ part is optional. The -value returned from a conditional expression is -$\hbox{\it expression}_{1}$ if the predicate evaluates to {\tt true} and -$\hbox{\it expression}_{2}$ otherwise. If no {\tt else} clause is given, -the value is always the unique value of {\tt Void}.\\ -} - -An {\tt if-then-else} expression always returns a value. If the -{\tt else} clause is missing then the entire expression returns the unique -value of {\tt Void}. If both clauses are present, the type of the -value returned by {\tt if} is obtained by resolving the types of the -values of the two clauses. See \ref{ugTypesResolve} on -page~\pageref{ugTypesResolve} for more information. - -The predicate must evaluate to, or be convertible to, an object of -type {\tt Boolean}: {\tt true} or {\tt false}. By default, the equal -sign \spadopFrom{=}{Equation} creates \index{equation} an equation. - -This is an equation. \index{Equation} In particular, it is an object -of type {\tt Equation Polynomial Integer}. - -\spadcommand{x + 1 = y} -$$ -{x+1}=y -$$ -\returnType{Type: Equation Polynomial Integer} - -However, for predicates in {\tt if} expressions, Axiom \index{equality -testing} places a default target type of {\tt Boolean} on the -predicate and equality testing is performed. \index{Boolean} Thus you -need not qualify the ``{\tt =}'' in any way. In other contexts you -may need to tell Axiom that you want to test for equality rather than -create an equation. In those cases, use ``{\tt @}'' and a target type -of {\tt Boolean}. See section \ref{ugTypesPkgCall} on -page~\pageref{ugTypesPkgCall} for more information. - -The compound symbol meaning ``not equal'' in Axiom is -\index{inequality testing} ``{$\sim =$}''. \index{\_notequal@$\sim =$} -This can be used directly without a package call or a target -specification. The expression $a$~$\sim =$~$b$ is directly translated -into {\tt not}$(a = b)$. - -Many other functions have return values of type {\tt Boolean}. These -include ``{\tt <}'', ``{\tt <=}'', ``{\tt >}'', ``{\tt >=}'', -``{\tt $\sim$=}'' and ``{\bf member?}''. By convention, -operations with names ending in ``{\tt ?}'' return {\tt Boolean} values. - -The usual rules for piles are suspended for conditional expressions. -In {\bf .input} files, the {\tt then} and {\tt else} keywords can begin in the -same column as the corresponding {\tt if} but may also appear to the -right. Each of the following styles of writing {\tt if-then-else} -expressions is acceptable: -\begin{verbatim} -if i>0 then output("positive") else output("nonpositive") - -if i > 0 then output("positive") - else output("nonpositive") - -if i > 0 then output("positive") -else output("nonpositive") - -if i > 0 -then output("positive") -else output("nonpositive") - -if i > 0 - then output("positive") - else output("nonpositive") -\end{verbatim} - -A block can follow the {\tt then} or {\tt else} keywords. In the following -two assignments to {\tt a}, the {\tt then} and {\tt else} clauses each are -followed by two-line piles. The value returned in each is the value -of the second line. -\begin{verbatim} -a := - if i > 0 then - j := sin(i * pi()) - exp(j + 1/j) - else - j := cos(i * 0.5 * pi()) - log(abs(j)**5 + 1) - -a := - if i > 0 - then - j := sin(i * pi()) - exp(j + 1/j) - else - j := cos(i * 0.5 * pi()) - log(abs(j)**5 + 1) -\end{verbatim} - -These are both equivalent to the following: -\begin{verbatim} -a := - if i > 0 then (j := sin(i * pi()); exp(j + 1/j)) - else (j := cos(i * 0.5 * pi()); log(abs(j)**5 + 1)) -\end{verbatim} - -\section{Loops} -\label{ugLangLoops} - -A {\it loop} is an expression that contains another expression, -\index{loop} called the {\it loop body}, which is to be evaluated zero -or more \index{loop!body} times. All loops contain the {\tt repeat} -keyword and return the unique value of {\tt Void}. Loops can contain -inner loops to any depth. - -\boxed{4.6in}{ -\vskip 0.1cm -The most basic loop is of the form -\begin{center} -{\tt repeat\ }{\it loopBody} -\end{center} - -Unless {\it loopBody} contains a {\tt break} or {\tt return} expression, the -loop repeats forever. The value returned by the loop is the unique -value of {\tt Void}.\\ -} - -\subsection{Compiling vs. Interpreting Loops} -\label{ugLangLoopsCompInt} - -Axiom tries to determine completely the type of every object in a loop -and then to translate the loop body to LISP or even to machine code. -This translation is called compilation. - -If Axiom decides that it cannot compile the loop, it issues a -\index{loop!compilation} message stating the problem and then the -following message: -\begin{center} -{\bf We will attempt to step through and interpret the code.} -\end{center} - -It is still possible that Axiom can evaluate the loop but in {\it -interpret-code mode}. See section \ref{ugUserCompInt} on -page~\pageref{ugUserCompInt} where this is discussed in terms -\index{panic!avoiding} of compiling versus interpreting functions. - -\subsection{return in Loops} -\label{ugLangLoopsReturn} - -A {\tt return} expression is used to exit a function with -\index{loop!leaving via return} a particular value. In particular, if -a {\tt return} is in a loop within the \index{return} function, the loop -is terminated whenever the {\tt return} is evaluated. -%> This is a bug! The compiler should never accept allow -%> Void to be the return type of a function when it has to use -%> resolve to determine it. - -Suppose we start with this. -\begin{verbatim} -f() == - i := 1 - repeat - if factorial(i) > 1000 then return i - i := i + 1 -\end{verbatim} -\returnType{Type: Void} - -When {\tt factorial(i)} is big enough, control passes from inside the loop -all the way outside the function, returning the value of $i$ (or so we -think). -\spadcommand{f()} -\returnType{Type: Void} - -What went wrong? Isn't it obvious that this function should return an -integer? Well, Axiom makes no attempt to analyze the structure of a -loop to determine if it always returns a value because, in general, -this is impossible. So Axiom has this simple rule: the type of the -function is determined by the type of its body, in this case a block. -The normal value of a block is the value of its last expression, in -this case, a loop. And the value of every loop is the unique value of -{\tt Void}.! So the return type of {\bf f} is {\tt Void}. - -There are two ways to fix this. The best way is for you to tell Axiom -what the return type of $f$ is. You do this by giving $f$ a -declaration {\tt f:()~->~Integer} prior to calling for its value. This -tells Axiom: ``trust me---an integer is returned.'' We'll explain -more about this in the next chapter. Another clumsy way is to add a -dummy expression as follows. - -Since we want an integer, let's stick in a dummy final expression that is -an integer and will never be evaluated. -\begin{verbatim} -f() == - i := 1 - repeat - if factorial(i) > 1000 then return i - i := i + 1 - 0 -\end{verbatim} -\returnType{Type: Void} - -When we try {\bf f} again we get what we wanted. See -\ref{ugUserBlocks} on page~\pageref{ugUserBlocks} for more information. - -\spadcommand{f()} -\begin{verbatim} - Compiling function f with type () -> NonNegativeInteger -\end{verbatim} -$$ -7 -$$ -\returnType{Type: PositiveInteger} - -\subsection{break in Loops} -\label{ugLangLoopsBreak} - -The {\tt break} keyword is often more useful \index{break} in terminating -\index{loop!leaving via break} a loop. A {\tt break} causes control to -transfer to the expression immediately following the loop. As loops -always return the unique value of {\tt Void}., you cannot return a -value with {\tt break}. That is, {\tt break} takes no argument. - -This example is a modification of the last example in the previous -section \ref{ugLangLoopsReturn} on page~\pageref{ugLangLoopsReturn}. -Instead of using {\tt return}, we'll use {\tt break}. - -\begin{verbatim} -f() == - i := 1 - repeat - if factorial(i) > 1000 then break - i := i + 1 - i -\end{verbatim} -\begin{verbatim} - Compiled code for f has been cleared. - 1 old definition(s) deleted for function or rule f -\end{verbatim} -\returnType{Type: Void} - -The loop terminates when {\tt factorial(i)} gets big enough, the last line -of the function evaluates to the corresponding ``good'' value of $i$, -and the function terminates, returning that value. - -\spadcommand{f()} -\begin{verbatim} - Compiling function f with type () -> PositiveInteger - -+++ |*0;f;1;G82322| redefined -\end{verbatim} -$$ -7 -$$ -\returnType{Type: PositiveInteger} - -You can only use {\tt break} to terminate the evaluation of one loop. -Let's consider a loop within a loop, that is, a loop with a nested -loop. First, we initialize two counter variables. - -\spadcommand{(i,j) := (1, 1)} -$$ -1 -$$ -\returnType{Type: PositiveInteger} - -Nested loops must have multiple {\tt break} \index{loop!nested} -expressions at the appropriate nesting level. How would you rewrite -this so {\tt (i + j) > 10} is only evaluated once? -\begin{verbatim} -repeat - repeat - if (i + j) > 10 then break - j := j + 1 - if (i + j) > 10 then break - i := i + 1 -\end{verbatim} -\returnType{Type: Void} - -\subsection{break vs. {\tt =>} in Loop Bodies} -\label{ugLangLoopsBreakVs} - -Compare the following two loops: -\begin{verbatim} -i := 1 i := 1 -repeat repeat - i := i + 1 i := i + 1 - i > 3 => i if i > 3 then break - output(i) output(i) -\end{verbatim} - -In the example on the left, the values $2$ and $3$ for $i$ are -displayed but then the ``{\tt =>}'' does not allow control to reach the -call to \spadfunFrom{output}{OutputForm} again. The loop will not -terminate until you run out of space or interrupt the execution. The -variable $i$ will continue to be incremented because the ``{\tt =>}'' only -means to leave the {\it block}, not the loop. - -In the example on the right, upon reaching $4$, the {\tt break} will be -executed, and both the block and the loop will terminate. This is one -of the reasons why both ``{\tt =>}'' and {\tt break} are provided. Using a -{\tt while} clause (see below) with the ``{\tt =>}'' \index{while} lets you -simulate the action of {\tt break}. - -\subsection{More Examples of break} -\label{ugLangLoopsBreakMore} - -Here we give four examples of {\tt repeat} loops that terminate when a -value exceeds a given bound. - -First, initialize $i$ as the loop counter. -\spadcommand{i := 0} -$$ -0 -$$ -\returnType{Type: NonNegativeInteger} - -Here is the first loop. When the square of $i$ exceeds $100$, the -loop terminates. -\begin{verbatim} -repeat - i := i + 1 - if i**2 > 100 then break -\end{verbatim} -\returnType{Type: Void} - -Upon completion, $i$ should have the value $11$. -\spadcommand{i} -$$ -11 -$$ -\returnType{Type: NonNegativeInteger} - -Do the same thing except use ``{\tt =>}'' instead an {\tt if-then} expression. - -\spadcommand{i := 0} -$$ -0 -$$ -\returnType{Type: NonNegativeInteger} - -\begin{verbatim} -repeat - i := i + 1 - i**2 > 100 => break -\end{verbatim} -\returnType{Type: Void} - -\spadcommand{i} -$$ -11 -$$ -\returnType{Type: NonNegativeInteger} - -As a third example, we use a simple loop to compute $n!$. -\spadcommand{(n, i, f) := (100, 1, 1)} -$$ -1 -$$ -\returnType{Type: PositiveInteger} - -Use $i$ as the iteration variable and $f$ to compute the factorial. -\begin{verbatim} -repeat - if i > n then break - f := f * i - i := i + 1 -\end{verbatim} -\returnType{Type: Void} - -Look at the value of $f$. -\spadcommand{f} -\begin{verbatim} - 93326215443944152681699238856266700490715968264381621468_ - 59296389521759999322991560894146397615651828625369792082_ - 7223758251185210916864000000000000000000000000 -\end{verbatim} -\returnType{Type: PositiveInteger} - -Finally, we show an example of nested loops. First define a four by -four matrix. -\spadcommand{m := matrix [ [21,37,53,14], [8,-24,22,-16], [2,10,15,14], [26,33,55,-13] ]} -$$ -\left[ -\begin{array}{cccc} -{21} & {37} & {53} & {14} \\ -8 & -{24} & {22} & -{16} \\ -2 & {10} & {15} & {14} \\ -{26} & {33} & {55} & -{13} -\end{array} -\right] -$$ -\returnType{Type: Matrix Integer} - -Next, set row counter $r$ and column counter $c$ to $1$. Note: if we -were writing a function, these would all be local variables rather -than global workspace variables. -\spadcommand{(r, c) := (1, 1)} -$$ -1 -$$ -\returnType{Type: PositiveInteger} - -Also, let {\tt lastrow} and {\tt lastcol} be the final row and column index. - -\spadcommand{(lastrow, lastcol) := (nrows(m), ncols(m))} -$$ -4 -$$ -\returnType{Type: PositiveInteger} - -Scan the rows looking for the first negative element. We remark that -you can reformulate this example in a better, more concise form by -using a {\tt for} clause with {\tt repeat}. See -\ref{ugLangLoopsForIn} on page~\pageref{ugLangLoopsForIn} for more -information. - -\begin{verbatim} -repeat - if r > lastrow then break - c := 1 - repeat - if c > lastcol then break - if elt(m,r,c) < 0 then - output [r, c, elt(m,r,c)] - r := lastrow - break -- don't look any further - c := c + 1 - r := r + 1 - - [2,2,- 24] -\end{verbatim} -\returnType{Type: Void} - -\subsection{iterate in Loops} -\label{ugLangLoopsIterate} - -Axiom provides an {\tt iterate} expression that \index{iterate} skips over -the remainder of a loop body and starts the next loop iteration. - -We first initialize a counter. - -\spadcommand{i := 0} -$$ -0 -$$ -\returnType{Type: NonNegativeInteger} - -Display the even integers from $2$ to $5$. -\begin{verbatim} -repeat - i := i + 1 - if i > 5 then break - if odd?(i) then iterate - output(i) - - 2 - 4 -\end{verbatim} -\returnType{Type: Void} - -\subsection{while Loops} -\label{ugLangLoopsWhile} - -The {\tt repeat} in a loop can be modified by adding one or more {\tt while} -clauses. \index{while} Each clause contains a {\it predicate} -immediately following the {\tt while} keyword. The predicate is tested -{\it before} the evaluation of the body of the loop. The loop body is -evaluated whenever the predicates in a {\tt while} clause are all {\tt true}. - -\boxed{4.6in}{ -\vskip 0.1cm -The syntax for a simple loop using {\tt while} is -\begin{center} -{\tt while} {\it predicate} {\tt repeat} {\it loopBody} -\end{center} -The {\it predicate} is evaluated before {\it loopBody} is evaluated. -A {\tt while} loop terminates immediately when {\it predicate} evaluates -to {\tt false} or when a {\tt break} or {\tt return} expression is evaluated in -{\it loopBody}. The value returned by the loop is the unique value of -{\tt Void}.\\ -} - -Here is a simple example of using {\tt while} in a loop. We first -initialize the counter. -\spadcommand{i := 1} -$$ -1 -$$ -\returnType{Type: PositiveInteger} - -The steps involved in computing this example are\\ -(1) set $i$ to $1$,\\ -(2) test the condition $i < 1$ and determine that it is not {\tt true}, and\\ -(3) do not evaluate the loop body and therefore do not display $"hello"$. -\begin{verbatim} -while i < 1 repeat - output "hello" - i := i + 1 -\end{verbatim} -\returnType{Type: Void} - -If you have multiple predicates to be tested use the logical {\tt and} -operation to separate them. Axiom evaluates these predicates from -left to right. -\spadcommand{(x, y) := (1, 1)} -$$ -1 -$$ -\returnType{Type: PositiveInteger} - -\begin{verbatim} -while x < 4 and y < 10 repeat - output [x,y] - x := x + 1 - y := y + 2 - - [1,1] - [2,3] - [3,5] -\end{verbatim} -\returnType{Type: Void} - -A {\tt break} expression can be included in a loop body to terminate a -loop even if the predicate in any {\tt while} clauses are not {\tt false}. -\spadcommand{(x, y) := (1, 1)} -$$ -1 -$$ -\returnType{Type: PositiveInteger} - -This loop has multiple {\tt while} clauses and the loop terminates -before any one of their conditions evaluates to {\tt false}. -\begin{verbatim} -while x < 4 while y < 10 repeat - if x + y > 7 then break - output [x,y] - x := x + 1 - y := y + 2 - - [1,1] - [2,3] -\end{verbatim} -\returnType{Type: Void} - -Here's a different version of the nested loops that looked for the -first negative element in a matrix. -\spadcommand{m := matrix [ [21,37,53,14], [8,-24,22,-16], [2,10,15,14], [26,33,55,-13] ]} -$$ -\left[ -\begin{array}{cccc} -{21} & {37} & {53} & {14} \\ -8 & -{24} & {22} & -{16} \\ -2 & {10} & {15} & {14} \\ -{26} & {33} & {55} & -{13} -\end{array} -\right] -$$ -\returnType{Type: Matrix Integer} - -Initialized the row index to $1$ and get the number of rows and -columns. If we were writing a function, these would all be local -variables. -\spadcommand{r := 1} -$$ -1 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{(lastrow, lastcol) := (nrows(m), ncols(m))} -$$ -4 -$$ -\returnType{Type: PositiveInteger} - -Scan the rows looking for the first negative element. -\begin{verbatim} -while r <= lastrow repeat - c := 1 -- index of first column - while c <= lastcol repeat - if elt(m,r,c) < 0 then - output [r, c, elt(m,r,c)] - r := lastrow - break -- don't look any further - c := c + 1 - r := r + 1 - - [2,2,- 24] -\end{verbatim} -\returnType{Type: Void} - -\subsection{for Loops} -\label{ugLangLoopsForIn} - -Axiom provides the {\tt for} \index{for} and {\tt in\ } \index{in} keywords in -{\tt repeat} loops, allowing you to iterate across all \index{iteration} -elements of a list, or to have a variable take on integral values from -a lower bound to an upper bound. We shall refer to these modifying -clauses of {\tt repeat} loops as {\tt for} clauses. These clauses can be -present in addition to {\tt while} clauses. As with all other types of -{\tt repeat} loops, {\tt break} can \index{break} be used to prematurely -terminate the evaluation of the loop. - -\boxed{4.6in}{ -\vskip 0.1cm -The syntax for a simple loop using {\tt for} is -\begin{center} -{\tt for} {\it iterator} {\tt repeat} {\it loopBody} -\end{center} - -The {\it iterator} has several forms. Each form has an end test which -is evaluated before {\it loopBody} is evaluated. A {\tt for} loop -terminates immediately when the end test succeeds (evaluates to -{\tt true}) or when a {\tt break} or {\tt return} expression is evaluated -in {\it loopBody}. The value returned by the loop is the unique value -of {\tt Void}.\\ } - -\subsection{for i in n..m repeat} -\label{ugLangLoopsForInNM} - -If {\tt for} \index{for} is followed by a variable name, the {\tt in\ } -\index{in} keyword and then an integer segment of the form $n..m$, -\index{segment} the end test for this loop is the predicate $i > m$. -The body of the loop is evaluated $m-n+1$ times if this number is -greater than 0. If this number is less than or equal to 0, the loop -body is not evaluated at all. - -The variable $i$ has the value $n, n+1, ..., m$ for successive iterations -of the loop body.The loop variable is a {\it local variable} -within the loop body: its value is not available outside the loop body -and its value and type within the loop body completely mask any outer -definition of a variable with the same name. - -This loop prints the values of -${10}^3$, ${11}^3$, and $12^3$: -\spadcommand{for i in 10..12 repeat output(i**3)} -\begin{verbatim} - 1000 - 1331 - 1728 -\end{verbatim} -\returnType{Type: Void} - -Here is a sample list. -\spadcommand{a := [1,2,3]} -$$ -\left[ -1, 2, 3 -\right] -$$ -\returnType{Type: List PositiveInteger} - -Iterate across this list, using ``{\tt .}'' to access the elements of -a list and the ``{\bf \#}'' operation to count its elements. - -\spadcommand{for i in 1..\#a repeat output(a.i)} -\begin{verbatim} - 1 - 2 - 3 -\end{verbatim} -\returnType{Type: Void} - -This type of iteration is applicable to anything that uses ``{\tt .}''. -You can also use it with functions that use indices to extract elements. - -Define $m$ to be a matrix. -\spadcommand{m := matrix [ [1,2],[4,3],[9,0] ]} -$$ -\left[ -\begin{array}{cc} -1 & 2 \\ -4 & 3 \\ -9 & 0 -\end{array} -\right] -$$ -\returnType{Type: Matrix Integer} - -Display the rows of $m$. -\spadcommand{for i in 1..nrows(m) repeat output row(m,i)} -\begin{verbatim} - [1,2] - [4,3] - [9,0] -\end{verbatim} -\returnType{Type: Void} - -You can use {\tt iterate} with {\tt for}-loops.\index{iterate} - -Display the even integers in a segment. -\begin{verbatim} -for i in 1..5 repeat - if odd?(i) then iterate - output(i) - - 2 - 4 -\end{verbatim} -\returnType{Type: Void} - -See section \ref{SegmentXmpPage} on page~\pageref{SegmentXmpPage} for -more information about segments. - -\subsection{for i in n..m by s repeat} -\label{ugLangLoopsForInNMS} - -By default, the difference between values taken on by a variable in -loops such as {\tt for i in n..m repeat ...} is $1$. It is possible to -supply another, possibly negative, step value by using the {\tt by} -\index{by} keyword along with {\tt for} and {\tt in\ }. Like the upper and -lower bounds, the step value following the {\tt by} keyword must be an -integer. Note that the loop {\tt for i in 1..2 by 0 repeat output(i)} -will not terminate by itself, as the step value does not change the -index from its initial value of $1$. - -This expression displays the odd integers between two bounds. -\spadcommand{for i in 1..5 by 2 repeat output(i)} -\begin{verbatim} - 1 - 3 - 5 -\end{verbatim} -\returnType{Type: Void} - -Use this to display the numbers in reverse order. -\spadcommand{for i in 5..1 by -2 repeat output(i)} -\begin{verbatim} - 5 - 3 - 1 -\end{verbatim} -\returnType{Type: Void} - -\subsection{for i in n.. repeat} -\label{ugLangLoopsForInN} - -If the value after the ``{\tt ..}'' is omitted, the loop has no end test. -A potentially infinite loop is thus created. The variable is given -the successive values ${n}, {n+1}, {n+2}, ...$ and the loop is terminated -only if a {\tt break} or {\tt return} expression is evaluated in the loop -body. However you may also add some other modifying clause on the -{\tt repeat} (for example, a {\tt while} clause) to stop the loop. - -This loop displays the integers greater than or equal to $15$ -and less than the first prime greater than $15$. -\spadcommand{for i in 15.. while not prime?(i) repeat output(i)} -\begin{verbatim} - 15 - 16 -\end{verbatim} -\returnType{Type: Void} - -\subsection{for x in l repeat} -\label{ugLangLoopsForInXL} - -Another variant of the {\tt for} loop has the form: -\begin{center} -{\it {\tt for} x {\tt in\ } list {\tt repeat} loopBody} -\end{center} - -This form is used when you want to iterate directly over the elements -of a list. In this form of the {\tt for} loop, the variable {\tt x} takes on -the value of each successive element in {\tt l}. The end test is most -simply stated in English: ``are there no more {\tt x} in {\tt l}?'' - -If {\tt l} is this list, -\spadcommand{l := [0,-5,3]} -$$ -\left[ -0, -5, 3 -\right] -$$ -\returnType{Type: List Integer} - -display all elements of {\tt l}, one per line. -\spadcommand{for x in l repeat output(x)} -\begin{verbatim} - 0 - - 5 - 3 -\end{verbatim} -\returnType{Type: Void} - -Since the list constructing expression {\bf expand}{\tt [n..m]} creates the -list $[n, {n+1}, ..., m]$. Note that this list is empty if $n > m$. You -might be tempted to think that the loops -\begin{verbatim} -for i in n..m repeat output(i) -\end{verbatim} - -and -\begin{verbatim} -for x in expand [n..m] repeat output(x) -\end{verbatim} - -are equivalent. The second form first creates the list {\bf -expand}{\tt [n..m]} (no matter how large it might be) and then does -the iteration. The first form potentially runs in much less space, as -the index variable $i$ is simply incremented once per loop and the -list is not actually created. Using the first form is much more -efficient. - -Of course, sometimes you really want to iterate across a specific list. -This displays each of the factors of $2400000$. -\spadcommand{for f in factors(factor(2400000)) repeat output(f)} -\begin{verbatim} - [factor= 2,exponent= 8] - [factor= 3,exponent= 1] - [factor= 5,exponent= 5] -\end{verbatim} -\returnType{Type: Void} - -\subsection{``Such that'' Predicates} -\label{ugLangLoopsForInPred} - -A {\tt for} loop can be followed by a ``{\tt |}'' and then a predicate. The -predicate qualifies the use of the values from the iterator following -the {\tt for}. Think of the vertical bar ``{\tt |}'' as the phrase ``such -that.'' - -This loop expression prints out the integers $n$ in the given segment -such that $n$ is odd. -\spadcommand{for n in 0..4 | odd? n repeat output n} -\begin{verbatim} - 1 - 3 -\end{verbatim} -\returnType{Type: Void} - -\boxed{4.6in}{ -\vskip 0.1cm -A {\tt for} loop can also be written -$$ -{\rm for} {\it \ iterator\ } | {\it \ predicate\ } -{\rm repeat} {\it \ loopBody\ } -$$ - -which is equivalent to: -$$ -{\rm for} {\it \ iterator\ } {\rm repeat\ if} -{\it \ predicate\ } {\rm then} {\it \ loopBody\ } {\rm else\ }iterate -$$ -} - -The predicate need not refer only to the variable in the {\tt for} clause: -any variable in an outer scope can be part of the predicate. - -In this example, the predicate on the inner {\tt for} loop uses $i$ from -the outer loop and the $j$ from the {\tt for} \index{iteration!nested} -clause that it directly modifies. -\begin{verbatim} -for i in 1..50 repeat - for j in 1..50 | factorial(i+j) < 25 repeat - output [i,j] - - [1,1] - [1,2] - [1,3] - [2,1] - [2,2] - [3,1] -\end{verbatim} -\returnType{Type: Void} - -\subsection{Parallel Iteration} -\label{ugLangLoopsPar} - -The last example of the previous section -\ref{ugLangLoopsForInPred} on page~\pageref{ugLangLoopsForInPred} -gives an example of {\it nested iteration}: a loop is contained -\index{iteration!nested} in another loop. \index{iteration!parallel} -Sometimes you want to iterate across two lists in parallel, or perhaps -you want to traverse a list while incrementing a variable. - -\boxed{4.6in}{ -\vskip 0.1cm -The general syntax of a repeat loop is -$$ -iterator_1\ iterator_2\ \ldots\ iterator_N {\rm \ repeat\ } loopBody -$$ -where each {\it iterator} is either a {\tt for} or a {\tt while} clause. The -loop terminates immediately when the end test of any {\it iterator} -succeeds or when a {\tt break} or {\tt return} expression is evaluated in {\it -loopBody}. The value returned by the loop is the unique value of {\tt -Void}.\\ -} - -Here we write a loop to iterate across two lists, computing the sum of -the pairwise product of elements. Here is the first list. -\spadcommand{l := [1,3,5,7]} -$$ -\left[ -1, 3, 5, 7 -\right] -$$ -\returnType{Type: List PositiveInteger} - -And the second. -\spadcommand{m := [100,200]} -$$ -\left[ -{100}, {200} -\right] -$$ -\returnType{Type: List PositiveInteger} - -The initial value of the sum counter. -\spadcommand{sum := 0} -$$ -0 -$$ -\returnType{Type: NonNegativeInteger} - -The last two elements of $l$ are not used in the calculation because -$m$ has two fewer elements than $l$. -\begin{verbatim} -for x in l for y in m repeat - sum := sum + x*y -\end{verbatim} -\returnType{Type: Void} - -Display the ``dot product.'' -\spadcommand{sum} -$$ -700 -$$ -\returnType{Type: NonNegativeInteger} - -Next, we write a loop to compute the sum of the products of the loop -elements with their positions in the loop. -\spadcommand{l := [2,3,5,7,11,13,17,19,23,29,31,37]} -$$ -\left[ -2, 3, 5, 7, {11}, {13}, {17}, {19}, {23}, {29}, -{31}, {37} -\right] -$$ -\returnType{Type: List PositiveInteger} - -The initial sum. -\spadcommand{sum := 0} -$$ -0 -$$ -\returnType{Type: NonNegativeInteger} - -Here looping stops when the list $l$ is exhausted, even though -the $for i in 0..$ specifies no terminating condition. - -\spadcommand{for i in 0.. for x in l repeat sum := i * x} -\returnType{Type: Void} - -Display this weighted sum. -\spadcommand{sum} -$$ -407 -$$ -\returnType{Type: NonNegativeInteger} - -When ``{\tt |}'' is used to qualify any of the {\tt for} clauses in a parallel -iteration, the variables in the predicates can be from an outer scope -or from a {\tt for} clause in or to the left of a modified clause. - -This is correct: -% output from following is too long to show -\begin{verbatim} -for i in 1..10 repeat - for j in 200..300 | odd? (i+j) repeat - output [i,j] -\end{verbatim} - -This is not correct since the variable $j$ has not been defined -outside the inner loop. -\begin{verbatim} -for i in 1..10 | odd? (i+j) repeat -- wrong, j not defined - for j in 200..300 repeat - output [i,j] -\end{verbatim} - -\subsection{Mixing Loop Modifiers} -\label{ugLangLoopsMix} - -This example shows that it is possible to mix several of the -\index{loop!mixing modifiers} forms of {\tt repeat} modifying clauses on a loop. -\begin{verbatim} -for i in 1..10 - for j in 151..160 | odd? j - while i + j < 160 repeat - output [i,j] - - [1,151] - [3,153] -\end{verbatim} -\returnType{Type: Void} - -Here are useful rules for composing loop expressions: -\begin{enumerate} -\item {\tt while} predicates can only refer to variables that -are global (or in an outer scope) -or that are defined in {\tt for} clauses to the left of the -predicate. -\item A ``such that'' predicate (something following ``{\tt |}'') -must directly follow a {\tt for} clause and can only refer to -variables that are global (or in an outer scope) -or defined in the modified {\tt for} clause -or any {\tt for} clause to the left. -\end{enumerate} - -\section{Creating Lists and Streams with Iterators} -\label{ugLangIts} - -All of what we did for loops in -\ref{ugLangLoops} on page~\pageref{ugLangLoops} \index{iteration} -can be transformed into expressions that create lists -\index{list!created by iterator} and streams. \index{stream!created -by iterator} The {\tt repeat}, {\tt break} or {\tt iterate} words are not used but -all the other ideas carry over. Before we give you the general rule, -here are some examples which give you the idea. - -This creates a simple list of the integers from $1$ to $10$. -\spadcommand{list := [i for i in 1..10]} -$$ -\left[ -1, 2, 3, 4, 5, 6, 7, 8, 9, {10} -\right] -$$ -\returnType{Type: List PositiveInteger} - -Create a stream of the integers greater than or equal to $1$. -\spadcommand{stream := [i for i in 1..]} -$$ -\left[ -1, 2, 3, 4, 5, 6, 7, 8, 9, {10}, \ldots -\right] -$$ -\returnType{Type: Stream PositiveInteger} - -This is a list of the prime integers between $1$ and $10$, inclusive. -\spadcommand{[i for i in 1..10 | prime? i]} -$$ -\left[ -2, 3, 5, 7 -\right] -$$ -\returnType{Type: List PositiveInteger} - -This is a stream of the prime integers greater than or equal to $1$. -\spadcommand{[i for i in 1.. | prime? i]} -$$ -\left[ -2, 3, 5, 7, {11}, {13}, {17}, {19}, {23}, {29}, -\ldots -\right] -$$ -\returnType{Type: Stream PositiveInteger} - -This is a list of the integers between $1$ and $10$, inclusive, whose -squares are less than $700$. -\spadcommand{[i for i in 1..10 while i*i < 700]} -$$ -\left[ -1, 2, 3, 4, 5, 6, 7, 8, 9, {10} -\right] -$$ -\returnType{Type: List PositiveInteger} - -This is a stream of the integers greater than or equal to $1$ -whose squares are less than $700$. -\spadcommand{[i for i in 1.. while i*i < 700]} -$$ -\left[ -1, 2, 3, 4, 5, 6, 7, 8, 9, {10}, \ldots -\right] -$$ -\returnType{Type: Stream PositiveInteger} - -Here is the general rule. -\index{collection} - -\boxed{4.6in}{ -\vskip 0.1cm -The general syntax of a collection is -\begin{center} -{\tt [ {\it collectExpression} $\hbox{\it iterator}_{1}$ -$\hbox{\it iterator}_{2}$ \ldots $\hbox{\it iterator}_{N}$ ]} -\end{center} - -where each $\hbox{\it iterator}_{i}$ is either a {\tt for} or a {\tt while} -clause. The loop terminates immediately when the end test of any -$\hbox{\it iterator}_{i}$ succeeds or when a {\tt return} expression is -evaluated in {\it collectExpression}. The value returned by the -collection is either a list or a stream of elements, one for each -iteration of the {\it collectExpression}.\\ -} - -Be careful when you use {\tt while} -\index{stream!using while @{using {\tt while}}} -to create a stream. By default, Axiom tries to compute and -display the first ten elements of a stream. If the {\tt while} condition -is not satisfied quickly, Axiom can spend a long (possibly infinite) -time trying to compute \index{stream!number of elements computed} the -elements. Use {\tt )set streams calculate} to change the default to -something else. \index{set streams calculate} This also affects the -number of terms computed and displayed for power series. For the -purposes of this book, we have used this system command to display -fewer than ten terms. - -Use nested iterators to create lists of \index{iteration!nested} lists -which can then be given as an argument to {\bf matrix}. -\spadcommand{matrix [ [x**i+j for i in 1..3] for j in 10..12]} -$$ -\left[ -\begin{array}{ccc} -{x+{10}} & {{x \sp 2}+{10}} & {{x \sp 3}+{10}} \\ -{x+{11}} & {{x \sp 2}+{11}} & {{x \sp 3}+{11}} \\ -{x+{12}} & {{x \sp 2}+{12}} & {{x \sp 3}+{12}} -\end{array} -\right] -$$ -\returnType{Type: Matrix Polynomial Integer} - -You can also create lists of streams, streams of lists and streams of -streams. Here is a stream of streams. -\spadcommand{[ [i/j for i in j+1..] for j in 1..]} -$$ -\begin{array}{@{}l} -\left[ -{\left[ 2, 3, 4, 5, 6, 7, 8, 9, {10}, {11}, -\ldots -\right]}, - {\left[ {3 \over 2}, 2, {5 \over 2}, 3, {7 \over 2}, 4, -{9 \over 2}, 5, {{11} \over 2}, 6, \ldots -\right]}, -\right. -\\ -\\ -\displaystyle - {\left[ {4 \over 3}, {5 \over 3}, 2, {7 \over 3}, {8 \over 3}, - 3, {{10} \over 3}, {{11} \over 3}, 4, {{13} \over 3}, -\ldots -\right]}, - {\left[ {5 \over 4}, {3 \over 2}, {7 \over 4}, 2, {9 \over 4}, - {5 \over 2}, {{11} \over 4}, 3, {{13} \over 4}, {7 \over 2}, - \ldots -\right]}, -\\ -\\ -\displaystyle - {\left[ {6 \over 5}, {7 \over 5}, {8 \over 5}, {9 \over 5}, 2, - {{11} \over 5}, {{12} \over 5}, {{13} \over 5}, {{14} \over 5}, - 3, \ldots -\right]}, - {\left[ {7 \over 6}, {4 \over 3}, {3 \over 2}, {5 \over 3}, -{{11} \over 6}, 2, {{13} \over 6}, {7 \over 3}, {5 \over 2}, -{8 \over 3}, \ldots -\right]}, -\\ -\\ -\displaystyle - {\left[ {8 \over 7}, {9 \over 7}, {{10} \over 7}, {{11} \over 7}, - {{12} \over 7}, {{13} \over 7}, 2, {{15} \over 7}, {{16} \over -7}, {{17} \over 7}, \ldots -\right]}, - {\left[ {9 \over 8}, {5 \over 4}, {{11} \over 8}, {3 \over 2}, -{{13} \over 8}, {7 \over 4}, {{15} \over 8}, 2, {{17} \over 8}, - {9 \over 4}, \ldots -\right]}, -\\ -\\ -\displaystyle - {\left[ {{10} \over 9}, {{11} \over 9}, {4 \over 3}, {{13} \over -9}, {{14} \over 9}, {5 \over 3}, {{16} \over 9}, {{17} \over 9}, - 2, {{19} \over 9}, \ldots -\right]}, -\\ -\\ -\displaystyle -\left. - {\left[ {{11} \over {10}}, {6 \over 5}, {{13} \over {10}}, {7 -\over 5}, {3 \over 2}, {8 \over 5}, {{17} \over {10}}, {9 \over -5}, {{19} \over {10}}, 2, \ldots -\right]}, - \ldots -\right] -\end{array} -$$ -\returnType{Type: Stream Stream Fraction Integer} - -You can use parallel iteration across lists and streams to create -\index{iteration!parallel} new lists. -\spadcommand{[i/j for i in 3.. by 10 for j in 2..]} -$$ -\left[ -{3 \over 2}, {{13} \over 3}, {{23} \over 4}, {{33} \over 5}, -{{43} \over 6}, {{53} \over 7}, {{63} \over 8}, {{73} \over 9}, -{{83} \over {10}}, {{93} \over {11}}, \ldots -\right] -$$ -\returnType{Type: Stream Fraction Integer} - -Iteration stops if the end of a list or stream is reached. -\spadcommand{[i**j for i in 1..7 for j in 2.. ]} -$$ -\left[ -1, 8, {81}, {1024}, {15625}, {279936}, {5764801} -\right] -$$ -\returnType{Type: Stream Integer} - -%or a while condition fails. -%\spadcommand{[i**j for i in 1.. for j in 2.. while i + j < 5 ]} -%tpdhere -% There are no library operations named swhile -% Use HyperDoc Browse or issue -% )what op swhile -% to learn if there is any operation containing " swhile " in its -% name. -% -% Cannot find a definition or applicable library operation named -% swhile with argument type(s) -% (Record(part1: PositiveInteger,part2: PositiveInteger) -> Boolean) -% InfiniteTuple Record(part1: PositiveInteger,part2: PositiveInteger) -% -% Perhaps you should use "@" to indicate the required return type, -% or "$" to specify which version of the function you need. - -As with loops, you can combine these modifiers to make very -complicated conditions. -\spadcommand{[ [ [i,j] for i in 10..15 | prime? i] for j in 17..22 | j = squareFreePart j]} -$$ -\left[ -{\left[ {\left[ {11}, {17} -\right]}, - {\left[ {13}, {17} -\right]} -\right]}, - {\left[ {\left[ {11}, {19} -\right]}, - {\left[ {13}, {19} -\right]} -\right]}, - {\left[ {\left[ {11}, {21} -\right]}, - {\left[ {13}, {21} -\right]} -\right]}, - {\left[ {\left[ {11}, {22} -\right]}, - {\left[ {13}, {22} -\right]} -\right]} -\right] -$$ -\returnType{Type: List List List PositiveInteger} - -See List -(section \ref{ListXmpPage} on page~\pageref{ListXmpPage}) and Stream -(section \ref{StreamXmpPage} on page~\pageref{StreamXmpPage}) -for more information on creating and -manipulating lists and streams, respectively. - -\section{An Example: Streams of Primes} -\label{ugLangStreamsPrimes} - -We conclude this chapter with an example of the creation and -manipulation of infinite streams of prime integers. This might be -useful for experiments with numbers or other applications where you -are using sequences of primes over and over again. As for all -streams, the stream of primes is only computed as far out as you need. -Once computed, however, all the primes up to that point are saved for -future reference. - -Two useful operations provided by the Axiom library are -\spadfunFrom{prime?}{IntegerPrimesPackage} and -\spadfunFrom{nextPrime}{IntegerPrimesPackage}. A straight-forward way -to create a stream of prime numbers is to start with the stream of -positive integers $[2,..]$ and filter out those that are prime. - -Create a stream of primes. -\spadcommand{primes : Stream Integer := [i for i in 2.. | prime? i]} -$$ -\left[ -2, 3, 5, 7, {11}, {13}, {17}, {19}, {23}, {29}, -\ldots -\right] -$$ -\returnType{Type: Stream Integer} - -A more elegant way, however, is to use the -\spadfunFrom{generate}{Stream} operation from {\tt Stream}. Given an -initial value $a$ and a function $f$, \spadfunFrom{generate}{Stream} -constructs the stream $[a, f(a), f(f(a)), ...]$. This function gives -you the quickest method of getting the stream of primes. - -This is how you use \spadfunFrom{generate}{Stream} to generate an -infinite stream of primes. -\spadcommand{primes := generate(nextPrime,2)} -$$ -\left[ -2, 3, 5, 7, {11}, {13}, {17}, {19}, {23}, {29}, -\ldots -\right] -$$ -\returnType{Type: Stream Integer} - -Once the stream is generated, you might only be interested in primes -starting at a particular value. -\spadcommand{smallPrimes := [p for p in primes | p > 1000]} -$$ -\left[ -{1009}, {1013}, {1019}, {1021}, {1031}, {1033}, {1039}, -{1049}, {1051}, {1061}, \ldots -\right] -$$ -\returnType{Type: Stream Integer} - -Here are the first 11 primes greater than 1000. -\spadcommand{[p for p in smallPrimes for i in 1..11]} -$$ -\left[ -{1009}, {1013}, {1019}, {1021}, {1031}, {1033}, {1039}, -{1049}, {1051}, {1061}, \ldots -\right] -$$ -\returnType{Type: Stream Integer} - -Here is a stream of primes between 1000 and 1200. -\spadcommand{[p for p in smallPrimes while p < 1200]} -$$ -\left[ -{1009}, {1013}, {1019}, {1021}, {1031}, {1033}, {1039}, -{1049}, {1051}, {1061}, \ldots -\right] -$$ -\returnType{Type: Stream Integer} - -To get these expanded into a finite stream, you call -\spadfunFrom{complete}{Stream} on the stream. -\spadcommand{complete \%} -$$ -\left[ -{1009}, {1013}, {1019}, {1021}, {1031}, {1033}, {1039}, -{1049}, {1051}, {1061}, \ldots -\right] -$$ -\returnType{Type: Stream Integer} - -Twin primes are consecutive odd number pairs which are prime. -Here is the stream of twin primes. -\spadcommand{twinPrimes := [ [p,p+2] for p in primes | prime?(p + 2)]} -$$ -\begin{array}{@{}l} -\left[ -{\left[ 3, 5 \right]}, -{\left[ 5, 7 \right]}, -{\left[ {11}, {13} \right]}, -{\left[ {17}, {19} \right]}, -{\left[ {29}, {31} \right]}, -{\left[ {41}, {43} \right]}, -{\left[ {59}, {61} \right]}, -{\left[ {71}, {73} \right]}, -\right. -\\ -\\ -\displaystyle -\left. -{\left[ {101}, {103} \right]}, -{\left[ {107}, {109} \right]}, - \ldots -\right] -\end{array} -$$ -\returnType{Type: Stream List Integer} - -Since we already have the primes computed we can avoid the call to -\spadfunFrom{prime?}{IntegerPrimesPackage} by using a double -iteration. This time we'll just generate a stream of the first of the -twin primes. -\spadcommand{firstOfTwins:= [p for p in primes for q in rest primes | q=p+2]} -$$ -\left[ -3, 5, {11}, {17}, {29}, {41}, {59}, {71}, {101}, -{107}, \ldots -\right] -$$ -\returnType{Type: Stream Integer} - -Let's try to compute the infinite stream of triplet primes, the set of -primes $p$ such that $[p,p+2,p+4]$ are primes. For example, $[3,5,7]$ -is a triple prime. We could do this by a triple {\tt for} iteration. A -more economical way is to use {\bf firstOfTwins}. This time however, -put a semicolon at the end of the line. - -Create the stream of firstTriplets. Put a semicolon at the end so -that no elements are computed. -\spadcommand{firstTriplets := [p for p in firstOfTwins for q in rest firstOfTwins | q = p+2];} -\returnType{Type: Stream Integer} - -What happened? As you know, by default Axiom displays the first ten -elements of a stream when you first display it. And, therefore, it -needs to compute them! If you want {\it no} elements computed, just -terminate the expression by a semicolon (``{\tt ;}''). The semi-colon -prevents the display of the result of evaluating the expression. -Since no stream elements are needed for display (or anything else, so -far), none are computed. - -Compute the first triplet prime. -\spadcommand{firstTriplets.1} -$$ -3 -$$ -\returnType{Type: PositiveInteger} - -If you want to compute another, just ask for it. But wait a second! -Given three consecutive odd integers, one of them must be divisible by -$3$. Thus there is only one triplet prime. But suppose that you did not -know this and wanted to know what was the tenth triplet prime. -\begin{verbatim} -firstTriples.10 -\end{verbatim} - -To compute the tenth triplet prime, Axiom first must compute the -second, the third, and so on. But since there isn't even a second -triplet prime, Axiom will compute forever. Nonetheless, this effort -can produce a useful result. After waiting a bit, hit \fbox{\bf Ctrl-c}. -The system responds as follows. -\begin{verbatim} - >> System error: - Console interrupt. - You are being returned to the top level of - the interpreter. -\end{verbatim} - -If you want to know how many primes have been computed, type: -\begin{verbatim} -numberOfComputedEntries primes -\end{verbatim} - -and, for this discussion, let's say that the result is $2045$. -How big is the $2045$-th prime? -\spadcommand{primes.2045} -$$ -17837 -$$ -\returnType{Type: PositiveInteger} - -What you have learned is that there are no triplet primes between 5 -and 17837. Although this result is well known (some might even say -trivial), there are many experiments you could make where the result -is not known. What you see here is a paradigm for testing of -hypotheses. Here our hypothesis could have been: ``there is more than -one triplet prime.'' We have tested this hypothesis for 17837 cases. -With streams, you can let your machine run, interrupt it to see how -far it has progressed, then start it up and let it continue from where -it left off. - -\setcounter{chapter}{5} - -\chapter{User-Defined Functions, Macros and Rules} -\label{ugUser} - -In this chapter we show you how to write functions and macros, -and we explain how Axiom looks for and applies them. -We show some simple one-line examples of functions, together -with larger ones that are defined piece-by-piece or through the use of -piles. - -\section{Functions vs. Macros} -\label{ugUserFunMac} - -A function is a program to perform some \index{function!vs. macro} -computation. \index{macro!vs. function} Most functions have names so -that it is easy to refer to them. A simple example of a function is -one named \spadfunFrom{abs}{Integer} which computes the absolute value -of an integer. - -This is a use of the ``absolute value'' library function for integers. -\spadcommand{abs(-8)} -$$ -8 -$$ -\returnType{Type: PositiveInteger} - -This is an unnamed function that does the same thing, using the -``maps-to'' syntax {\tt +->} that we discuss in -section \ref{ugUserAnon} on page~\pageref{ugUserAnon}. -\spadcommand{(x +-> if x < 0 then -x else x)(-8)} -$$ -8 -$$ -\returnType{Type: PositiveInteger} - -Functions can be used alone or serve as the building blocks for larger -programs. Usually they return a value that you might want to use in -the next stage of a computation, but not always (for example, see -\ref{ExitXmpPage} on page~\pageref{ExitXmpPage} and \ref{VoidXmpPage} -on page~\pageref{VoidXmpPage}). They may also read data from your -keyboard, move information from one place to another, or format and -display results on your screen. - -In Axiom, as in mathematics, functions \index{function!parameters} are -usually parameterized. Each time you {\it call} (some people say {\it -apply} or invoke) a function, you give \index{parameters to a -function} values to the parameters (variables). Such a value is -called an {\it argument} of \index{function!arguments} the function. -Axiom uses the arguments for the computation. In this way you get -different results depending on what you ``feed'' the function. - -Functions can have local variables or refer to global variables in the -workspace. Axiom can often compile functions so that they execute -very efficiently. Functions can be passed as arguments to other -functions. - -Macros are textual substitutions. They are used to clarify the -meaning of constants or expressions and to be templates for frequently -used expressions. Macros can be parameterized but they are not -objects that can be passed as arguments to functions. In effect, -macros are extensions to the Axiom expression parser. - -\section{Macros} -\label{ugUserMacros} - -A {\it macro} provides general textual substitution of \index{macro} -an Axiom expression for a name. You can think of a macro as being a -generalized abbreviation. You can only have one macro in your -workspace with a given name, no matter how many arguments it has. - -\boxed{4.6in}{ -\vskip 0.1cm -The two general forms for macros are -\begin{center} -{\tt macro} {\it name} {\tt ==} {\it body} \\ -{\tt macro} {\it name(arg1,...)} {\tt ==} {\it body} -\end{center} -where the body of the macro can be any Axiom expression.\\ -} - -For example, suppose you decided that you like to use {\tt df} for -{\tt D}. You define the macro {\tt df} like this. -\spadcommand{macro df == D} -\returnType{Type: Void} - -Whenever you type {\tt df}, the system expands it to {\tt D}. -\spadcommand{df(x**2 + x + 1,x)} -$$ -{2 \ x}+1 -$$ -\returnType{Type: Polynomial Integer} - -Macros can be parameterized and so can be used for many different -kinds of objects. -\spadcommand{macro ff(x) == x**2 + 1} -\returnType{Type: Void} - -Apply it to a number, a symbol, or an expression. -\spadcommand{ff z} -$$ -{z \sp 2}+1 -$$ -\returnType{Type: Polynomial Integer} - -Macros can also be nested, but you get an error message if you -run out of space because of an infinite nesting loop. -\spadcommand{macro gg(x) == ff(2*x - 2/3)} -\returnType{Type: Void} - -This new macro is fine as it does not produce a loop. -\spadcommand{gg(1/w)} -$$ -{{{13} \ {w \sp 2}} -{{24} \ w}+{36}} \over {9 \ {w \sp 2}} -$$ -\returnType{Type: Fraction Polynomial Integer} - -This, however, loops since {\tt gg} is defined in terms of {\tt ff}. -\spadcommand{macro ff(x) == gg(-x)} -\returnType{Type: Void} - -The body of a macro can be a block. -\spadcommand{macro next == (past := present; present := future; future := past + present)} -\returnType{Type: Void} - -Before entering {\tt next}, we need values for {\tt present} and {\tt future}. -\spadcommand{present : Integer := 0} -$$ -0 -$$ -\returnType{Type: Integer} - -\spadcommand{future : Integer := 1} -$$ -1 -$$ -\returnType{Type: Integer} - -Repeatedly evaluating {\tt next} produces the next Fibonacci number. -\spadcommand{next} -$$ -1 -$$ -\returnType{Type: Integer} - -And the next one. -\spadcommand{next} -$$ -2 -$$ -\returnType{Type: Integer} - -Here is the infinite stream of the rest of the Fibonacci numbers. -\spadcommand{[next for i in 1..]} -$$ -\left[ -3, 5, 8, {13}, {21}, {34}, {55}, {89}, {144}, -{233}, \ldots -\right] -$$ -\returnType{Type: Stream Integer} - -Bundle all the above lines into a single macro. -\begin{verbatim} -macro fibStream == - present : Integer := 1 - future : Integer := 1 - [next for i in 1..] where - macro next == - past := present - present := future - future := past + present -\end{verbatim} -\returnType{Type: Void} - -Use \spadfunFrom{concat}{Stream} to start with the first two -\index{Fibonacci numbers} Fibonacci numbers. -\spadcommand{concat([0,1],fibStream)} -$$ -\left[ -0, 1, 2, 3, 5, 8, {13}, {21}, {34}, {55}, -\ldots -\right] -$$ -\returnType{Type: Stream Integer} - -The library operation {\bf fibonacci} is an easier way to compute -these numbers. - -\spadcommand{[fibonacci i for i in 1..]} -$$ -\left[ -1, 1, 2, 3, 5, 8, {13}, {21}, {34}, {55}, -\ldots -\right] -$$ -\returnType{Type: Stream Integer} - -\section{Introduction to Functions} -\label{ugUserIntro} - -Each name in your workspace can refer to a single object. This may be -any kind of object including a function. You can use interactively -any function from the library or any that you define in the workspace. -In the library the same name can have very many functions, but you can -have only one function with a given name, although it can have any -number of arguments that you choose. - -If you define a function in the workspace that has the same name and -number of arguments as one in the library, then your definition takes -precedence. In fact, to get the library function you must -{\sl package-call} it -(see section \ref{ugTypesPkgCall} on page~\pageref{ugTypesPkgCall}). - -To use a function in Axiom, you apply it to its arguments. Most -functions are applied by entering the name of the function followed by -its argument or arguments. -\spadcommand{factor(12)} -$$ -{2 \sp 2} \ 3 -$$ -\returnType{Type: Factored Integer} - -Some functions like ``{\tt +}'' have {\it infix} {\it operators} as names. -\spadcommand{3 + 4} -$$ -7 -$$ -\returnType{Type: PositiveInteger} - -The function ``{\tt +}'' has two arguments. When you give it more than -two arguments, Axiom groups the arguments to the left. This -expression is equivalent to $(1 + 2) + 7$. -\spadcommand{1 + 2 + 7} -$$ -10 -$$ -\returnType{Type: PositiveInteger} - -All operations, including infix operators, can be written in prefix -form, that is, with the operation name followed by the arguments in -parentheses. For example, $2 + 3$ can alternatively be written as -$+(2,3)$. But $+(2,3,4)$ is an error since {\tt +} takes only two -arguments. - -Prefix operations are generally applied before the infix operation. -Thus the form ${\bf factorial\ } 3 + 1$ means ${\bf factorial}(3) + 1$ -producing $7$, and $-2 + 5$ means $(-2) + 5$ producing $3$. An -example of a prefix operator is prefix ``{\tt -}''. For example, $- 2 + -5$ converts to $(- 2) + 5$ producing the value $3$. Any prefix -function taking two arguments can be written in an infix manner by -putting an ampersand ``{\tt \&}'' before the name. Thus ${\tt D}(2*x,x)$ can -be written as $2*x\ {\tt \&D} x$ returning $2$. - -Every function in Axiom is identified by a {\it name} and -{\it type}. (An exception is an ``anonymous function'' discussed in -\ref{ugUserAnon} on page~\pageref{ugUserAnon}.) -The type of a function is always a mapping of the -form \spadsig{Source}{Target} where {\tt Source} and {\tt Target} are types. -To enter a type from the keyboard, enter the arrow by using a hyphen -``{\tt -}'' followed by a greater-than sign ``{\tt >}'', e.g. -{\tt Integer -> Integer}. - -Let's go back to ``{\tt +}''. There are many ``{\tt +}'' functions in the -Axiom library: one for integers, one for floats, another for rational -numbers, and so on. These ``{\tt +}'' functions have different types and -thus are different functions. You've seen examples of this -{\it overloading} before---using the same name for different functions. -Overloading is the rule rather than the exception. You can add two -integers, two polynomials, two matrices or two power series. These -are all done with the same function name but with different functions. - -\section{Declaring the Type of Functions} -\label{ugUserDeclare} - -In \ref{ugTypesDeclare} on page~\pageref{ugTypesDeclare} we discussed -how to declare a variable to restrict the kind of values that can be -assigned to it. In this section we show how to declare a variable -that refers to function objects. - -\boxed{4.6in}{ -\vskip 0.1cm -A function is an object of type -\begin{center} -{\sf Source $\rightarrow$ Type} -\end{center} - -where {\tt Source} and {\tt Target} can be any type. A common type -for {\tt Source} is {\tt Tuple}($\hbox{\it T}_{1}$, \ldots, -$\hbox{\it T}_{n}$), usually written ($\hbox{\it T}_{1}$, \ldots, -$\hbox{\it T}_{n}$), to indicate a function of $n$ arguments.\\ -} - -If $g$ takes an {\tt Integer}, a {\tt Float} and another {\tt Integer}, -and returns a {\tt String}, the declaration is written: -\spadcommand{g: (Integer,Float,Integer) -> String} -\returnType{Type: Void} - -The types need not be written fully; using abbreviations, the above -declaration is: -\spadcommand{g: (INT,FLOAT,INT) -> STRING} -\returnType{Type: Void} - -It is possible for a function to take no arguments. If $ h$ takes no -arguments but returns a {\tt Polynomial} {\tt Integer}, any of the -following declarations is acceptable. -\spadcommand{h: () -> POLY INT} -\returnType{Type: Void} - -\spadcommand{h: () -> Polynomial INT} -\returnType{Type: Void} - -\spadcommand{h: () -> POLY Integer} -\returnType{Type: Void} - -\boxed{4.6in}{ -\vskip 0.1cm -Functions can also be declared when they are being defined. -The syntax for combined declaration/definition is: -\begin{center} -\frenchspacing{\tt {\it functionName}($\hbox{\it parm}_{1}$: -$\hbox{\it parmType}_{1}$, \ldots, $\hbox{\it parm}_{N}$: -$\hbox{\it parmType}_{N}$): {\it functionReturnType}} -\end{center} -{\ }%force a blank line -} - -The following definition fragments show how this can be done for -the functions $g$ and $h$ above. -\begin{verbatim} -g(arg1: INT, arg2: FLOAT, arg3: INT): STRING == ... - -h(): POLY INT == ... -\end{verbatim} - -A current restriction on function declarations is that they must -involve fully specified types (that is, cannot include modes involving -explicit or implicit ``{\tt ?}''). For more information on declaring -things in general, see \ref{ugTypesDeclare} on page~\pageref{ugTypesDeclare}. - -\section{One-Line Functions} -\label{ugUserOne} - -As you use Axiom, you will find that you will write many short -functions \index{function!one-line definition} to codify sequences of -operations that you often perform. In this section we write some -simple one-line functions. - -This is a simple recursive factorial function for positive integers. -\spadcommand{fac n == if n < 3 then n else n * fac(n-1)} -\returnType{Type: Void} - -\spadcommand{fac 10} -$$ -3628800 -$$ -\returnType{Type: PositiveInteger} - -This function computes $1 + 1/2 + 1/3 + ... + 1/n$. -\spadcommand{s n == reduce(+,[1/i for i in 1..n])} -\returnType{Type: Void} - -\spadcommand{s 50} -$$ -{13943237577224054960759} \over {3099044504245996706400} -$$ -\returnType{Type: Fraction Integer} - -This function computes a Mersenne number, several of which are prime. -\index{Mersenne number} -\spadcommand{mersenne i == 2**i - 1} -\returnType{Type: Void} - -If you type {\tt mersenne}, Axiom shows you the function definition. -\spadcommand{mersenne} -$$ -mersenne \ i \ == \ {{2 \sp i} -1} -$$ -\returnType{Type: FunctionCalled mersenne} - -Generate a stream of Mersenne numbers. -\spadcommand{[mersenne i for i in 1..]} -$$ -\left[ -1, 3, 7, {15}, {31}, {63}, {127}, {255}, {511}, -{1023}, \ldots -\right] -$$ -\returnType{Type: Stream Integer} - -Create a stream of those values of $i$ such that {\tt mersenne(i)} is prime. -\spadcommand{mersenneIndex := [n for n in 1.. | prime?(mersenne(n))]} -\begin{verbatim} - Compiling function mersenne with type PositiveInteger -> Integer -\end{verbatim} -$$ -\left[ -2, 3, 5, 7, {13}, {17}, {19}, {31}, {61}, {89}, -\ldots -\right] -$$ -\returnType{Type: Stream PositiveInteger} - -Finally, write a function that returns the $n$-th Mersenne prime. -\spadcommand{mersennePrime n == mersenne mersenneIndex(n)} -\returnType{Type: Void} - -\spadcommand{mersennePrime 5} -$$ -8191 -$$ -\returnType{Type: PositiveInteger} - -\section{Declared vs. Undeclared Functions} -\label{ugUserDecUndec} - -If you declare the type of a function, you can apply it to any data -that can be converted to the source type of the function. - -Define {\bf f} with type {\sf Integer $\rightarrow$ Integer}. -\spadcommand{f(x: Integer): Integer == x + 1} -\begin{verbatim} - Function declaration f : Integer -> Integer has been added to - workspace. -\end{verbatim} -\returnType{Type: Void} - -The function {\bf f} can be applied to integers, \ldots -\spadcommand{f 9} -\begin{verbatim} - Compiling function f with type Integer -> Integer -\end{verbatim} -$$ -10 -$$ -\returnType{Type: PositiveInteger} - -and to values that convert to integers, \ldots -\spadcommand{f(-2.0)} -$$ --1 -$$ -\returnType{Type: Integer} - -but not to values that cannot be converted to integers. -\spadcommand{f(2/3)} -\begin{verbatim} - Conversion failed in the compiled user function f . - - Cannot convert from type Fraction Integer to Integer for value - 2 - - - 3 -\end{verbatim} - -To make the function over a wide range of types, do not declare its type. -Give the same definition with no declaration. -\spadcommand{g x == x + 1} -\returnType{Type: Void} - -If $x + 1$ makes sense, you can apply {\bf g} to $x$. -\spadcommand{g 9} -\begin{verbatim} - Compiling function g with type PositiveInteger -> PositiveInteger -\end{verbatim} -$$ -10 -$$ -\returnType{Type: PositiveInteger} - -A version of {\bf g} with different argument types get compiled for -each new kind of argument used. -\spadcommand{g(2/3)} -\begin{verbatim} - Compiling function g with type Fraction Integer -> Fraction Integer -\end{verbatim} -$$ -5 \over 3 -$$ -\returnType{Type: Fraction Integer} - -Here $x+1$ for $x = "axiom"$ makes no sense. -\spadcommand{g("axiom")} -\begin{verbatim} - There are 11 exposed and 5 unexposed library operations named + - having 2 argument(s) but none was determined to be applicable. - Use HyperDoc Browse, or issue - )display op + - to learn more about the available operations. Perhaps - package-calling the operation or using coercions on the arguments - will allow you to apply the operation. - Cannot find a definition or applicable library operation named + - with argument type(s) - String - PositiveInteger - - Perhaps you should use "@" to indicate the required return type, - or "$" to specify which version of the function you need. - AXIOM will attempt to step through and interpret the code. - There are 11 exposed and 5 unexposed library operations named + - having 2 argument(s) but none was determined to be applicable. - Use HyperDoc Browse, or issue - )display op + - to learn more about the available operations. Perhaps - package-calling the operation or using coercions on the arguments - will allow you to apply the operation. - - Cannot find a definition or applicable library operation named + - with argument type(s) - String - PositiveInteger - - Perhaps you should use "@" to indicate the required return type, - or "$" to specify which version of the function you need. -\end{verbatim} - -As you will see in Chapter \ref{ugCategories} on -page~\pageref{ugCategories}, Axiom has a formal idea of categories for -what ``makes sense.'' - -\section{Functions vs. Operations} -\label{ugUserDecOpers} - -A function is an object that you can create, manipulate, pass to, and -return from functions (for some interesting examples of library -functions that manipulate functions, see \ref{MappingPackage1XmpPage} -on page~\pageref{MappingPackage1XmpPage}). Yet, we often seem to use -the term {\it operation} and {\it function} interchangeably in Axiom. What -is the distinction? - -First consider values and types associated with some variable $n$ in -your workspace. You can make the declaration {\tt n : Integer}, then -assign $n$ an integer value. You then speak of the integer $n$. -However, note that the integer is not the name $n$ itself, but the -value that you assign to $n$. - -Similarly, you can declare a variable $f$ in your workspace to have -type {\sf Integer $\rightarrow$ Integer}, then assign $f$, through a -definition or an assignment of an anonymous function. You then speak -of the function $f$. However, the function is not $f$, but the value -that you assign to $f$. - -A function is a value, in fact, some machine code for doing something. -Doing what? Well, performing some {\it operation}. Formally, an -operation consists of the constituent parts of $f$ in your workspace, -excluding the value; thus an operation has a name and a type. An -operation is what domains and packages export. Thus {\tt Ring} -exports one operation ``{\tt +}''. Every ring also exports this -operation. Also, the author of every ring in the system is obliged -under contract (see \ref{ugPackagesAbstract} on -page~\pageref{ugPackagesAbstract}) to provide an implementation for -this operation. - -This chapter is all about functions---how you create them -interactively and how you apply them to meet your needs. In Chapter -\ref{ugPackages} on page~\pageref{ugPackages} you will learn how to -create them for the Axiom library. Then in Chapter \ref{ugCategories} -on page~\pageref{ugCategories}, you will learn about categories and -exported operations. - -\section{Delayed Assignments vs. Functions with No Arguments} -\label{ugUserDelay} - -In \ref{ugLangAssign} on page~\pageref{ugLangAssign} we discussed the -difference between immediate and \index{function!with no arguments} -delayed assignments. In this section we show the difference between -delayed assignments and functions of no arguments. - -A function of no arguments is sometimes called a {\it nullary function.} -\spadcommand{sin24() == sin(24.0)} -\returnType{Type: Void} - -You must use the parentheses ``{\tt ()}'' to evaluate it. Like a -delayed assignment, the right-hand-side of a function evaluation is -not evaluated until the left-hand-side is used. -\spadcommand{sin24()} -\begin{verbatim} - Compiling function sin24 with type () -> Float -\end{verbatim} -$$ --{0.9055783620\ 0662384514} -$$ -\returnType{Type: Float} - -If you omit the parentheses, you just get the function definition. -\spadcommand{sin24} -$$ -sin24 \ {\left( -\right)} -\ == \ {\sin -\left( -{{24.0}} -\right)} -$$ -\returnType{Type: FunctionCalled sin24} - -You do not use the parentheses ``{\tt ()}'' in a delayed assignment\ldots - -\spadcommand{cos24 == cos(24.0)} -\returnType{Type: Void} - -nor in the evaluation. - -\spadcommand{cos24} -\begin{verbatim} - Compiling body of rule cos24 to compute value of type Float -\end{verbatim} -$$ -0.4241790073\ 3699697594 -$$ -\returnType{Type: Float} - -The only syntactic difference between delayed assignments -and nullary functions is that you use ``{\tt ()}'' in the latter case. - -\section{How Axiom Determines What Function to Use} -\label{ugUserUse} - -What happens if you define a function that has the same name as a -library function? Well, if your function has the same name and number -of arguments (we sometimes say {\it arity}) as another function in the -library, then your function covers up the library function. If you -want then to call the library function, you will have to {\sl package-call} -it. Axiom can use both the functions you write and those that come -from the library. Let's do a simple example to illustrate this. - -Suppose you (wrongly!) define {\bf sin} in this way. -\spadcommand{sin x == 1.0} -\returnType{Type: Void} - -The value $1.0$ is returned for any argument. -\spadcommand{sin 4.3} -\begin{verbatim} - Compiling function sin with type Float -> Float -\end{verbatim} -$$ -1.0 -$$ -\returnType{Type: Float} - -If you want the library operation, we have to package-call it -(see \ref{ugTypesPkgCall} on page~\pageref{ugTypesPkgCall} -for more information). -\spadcommand{sin(4.3)\$Float} -$$ --{0.9161659367 4945498404} -$$ -\returnType{Type: Float} - -\spadcommand{sin(34.6)\$Float} -$$ --{0.0424680347 1695010154 3} -$$ -\returnType{Type: Float} - -Even worse, say we accidentally used the same name as a library -function in the function. -\spadcommand{sin x == sin x} -\begin{verbatim} - Compiled code for sin has been cleared. - 1 old definition(s) deleted for function or rule sin -\end{verbatim} -\returnType{Type: Void} - -Then Axiom definitely does not understand us. -\spadcommand{sin 4.3} -\begin{verbatim} -AXIOM cannot determine the type of sin because it cannot analyze - the non-recursive part, if that exists. This may be remedied - by declaring the function. -\end{verbatim} - -Again, we could package-call the inside function. -\spadcommand{sin x == sin(x)\$Float} -\begin{verbatim} - 1 old definition(s) deleted for function or rule sin -\end{verbatim} -\returnType{Type: Void} - -\spadcommand{sin 4.3} -\begin{verbatim} - Compiling function sin with type Float -> Float - -+++ |*1;sin;1;G82322| redefined -\end{verbatim} -$$ --{0.9161659367 4945498404} -$$ -\returnType{Type: Float} - -Of course, you are unlikely to make such obvious errors. It is more -probable that you would write a function and in the body use a -function that you think is a library function. If you had also -written a function by that same name, the library function would be -invisible. - -How does Axiom determine what library function to call? It very much -depends on the particular example, but the simple case of creating the -polynomial $x + 2/3$ will give you an idea. -\begin{enumerate} -\item The $x$ is analyzed and its default type is -{\tt Variable(x)}. -\item The $2$ is analyzed and its default type is -{\tt PositiveInteger}. -\item The $3$ is analyzed and its default type is -{\tt PositiveInteger}. -\item Because the arguments to ``{\tt /}'' are integers, Axiom -gives the expression $2/3$ a default target type of -{\tt Fraction(Integer)}. -\item Axiom looks in {\tt PositiveInteger} for ``{\tt /}''. -It is not found. -\item Axiom looks in {\tt Fraction(Integer)} for ``{\tt /}''. -It is found for arguments of type {\tt Integer}. -\item The $2$ and $3$ are converted to objects of type -{\tt Integer} (this is trivial) and ``{\tt /}'' is applied, -creating an object of type {\tt Fraction(Integer)}. -\item No ``{\tt +}'' for arguments of types {\tt Variable(x)} and -{\tt Fraction(Integer)} are found in either domain. -\item Axiom resolves -\index{resolve} -(see \ref{ugTypesResolve} on page~\pageref{ugTypesResolve}) -the types and gets {\tt Polynomial (Fraction (Integer))}. -\item The $x$ and the $2/3$ are converted to objects of this -type and {\tt +} is applied, yielding the answer, an object of type -{\tt Polynomial (Fraction (Integer))}. -\end{enumerate} - -\section{Compiling vs. Interpreting} -\label{ugUserCompInt} - -When possible, Axiom completely determines the type of every object in -a function, then translates the function definition to Common Lisp or -to machine code (see the next section). This translation, -\index{function!compiler} called compilation, happens the first time -you call the function and results in a computational delay. -Subsequent function calls with the same argument types use the -compiled version of the code without delay. - -If Axiom cannot determine the type of everything, the function may -still be executed \index{function!interpretation} but -\index{interpret-code mode} in interpret-code mode: each statement in -the function is analyzed and executed as the control flow indicates. -This process is slower than executing a compiled function, but it -allows the execution of code that may involve objects whose types -change. - -\boxed{4.6in}{ -\vskip 0.1cm -If Axiom decides that it cannot compile the code, it issues a message -stating the problem and then the following message: -\begin{center} -{\bf We will attempt to step through and interpret the code.} -\end{center} - -This is not a time to panic. \index{panic!avoiding} Rather, it just -means that what you gave to Axiom is somehow ambiguous: either it is -not specific enough to be analyzed completely, or it is beyond Axiom's -present interactive compilation abilities.\\ -} - -This function runs in interpret-code mode, but it does not compile. -\begin{verbatim} -varPolys(vars) == - for var in vars repeat - output(1 :: UnivariatePolynomial(var,Integer)) -\end{verbatim} -\returnType{Type: Void} - -For $vars$ equal to $['x, 'y, 'z]$, this function displays $1$ three times. -\spadcommand{varPolys ['x,'y,'z]} -\begin{verbatim} -Cannot compile conversion for types involving local variables. - In particular, could not compile the expression involving :: - UnivariatePolynomial(var,Integer) - AXIOM will attempt to step through and interpret the code. - 1 - 1 - 1 -\end{verbatim} -\returnType{Type: Void} - -The type of the argument to {\bf output} changes in each iteration, so -Axiom cannot compile the function. In this case, even the inner loop -by itself would have a problem: -\begin{verbatim} -for var in ['x,'y,'z] repeat - output(1 :: UnivariatePolynomial(var,Integer)) -\end{verbatim} -\begin{verbatim} -Cannot compile conversion for types involving local variables. - In particular, could not compile the expression involving :: - UnivariatePolynomial(var,Integer) - AXIOM will attempt to step through and interpret the code. - 1 - 1 - 1 -\end{verbatim} -\returnType{Type: Void} - -Sometimes you can help a function to compile by using an extra -conversion or by using $pretend$. \index{pretend} See -\ref{ugTypesSubdomains} on page~\pageref{ugTypesSubdomains} for details. - -When a function is compilable, you have the choice of whether it is -compiled to Common Lisp and then interpreted by the Common Lisp -interpreter or then further compiled from Common Lisp to machine code. -\index{machine code} The option is controlled via -{\tt )set functions compile}. -\index{set function compile} Issue {\tt )set functions compile on} -to compile all the way to machine code. With the default -setting {\tt )set functions compile off}, Axiom has its Common Lisp -code interpreted because the overhead of further compilation is larger -than the run-time of most of the functions our users have defined. -You may find that selectively turning this option on and off will -\index{performance} give you the best performance in your particular -application. For example, if you are writing functions for graphics -applications where hundreds of points are being computed, it is almost -certainly true that you will get the best performance by issuing -{\tt )set functions compile on}. - -\section{Piece-Wise Function Definitions} -\label{ugUserPiece} - -To move beyond functions defined in one line, we introduce in this -section functions that are defined piece-by-piece. That is, we say -``use this definition when the argument is such-and-such and use this -other definition when the argument is that-and-that.'' - -\subsection{A Basic Example} -\label{ugUserPieceBasic} - -There are many other ways to define a factorial function for -nonnegative integers. You might -\index{function!piece-wise definition} -say \index{piece-wise function definition} factorial of -$0$ is $1$, otherwise factorial of $n$ is $n$ times factorial of -$n-1$. Here is one way to do this in Axiom. - -Here is the value for $n = 0$. -\spadcommand{fact(0) == 1} -\returnType{Type: Void} - -Here is the value for $n > 0$. The vertical bar ``{\tt |}'' means ``such -that''. \index{such that} -\spadcommand{fact(n | n > 0) == n * fact(n - 1)} -\returnType{Type: Void} - -What is the value for $n = 3$? -\spadcommand{fact(3)} -\begin{verbatim} - Compiling function fact with type Integer -> Integer - Compiling function fact as a recurrence relation. -\end{verbatim} -$$ -6 -$$ -\returnType{Type: PositiveInteger} - -What is the value for $n = -3$? -\spadcommand{fact(-3)} -\begin{verbatim} - You did not define fact for argument -3 . -\end{verbatim} - -Now for a second definition. Here is the value for $n = 0$. -\spadcommand{facto(0) == 1} -\returnType{Type: Void} - -Give an error message if $n < 0$. -\spadcommand{facto(n | n < 0) == error "arguments to facto must be non-negative"} -\returnType{Type: Void} - -Here is the value otherwise. -\spadcommand{facto(n) == n * facto(n - 1)} -\returnType{Type: Void} - -What is the value for $n = 7$? -\spadcommand{facto(3)} -\begin{verbatim} - Compiling function facto with type Integer -> Integer -\end{verbatim} -$$ -6 -$$ -\returnType{Type: PositiveInteger} - -What is the value for $n = -7$? -\spadcommand{facto(-7)} -\begin{verbatim} - Error signalled from user code in function facto: - arguments to facto must be non-negative -\end{verbatim} -\returnType{Type: PositiveInteger} - -To see the current piece-wise definition of a function, use -{\tt )display value}. -\spadcommand{)display value facto} -\begin{verbatim} - Definition: - facto 0 == 1 - facto (n | n < 0) == - error(arguments to facto must be non-negative) - facto n == n facto(n - 1) -\end{verbatim} - -In general a {\it piece-wise definition} of a function consists of two -or more parts. Each part gives a ``piece'' of the entire definition. -Axiom collects the pieces of a function as you enter them. When you -ask for a value of the function, it then ``glues'' the pieces together -to form a function. - -The two piece-wise definitions for the factorial function are examples -of recursive functions, that is, functions that are defined in terms -of themselves. Here is an interesting doubly-recursive function. -This function returns the value $11$ for all positive integer -arguments. - -Here is the first of two pieces. -\spadcommand{eleven(n | n < 1) == n + 11} -\returnType{Type: Void} - -And the general case. -\spadcommand{eleven(m) == eleven(eleven(m - 12))} -\returnType{Type: Void} - -Compute $elevens$, the infinite stream of values of $eleven$. -\spadcommand{elevens := [eleven(i) for i in 0..]} -$$ -\left[ -{11}, {11}, {11}, {11}, {11}, {11}, {11}, {11}, {11}, - {11}, \ldots -\right] -$$ -\returnType{Type: Stream Integer} - -What is the value at $n = 200$? -\spadcommand{elevens 200} -$$ -11 -$$ -\returnType{Type: PositiveInteger} - -What is the Axiom's definition of $eleven$? -\spadcommand{)display value eleven} -\begin{verbatim} - Definition: - eleven (m | m < 1) == m + 11 - eleven m == eleven(eleven(m - 12)) -\end{verbatim} - -\subsection{Picking Up the Pieces} -\label{ugUserPiecePicking} - -Here are the details about how Axiom creates a function from its -pieces. Axiom converts the $i$-th piece of a function definition -into a conditional expression of the form: -{\tt if} $\hbox{\it pred}_{i}$ {\tt then} $\hbox{\it expression}_{i}$. -If any new piece has a $\hbox{\it pred}_{i}$ that is -identical (after all variables are uniformly named) to -an earlier $\hbox{\it pred}_{j}$, the earlier piece is removed. -Otherwise, the new piece is always added at the end. - -\boxed{4.6in}{ -\vskip 0.1cm -If there are $n$ pieces to a function definition for $f$, the function -defined $f$ is: \newline -\hspace*{3pc} -{\tt if} $\hbox{\it pred}_{1}$ {\tt then} $\hbox{\it expression}_{1}$ {\tt else}\newline -\hspace*{6pc}. . . \newline -\hspace*{3pc} -{\tt if} $\hbox{\it pred}_{n}$ {\tt then} $\hbox{\it expression}_{n}$ {\tt else}\newline -\hspace*{3pc} -{\tt error "You did not define f for argument ."}\\ -} - -You can give definitions of any number of mutually recursive function -definitions, piece-wise or otherwise. No computation is done until -you ask for a value. When you do ask for a value, all the relevant -definitions are gathered, analyzed, and translated into separate -functions and compiled. - -Let's recall the definition of {\bf eleven} from -the previous section. -\spadcommand{eleven(n | n < 1) == n + 11} -\returnType{Type: Void} - -\spadcommand{eleven(m) == eleven(eleven(m - 12))} -\returnType{Type: Void} - -A similar doubly-recursive function below produces $-11$ for all -negative positive integers. If you haven't worked out why or how -{\bf eleven} works, the structure of this definition gives a clue. - -This definition we write as a block. -\begin{verbatim} -minusEleven(n) == - n >= 0 => n - 11 - minusEleven (5 + minusEleven(n + 7)) -\end{verbatim} -\returnType{Type: Void} - -Define $s(n)$ to be the sum of plus and minus ``eleven'' functions -divided by $n$. Since $11 - 11 = 0$, we define $s(0)$ to be $1$. -\spadcommand{s(0) == 1} -\returnType{Type: Void} - -And the general term. -\spadcommand{s(n) == (eleven(n) + minusEleven(n))/n} -\returnType{Type: Void} - -What are the first ten values of $s$? -\spadcommand{[s(n) for n in 0..]} -$$ -\left[ -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, \ldots -\right] -$$ -\returnType{Type: Stream Fraction Integer} - -%% interpreter puts the rule at the end - should fix - -% Oops! Evidently $s(0)$ should be $1$. -% Let's check the current definition of {\bf s} using {\tt )display}. - -% \spadcommand{)display value s} - -% Change the value at $n = 0$. - -% \spadcommand{s(0) == 1} - -% Now, what is the definition of {\bf s}? -% Note: {\it you can only replace a given piece if you give exactly the same -% predicate!} - -% \spadcommand{)display value s} - -Axiom can create infinite streams in the positive direction (for -example, for index values $0,1, \ldots$) or negative direction (for -example, for $0,-1,-2, \ldots$). Here we would like a -stream of values of $s(n)$ that is infinite in both directions. The -function $t(n)$ below returns the $n$-th term of the infinite stream -$$[s(0), s(1), s(-1), s(2), s(-2), \ldots]$$ -Its definition has three pieces. - -Define the initial term. -\spadcommand{t(1) == s(0)} -\returnType{Type: Void} - -The even numbered terms are the $s(i)$ for positive $i$. We use -``{\tt quo}'' rather than ``{\tt /}'' since we want the result to be -an integer. - -\spadcommand{t(n | even?(n)) == s(n quo 2)} -\returnType{Type: Void} - -Finally, the odd numbered terms are the $s(i)$ for negative $i$. In -piece-wise definitions, you can use different variables to define -different pieces. Axiom will not get confused. -\spadcommand{t(p) == s(- p quo 2)} -\returnType{Type: Void} - -Look at the definition of $t$. In the first piece, the variable $n$ -was used; in the second piece, $p$. Axiom always uses your last -variable to display your definitions back to you. -\spadcommand{)display value t} -\begin{verbatim} - Definition: - t 1 == s(0) - t (p | even?(p)) == s(p quo 2) - t p == s(- p quo 2) -\end{verbatim} - -Create a series of values of $s$ applied to -alternating positive and negative arguments. -\spadcommand{[t(i) for i in 1..]} -\begin{verbatim} - Compiling function s with type Integer -> Fraction Integer - Compiling function t with type PositiveInteger -> Fraction Integer -\end{verbatim} -$$ -\left[ -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, \ldots -\right] -$$ -\returnType{Type: Stream Fraction Integer} - -Evidently $t(n) = 1$ for all $i$. Check it at $n= 100$. - -\spadcommand{t(100)} -$$ -1 -$$ -\returnType{Type: Fraction Integer} - -\subsection{Predicates} -\label{ugUserPiecePred} - -We have already seen some examples of \index{function!predicate} -predicates \index{predicate!in function definition} -(\ref{ugUserPieceBasic} on page~\pageref{ugUserPieceBasic}). -Predicates are {\tt Boolean}-valued expressions and Axiom uses them -for filtering collections (see \ref{ugLangIts} on -page~\pageref{ugLangIts}) and for placing constraints on function -arguments. In this section we discuss their latter usage. - -The simplest use of a predicate is one you don't see at all. -\spadcommand{opposite 'right == 'left} -\returnType{Type: Void} - -Here is a longer way to give the ``opposite definition.'' -\spadcommand{opposite (x | x = 'left) == 'right} -\returnType{Type: Void} - -Try it out. -\spadcommand{for x in ['right,'left,'inbetween] repeat output opposite x} -\begin{verbatim} -Compiling function opposite with type - OrderedVariableList [right, left,inbetween] -> Symbol - left - right - -The function opposite is not defined for the given argument(s). -\end{verbatim} - -Explicit predicates tell Axiom that the given function definition -piece is to be applied if the predicate evaluates to {\tt true} for -the arguments to the function. You can use such ``constant'' -arguments for integers, \index{function!constant argument} strings, -and quoted symbols. \index{constant function argument} The {\tt -Boolean} values {\tt true} and {\tt false} can also be used if qualified with -``$@$'' or ``$\$$'' and {\tt Boolean}. The following are all valid -function definition fragments using constant arguments. -\begin{verbatim} -a(1) == ... -b("unramified") == ... -c('untested) == ... -d(true@Boolean) == ... -\end{verbatim} - -If a function has more than one argument, each argument can have its -own predicate. However, if a predicate involves two or more -arguments, it must be given {\it after} all the arguments mentioned in -the predicate have been given. You are always safe to give a single -predicate at the end of the argument list. - -A function involving predicates on two arguments. -\spadcommand{inFirstHalfQuadrant(x | x > 0,y | y < x) == true} -\returnType{Type: Void} - -This is incorrect as it gives a predicate on $y$ before the argument -$y$ is given. -\spadcommand{inFirstHalfQuadrant(x | x > 0 and y < x,y) == true} -\begin{verbatim} - 1 old definition(s) deleted for function or rule inFirstHalfQuadrant -\end{verbatim} -\returnType{Type: Void} - -It is always correct to write the predicate at the end. -\spadcommand{inFirstHalfQuadrant(x,y | x > 0 and y < x) == true} -\begin{verbatim} - 1 old definition(s) deleted for function or rule inFirstHalfQuadrant -\end{verbatim} -\returnType{Type: Void} - -Here is the rest of the definition. -\spadcommand{inFirstHalfQuadrant(x,y) == false} -\returnType{Type: Void} - -Try it out. -\spadcommand{[inFirstHalfQuadrant(i,3) for i in 1..5]} -\begin{verbatim} - Compiling function inFirstHalfQuadrant with type (PositiveInteger, - PositiveInteger) -> Boolean -\end{verbatim} -$$ -\left[ -{\tt false}, {\tt false}, {\tt false}, {\tt true}, {\tt true} -\right] -$$ -\returnType{Type: List Boolean} - -\section{Caching Previously Computed Results} -\label{ugUserCache} - -By default, Axiom does not save the values of any function. -\index{function!caching values} You can cause it to save values and -not to recompute unnecessarily \index{remembering function values} by -using {\tt )set functions cache}. \index{set functions cache} This -should be used before the functions are defined or, at least, before -they are executed. The word following ``cache'' should be $0$ to turn -off caching, a positive integer $n$ to save the last $n$ computed -values or ``all'' to save all computed values. If you then give a -list of names of functions, the caching only affects those functions. -Use no list of names or ``all'' when you want to define the default -behavior for functions not specifically mentioned in other -{\tt )set functions cache} statements. If you give no list of names, all -functions will have the caching behavior. If you explicitly turn on -caching for one or more names, you must explicitly turn off caching -for those names when you want to stop saving their values. - -This causes the functions {\bf f} and {\bf g} to have the last three -computed values saved. -\spadcommand{)set functions cache 3 f g} -\begin{verbatim} - function f will cache the last 3 values. - function g will cache the last 3 values. -\end{verbatim} - -This is a sample definition for {\bf f}. -\spadcommand{f x == factorial(2**x)} -\returnType{Type: Void} - -A message is displayed stating what {\bf f} will cache. -\spadcommand{f(4)} -\begin{verbatim} - Compiling function f with type PositiveInteger -> Integer - f will cache 3 most recently computed value(s). - -+++ |*1;f;1;G82322| redefined -\end{verbatim} -$$ -20922789888000 -$$ -\returnType{Type: PositiveInteger} - -This causes all other functions to have all computed values saved by default. -\spadcommand{)set functions cache all} -\begin{verbatim} - In general, interpreter functions will cache all values. -\end{verbatim} - -This causes all functions that have not been specifically cached in some way -to have no computed values saved. -\spadcommand{)set functions cache 0} -\begin{verbatim} - In general, functions will cache no returned values. -\end{verbatim} - -We also make {\bf f} and {\bf g} uncached. -\spadcommand{)set functions cache 0 f g} -\begin{verbatim} - Caching for function f is turned off - Caching for function g is turned off -\end{verbatim} - -\boxed{4.6in}{ -\vskip 0.1cm -Be careful about caching functions that have side effects. Such a -function might destructively modify the elements of an array or issue -a {\bf draw} command, for example. A function that you expect to -execute every time it is called should not be cached. Also, it is -highly unlikely that a function with no arguments should be cached.\\ -} - -You should also be careful about caching functions that depend on free -variables. See \ref{ugUserFreeLocal} on -page~\pageref{ugUserFreeLocal} for an example. - -\section{Recurrence Relations} -\label{ugUserRecur} - -One of the most useful classes of function are those defined via a -``recurrence relation.'' A {\it recurrence relation} makes each -successive \index{recurrence relation} value depend on some or all of -the previous values. A simple example is the ordinary ``factorial'' function: -\begin{verbatim} -fact(0) == 1 -fact(n | n > 0) == n * fact(n-1) -\end{verbatim} - -The value of $fact(10)$ depends on the value of $fact(9)$, $fact(9)$ -on $fact(8)$, and so on. Because it depends on only one previous -value, it is usually called a {\it first order recurrence relation.} -You can easily imagine a function based on two, three or more previous -values. The Fibonacci numbers are probably the most famous function -defined by a \index{Fibonacci numbers} second order recurrence relation. - -The library function {\bf fibonacci} computes Fibonacci numbers. -It is obviously optimized for speed. -\spadcommand{[fibonacci(i) for i in 0..]} -$$ -\left[ -0, 1, 1, 2, 3, 5, 8, {13}, {21}, {34}, \ldots -\right] -$$ -\returnType{Type: Stream Integer} - -Define the Fibonacci numbers ourselves using a piece-wise definition. -\spadcommand{fib(1) == 1} -\returnType{Type: Void} - -\spadcommand{fib(2) == 1} -\returnType{Type: Void} - -\spadcommand{fib(n) == fib(n-1) + fib(n-2)} -\returnType{Type: Void} - -As defined, this recurrence relation is obviously doubly-recursive. -To compute $fib(10)$, we need to compute $fib(9)$ and $fib(8)$. And -to $fib(9)$, we need to compute $fib(8)$ and $fib(7)$. And so on. It -seems that to compute $fib(10)$ we need to compute $fib(9)$ once, -$fib(8)$ twice, $fib(7)$ three times. Look familiar? The number of -function calls needed to compute {\it any} second order recurrence -relation in the obvious way is exactly $fib(n)$. These numbers grow! -For example, if Axiom actually did this, then $fib(500)$ requires more -than $10^{104}$ function calls. And, given all -this, our definition of {\bf fib} obviously could not be used to -calculate the five-hundredth Fibonacci number. - -Let's try it anyway. -\spadcommand{fib(500)} -\begin{verbatim} - Compiling function fib with type Integer -> PositiveInteger - Compiling function fib as a recurrence relation. - -13942322456169788013972438287040728395007025658769730726410_ -8962948325571622863290691557658876222521294125 -\end{verbatim} - -\returnType{Type: PositiveInteger} - -Since this takes a short time to compute, it obviously didn't do as -many as $10^{104}$ operations! By default, Axiom transforms any -recurrence relation it recognizes into an iteration. Iterations are -efficient. To compute the value of the $n$-th term of a recurrence -relation using an iteration requires only $n$ function calls. Note -that if you compare the speed of our {\bf fib} function to the library -function, our version is still slower. This is because the library -\spadfunFrom{fibonacci}{IntegerNumberTheoryFunctions} uses a -``powering algorithm'' with a computing time proportional to -$\log^3(n)$ to compute {\tt fibonacci(n)}. - -To turn off this special recurrence relation compilation, issue -\index{set function recurrence} -\begin{verbatim} -)set functions recurrence off -\end{verbatim} -To turn it back on, substitute ``{\tt on}'' for ``{\tt off}''. - -The transformations that Axiom uses for {\bf fib} caches the last two -values. For a more general $k$-th order recurrence relation, Axiom -caches the last $k$ values. If, after computing a value for {\bf -fib}, you ask for some larger value, Axiom picks up the cached values -and continues computing from there. See \ref{ugUserFreeLocal} on -page~\pageref{ugUserFreeLocal} for an example of a function definition -that has this same behavior. Also see \ref{ugUserCache} on -page~\pageref{ugUserCache} for a more general discussion of how you -can cache function values. - -Recurrence relations can be used for defining recurrence relations -involving polynomials, rational functions, or anything you like. -Here we compute the infinite stream of Legendre polynomials. - -The Legendre polynomial of degree $0.$ -\spadcommand{p(0) == 1} -\returnType{Type: Void} - -The Legendre polynomial of degree $1.$ -\spadcommand{p(1) == x} -\returnType{Type: Void} - -The Legendre polynomial of degree $n$. -\spadcommand{p(n) == ((2*n-1)*x*p(n-1) - (n-1)*p(n-2))/n} -\returnType{Type: Void} - -Compute the Legendre polynomial of degree $6.$ -\spadcommand{p(6)} -\begin{verbatim} - Compiling function p with type Integer -> Polynomial Fraction - Integer - Compiling function p as a recurrence relation. -\end{verbatim} -$$ -{{{231} \over {16}} \ {x \sp 6}} -{{{315} \over {16}} \ {x \sp 4}}+{{{105} -\over {16}} \ {x \sp 2}} -{5 \over {16}} -$$ -\returnType{Type: Polynomial Fraction Integer} - -\section{Making Functions from Objects} -\label{ugUserMake} - -There are many times when you compute a complicated expression and -then wish to use that expression as the body of a function. Axiom -provides an operation called {\bf function} to do \index{function!from -an object} this. \index{function!made by function @{made by -{\bf function}}} It creates a function object and places it into the -workspace. There are several versions, depending on how many -arguments the function has. The first argument to {\bf function} is -always the expression to be converted into the function body, and the -second is always the name to be used for the function. For more -information, see section \ref{MakeFunctionXmpPage} on -page~\pageref{MakeFunctionXmpPage}. - -Start with a simple example of a polynomial in three variables. -\spadcommand{p := -x + y**2 - z**3} -$$ --{z \sp 3}+{y \sp 2} -x -$$ -\returnType{Type: Polynomial Integer} - -To make this into a function of no arguments that simply returns the -polynomial, use the two argument form of {\bf function}. -\spadcommand{function(p,'f0)} -$$ -f0 -$$ -\returnType{Type: Symbol} - -To avoid possible conflicts (see below), it is a good idea to -quote always this second argument. -\spadcommand{f0} -$$ -f0 \ {\left( -\right)} -\ == \ {-{z \sp 3}+{y \sp 2} -x} -$$ -\returnType{Type: FunctionCalled f0} - -This is what you get when you evaluate the function. -\spadcommand{f0()} -$$ --{z \sp 3}+{y \sp 2} -x -$$ -\returnType{Type: Polynomial Integer} - -To make a function in $x$, use a version of {\bf function} that takes -three arguments. The last argument is the name of the variable to use -as the parameter. Typically, this variable occurs in the expression -and, like the function name, you should quote it to avoid possible confusion. -\spadcommand{function(p,'f1,'x)} -$$ -f1 -$$ -\returnType{Type: Symbol} - -This is what the new function looks like. -\spadcommand{f1} -$$ -f1 \ x \ == \ {-{z \sp 3}+{y \sp 2} -x} -$$ -\returnType{Type: FunctionCalled f1} - -This is the value of {\bf f1} at $x = 3$. Notice that the return type -of the function is {\tt Polynomial (Integer)}, the same as $p$. -\spadcommand{f1(3)} -\begin{verbatim} - Compiling function f1 with type PositiveInteger -> Polynomial - Integer -\end{verbatim} -$$ --{z \sp 3}+{y \sp 2} -3 -$$ -\returnType{Type: Polynomial Integer} - -To use $x$ and $y$ as parameters, use the four argument form of {\bf function}. -\spadcommand{function(p,'f2,'x,'y)} -$$ -f2 -$$ -\returnType{Type: Symbol} - -\spadcommand{f2} -$$ -f2 \ {\left( x, y -\right)} -\ == \ {-{z \sp 3}+{y \sp 2} -x} -$$ -\returnType{Type: FunctionCalled f2} - -Evaluate $f2$ at $x = 3$ and $y = 0$. The return type of {\bf f2} is -still {\tt Polynomial(Integer)} because the variable $z$ is still -present and not one of the parameters. -\spadcommand{f2(3,0)} -$$ --{z \sp 3} -3 -$$ -\returnType{Type: Polynomial Integer} - -Finally, use all three variables as parameters. There is no five -argument form of {\bf function}, so use the one with three arguments, -the third argument being a list of the parameters. -\spadcommand{function(p,'f3,['x,'y,'z])} -$$ -f3 -$$ -\returnType{Type: Symbol} - -Evaluate this using the same values for $x$ and $y$ as above, but let -$z$ be $-6$. The result type of {\bf f3} is {\tt Integer}. -\spadcommand{f3} -$$ -f3 \ {\left( x, y, z -\right)} -\ == \ {-{z \sp 3}+{y \sp 2} -x} -$$ -\returnType{Type: FunctionCalled f3} - -\spadcommand{f3(3,0,-6)} -\begin{verbatim} - Compiling function f3 with type (PositiveInteger,NonNegativeInteger, - Integer) -> Integer -\end{verbatim} -$$ -213 -$$ -\returnType{Type: PositiveInteger} - -The four functions we have defined via $p$ have been undeclared. To -declare a function whose body is to be generated by -\index{function!declaring} {\bf function}, issue the declaration -{\it before} the function is created. -\spadcommand{g: (Integer, Integer) -> Float} -\returnType{Type: Void} - -\spadcommand{D(sin(x-y)/cos(x+y),x)} -$$ -{-{{\sin -\left( -{{y -x}} -\right)} -\ {\sin -\left( -{{y+x}} -\right)}}+{{\cos -\left( -{{y -x}} -\right)} -\ {\cos -\left( -{{y+x}} -\right)}}} -\over {{\cos -\left( -{{y+x}} -\right)} -\sp 2} -$$ -\returnType{Type: Expression Integer} - -\spadcommand{function(\%,'g,'x,'y)} -$$ -g -$$ -\returnType{Type: Symbol} - -\spadcommand{g} -$$ -g \ {\left( x, y -\right)} -\ == \ {{-{{\sin -\left( -{{y -x}} -\right)} -\ {\sin -\left( -{{y+x}} -\right)}}+{{\cos -\left( -{{y -x}} -\right)} -\ {\cos -\left( -{{y+x}} -\right)}}} -\over {{\cos -\left( -{{y+x}} -\right)} -\sp 2}} -$$ -\returnType{Type: FunctionCalled g} - -It is an error to use $g$ without the quote in the penultimate -expression since $g$ had been declared but did not have a value. -Similarly, since it is common to overuse variable names like $x$, $y$, -and so on, you avoid problems if you always quote the variable names -for {\bf function}. In general, if $x$ has a value and you use $x$ -without a quote in a call to {\bf function}, then Axiom does not know -what you are trying to do. - -What kind of object is allowable as the first argument to -{\bf function}? Let's use the Browse facility of HyperDoc to find out. -\index{Browse@Browse} At the main Browse menu, enter the string -{\tt function} and then click on {\bf Operations.} The exposed operations -called {\bf function} all take an object whose type belongs to -category {\tt ConvertibleTo InputForm}. What domains are those? Go -back to the main Browse menu, erase {\tt function}, enter -{\tt ConvertibleTo} in the input area, and click on {\bf categories} on the -{\tt Constructors} line. At the bottom of the page, enter -{\tt InputForm} in the input area following {\bf S =}. Click on -{\tt Cross Reference} and then on {\tt Domains}. -The list you see contains over forty domains that belong to the -category {\tt ConvertibleTo InputForm}. Thus you can use {\bf function} -for {\tt Integer}, {\tt Float}, {\tt String}, {\tt Complex}, -{\tt Expression}, and so on. - -\section{Functions Defined with Blocks} -\label{ugUserBlocks} - -You need not restrict yourself to functions that only fit on one line -or are written in a piece-wise manner. The body of the function can -be a block, as discussed in \ref{ugLangBlocks} on page~\pageref{ugLangBlocks}. - -Here is a short function that swaps two elements of a list, array or vector. -\begin{verbatim} -swap(m,i,j) == - temp := m.i - m.i := m.j - m.j := temp -\end{verbatim} -\returnType{Type: Void} - -The significance of {\bf swap} is that it has a destructive -effect on its first argument. -\spadcommand{k := [1,2,3,4,5]} -$$ -\left[ -1, 2, 3, 4, 5 -\right] -$$ -\returnType{Type: List PositiveInteger} - -\spadcommand{swap(k,2,4)} -\begin{verbatim} - Compiling function swap with type (List PositiveInteger, - PositiveInteger,PositiveInteger) -> PositiveInteger -\end{verbatim} -$$ -2 -$$ -\returnType{Type: PositiveInteger} - -You see that the second and fourth elements are interchanged. -\spadcommand{k} -$$ -\left[ -1, 4, 3, 2, 5 -\right] -$$ -\returnType{Type: List PositiveInteger} - -Using this, we write a couple of different sort functions. First, a -simple bubble sort. \index{sort!bubble} The operation -\spadopFrom{\#}{List} returns the number of elements in an aggregate. -\begin{verbatim} -bubbleSort(m) == - n := #m - for i in 1..(n-1) repeat - for j in n..(i+1) by -1 repeat - if m.j < m.(j-1) then swap(m,j,j-1) - m -\end{verbatim} -\returnType{Type: Void} - -Let this be the list we want to sort. -\spadcommand{m := [8,4,-3,9]} -$$ -\left[ -8, 4, -3, 9 -\right] -$$ -\returnType{Type: List Integer} - -This is the result of sorting. -\spadcommand{bubbleSort(m)} -\begin{verbatim} - Compiling function swap with type (List Integer,Integer,Integer) -> - Integer - -+++ |*3;swap;1;G82322| redefined - Compiling function bubbleSort with type List Integer -> List Integer -\end{verbatim} -$$ -\left[ --3, 4, 8, 9 -\right] -$$ -\returnType{Type: List Integer} - -Moreover, $m$ is destructively changed to be the sorted version. -\spadcommand{m} -$$ -\left[ --3, 4, 8, 9 -\right] -$$ -\returnType{Type: List Integer} - -This function implements an insertion sort. \index{sort!insertion} -The basic idea is to traverse the list and insert the $i$-th element -in its correct position among the $i-1$ previous elements. Since we -start at the beginning of the list, the list elements before the -$i$-th element have already been placed in ascending order. -\begin{verbatim} -insertionSort(m) == - for i in 2..#m repeat - j := i - while j > 1 and m.j < m.(j-1) repeat - swap(m,j,j-1) - j := j - 1 - m -\end{verbatim} -\returnType{Type: Void} - -As with our bubble sort, this is a destructive function. -\spadcommand{m := [8,4,-3,9]} -$$ -\left[ -8, 4, -3, 9 -\right] -$$ -\returnType{Type: List Integer} - -\spadcommand{insertionSort(m)} -\begin{verbatim} - Compiling function insertionSort with type List Integer -> List - Integer -\end{verbatim} -$$ -\left[ --3, 4, 8, 9 -\right] -$$ -\returnType{Type: List Integer} - -\spadcommand{m} -$$ -\left[ --3, 4, 8, 9 -\right] -$$ -\returnType{Type: List Integer} - -Neither of the above functions is efficient for sorting large lists -since they reference elements by asking for the $j$-th element of the -structure $m$. - -Here is a more efficient bubble sort for lists. -\begin{verbatim} -bubbleSort2(m: List Integer): List Integer == - null m => m - l := m - while not null (r := l.rest) repeat - r := bubbleSort2 r - x := l.first - if x < r.first then - l.first := r.first - r.first := x - l.rest := r - l := l.rest - m - - Function declaration bubbleSort2 : List Integer -> List Integer has - been added to workspace. -\end{verbatim} -\returnType{Type: Void} - -Try it out. -\spadcommand{bubbleSort2 [3,7,2]} -$$ -\left[ -7, 3, 2 -\right] -$$ -\returnType{Type: List Integer} - -This definition is both recursive and iterative, and is tricky! -Unless you are {\it really} curious about this definition, we suggest -you skip immediately to the next section. - -Here are the key points in the definition. First notice that if you -are sorting a list with less than two elements, there is nothing to -do: just return the list. This definition returns immediately if -there are zero elements, and skips the entire {\tt while} loop if there is -just one element. - -The second point to realize is that on each outer iteration, the -bubble sort ensures that the minimum element is propagated leftmost. -Each iteration of the {\tt while} loop calls {\bf bubbleSort2} recursively -to sort all but the first element. When finished, the minimum element -is either in the first or second position. The conditional expression -ensures that it comes first. If it is in the second, then a swap -occurs. In any case, the {\bf rest} of the original list must be -updated to hold the result of the recursive call. - -\section{Free and Local Variables} -\label{ugUserFreeLocal} - -When you want to refer to a variable that is not local to your -function, use a ``{\tt free}'' declaration. \index{free} Variables -declared to be {\tt free} \index{free variable} are assumed to be defined -globally \index{variable!free} in the \index{variable!global} -workspace. \index{global variable} - -This is a global workspace variable. -\spadcommand{counter := 0} -$$ -0 -$$ -\returnType{Type: NonNegativeInteger} - -This function refers to the global $counter$. -\begin{verbatim} -f() == - free counter - counter := counter + 1 -\end{verbatim} -\returnType{Type: Void} - -The global $counter$ is incremented by $1$. -\spadcommand{f()} -\begin{verbatim} - Compiling function f with type () -> NonNegativeInteger - -+++ |*0;f;1;G82322| redefined -\end{verbatim} -$$ -1 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{counter} -$$ -1 -$$ -\returnType{Type: NonNegativeInteger} - - -Usually Axiom can tell that you mean to refer to a global variable and -so {\tt free} isn't always necessary. However, for clarity and the sake -of self-documentation, we encourage you to use it. - -Declare a variable to be ``{\tt local}'' when you do not want to refer to -\index{variable!local} a global variable by the same name. -\index{local variable} - -This function uses $counter$ as a local variable. -\begin{verbatim} -g() == - local counter - counter := 7 -\end{verbatim} -\returnType{Type: Void} - -Apply the function. -\spadcommand{g()} -$$ -7 -$$ -\returnType{Type: PositiveInteger} - -Check that the global value of $counter$ is unchanged. -\spadcommand{counter} -$$ -1 -$$ -\returnType{Type: NonNegativeInteger} - -Parameters to a function are local variables in the function. Even if -you issue a {\tt free} declaration for a parameter, it is still local. - -What happens if you do not declare that a variable $x$ in the body of -your function is {\tt local} or {\tt free}? Well, Axiom decides on this basis: -\begin{enumerate} -\item Axiom scans your function line-by-line, from top-to-bottom. -The right-hand side of an assignment is looked at before the left-hand -side. -\item If $x$ is referenced before it is assigned a value, it is a -{\tt free} (global) variable. -\item If $x$ is assigned a value before it is referenced, it is a -{\tt local} variable. -\end{enumerate} - -Set two global variables to 1. -\spadcommand{a := b := 1} -$$ -1 -$$ -\returnType{Type: PositiveInteger} - -Refer to $a$ before it is assigned a value, but assign a value to $b$ -before it is referenced. -\begin{verbatim} -h() == - b := a + 1 - a := b + a -\end{verbatim} -\returnType{Type: Void} - -Can you predict this result? -\spadcommand{h()} -\begin{verbatim} - Compiling function h with type () -> PositiveInteger - -+++ |*0;h;1;G82322| redefined -\end{verbatim} -$$ -3 -$$ -\returnType{Type: PositiveInteger} - -How about this one? -\spadcommand{[a, b]} -$$ -\left[ -3, 1 -\right] -$$ -\returnType{Type: List PositiveInteger} - -What happened? In the first line of the function body for $h$, $a$ is -referenced on the right-hand side of the assignment. Thus $a$ is a -free variable. The variable $b$ is not referenced in that line, but -it is assigned a value. Thus $b$ is a local variable and is given the -value $a + 1 = 2$. In the second line, the free variable $a$ is -assigned the value $b + a$ which equals $2 + 1 = 3.$ This is the value -returned by the function. Since $a$ was free in {\bf h}, the global -variable $a$ has value $3.$ Since $b$ was local in {\bf h}, the global -variable $b$ is unchanged---it still has the value $1$. - -It is good programming practice always to declare global variables. -However, by far the most common situation is to have local variables -in your functions. No declaration is needed for this situation, but -be sure to initialize their values. - -Be careful if you use free variables and you cache the value of your -function (see \ref{ugUserCache} on page~\pageref{ugUserCache}). -Caching {\it only} checks if the values of the function arguments are -the same as in a function call previously seen. It does not check if -any of the free variables on which the function depends have changed -between function calls. - -Turn on caching for {\bf p}. -\spadcommand{)set fun cache all p} -\begin{verbatim} - function p will cache all values. -\end{verbatim} - -Define {\bf p} to depend on the free variable $N$. -\spadcommand{p(i,x) == ( free N; reduce( + , [ (x-i)**n for n in 1..N ] ) )} -\returnType{Type: Void} - -Set the value of $N$. -\spadcommand{N := 1} -$$ -1 -$$ -\returnType{Type: PositiveInteger} - -Evaluate {\bf p} the first time. -\spadcommand{p(0, x)} -$$ -x -$$ -\returnType{Type: Polynomial Integer} - -Change the value of $N$. -\spadcommand{N := 2} -$$ -2 -$$ -\returnType{Type: PositiveInteger} - -Evaluate {\bf p} the second time. -\spadcommand{p(0, x)} -$$ -x -$$ -\returnType{Type: Polynomial Integer} - -If caching had been turned off, the second evaluation would have -reflected the changed value of $N$. - -Turn off caching for {\bf p}. -\spadcommand{)set fun cache 0 p} -\begin{verbatim} - Caching for function p is turned off -\end{verbatim} - -Axiom does not allow {\it fluid variables}, that is, variables -\index{variable!fluid} bound by a function $f$ that can be referenced -by functions called by $f$. \index{fluid variable} - -Values are passed to functions by {\it reference}: a pointer to the -value is passed rather than a copy of the value or a pointer to a -copy. - -This is a global variable that is bound to a record object. -\spadcommand{r : Record(i : Integer) := [1]} -$$ -\left[ -{i=1} -\right] -$$ -\returnType{Type: Record(i: Integer)} - -This function first modifies the one component of its record argument -and then rebinds the parameter to another record. -\begin{verbatim} -resetRecord rr == - rr.i := 2 - rr := [10] -\end{verbatim} -\returnType{Type: Void} - -Pass $r$ as an argument to {\bf resetRecord}. -\spadcommand{resetRecord r} -$$ -\left[ -{i={10}} -\right] -$$ -\returnType{Type: Record(i: Integer)} - -The value of $r$ was changed by the expression $rr.i := 2$ but not by -$rr := [10]$. -\spadcommand{r} -$$ -\left[ -{i=2} -\right] -$$ -\returnType{Type: Record(i: Integer)} - -To conclude this section, we give an iterative definition of -\index{Fibonacci numbers} a function that computes Fibonacci numbers. -This definition approximates the definition into which Axiom -transforms the recurrence relation definition of {\bf fib} in -\ref{ugUserRecur} on page~\pageref{ugUserRecur}. - -Global variables {\tt past} and {\tt present} are used to hold the last -computed Fibonacci numbers. -\spadcommand{past := present := 1} -$$ -1 -$$ -\returnType{Type: PositiveInteger} - -Global variable $index$ gives the current index of $present$. -\spadcommand{index := 2} -$$ -2 -$$ -\returnType{Type: PositiveInteger} - -Here is a recurrence relation defined in terms of these three global -variables. -\begin{verbatim} -fib(n) == - free past, present, index - n < 3 => 1 - n = index - 1 => past - if n < index-1 then - (past,present) := (1,1) - index := 2 - while (index < n) repeat - (past,present) := (present, past+present) - index := index + 1 - present -\end{verbatim} -\returnType{Type: Void} - -Compute the infinite stream of Fibonacci numbers. -\spadcommand{fibs := [fib(n) for n in 1..]} -$$ -\left[ -1, 1, 2, 3, 5, 8, {13}, {21}, {34}, {55}, -\ldots -\right] -$$ -\returnType{Type: Stream PositiveInteger} - -What is the 1000th Fibonacci number? -\spadcommand{fibs 1000} -\begin{verbatim} - 434665576869374564356885276750406258025646605173717804024_ - 8172908953655541794905189040387984007925516929592259308_ - 0322634775209689623239873322471161642996440906533187938_ - 298969649928516003704476137795166849228875 -\end{verbatim} -\returnType{Type: PositiveInteger} - -As an exercise, we suggest you write a function in an iterative style -that computes the value of the recurrence relation -$p(n) = p(n-1) - 2 \, p(n-2) + 4 \, p(n-3)$ -having the initial values -$p(1) = 1,\, p(2) = 3 \hbox{ and } p(3) = 9.$ -How would you write the function using an element {\tt OneDimensionalArray} -or {\tt Vector} to hold the previously computed values? - -\section{Anonymous Functions} -\label{ugUserAnon} - -\boxed{4.6in}{ -\vskip 0.1cm -An {\it anonymous function} is a function that is -\index{function!anonymous} defined \index{anonymous function} by -giving a list of parameters, the ``maps-to'' compound -\index{+-> @{\tt +->}} symbol ``{\tt +->}'' -(from the mathematical symbol $\mapsto$), and -by an expression involving the parameters, the evaluation of which -determines the return value of the function. - -\begin{center} -{\tt ( $\hbox{\it parm}_{1}$, $\hbox{\it parm}_{2}$, \ldots, -$\hbox{\it parm}_{N}$ ) {\tt +->} {\it expression}} -\end{center} -{\ }%force a blank line -} - -You can apply an anonymous function in several ways. -\begin{enumerate} -\item Place the anonymous function definition in parentheses -directly followed by a list of arguments. -\item Assign the anonymous function to a variable and then -use the variable name when you would normally use a function name. -\item Use ``{\tt ==}'' to use the anonymous function definition as -the arguments and body of a regular function definition. -\item Have a named function contain a declared anonymous function and -use the result returned by the named function. -\end{enumerate} - -\subsection{Some Examples} -\label{ugUserAnonExamp} - -Anonymous functions are particularly useful for defining functions -``on the fly.'' That is, they are handy for simple functions that are -used only in one place. In the following examples, we show how to -write some simple anonymous functions. - -This is a simple absolute value function. -\spadcommand{x +-> if x < 0 then -x else x} -$$ -x \mapsto {if \ {x<0} \ {\begin{array}{l} {then \ -x} \\ -{else \ x} -\end{array} -}} -$$ -\returnType{Type: AnonymousFunction} - -\spadcommand{abs1 := \%} -$$ -x \mapsto {if \ {x<0} \ {\begin{array}{l} {then \ -x} \\ -{else \ x} -\end{array} -}} -$$ -\returnType{Type: AnonymousFunction} - -This function returns {\tt true} if the absolute value of -the first argument is greater than the absolute value of the -second, {\tt false} otherwise. - -\spadcommand{(x,y) +-> abs1(x) > abs1(y)} -$$ -{\left( x, y -\right)} -\mapsto {{abs1 -\left( -{y} -\right)}<{abs1 -\left( -{x} -\right)}} -$$ -\returnType{Type: AnonymousFunction} - -We use the above function to ``sort'' a list of integers. -\spadcommand{sort(\%,[3,9,-4,10,-3,-1,-9,5])} -$$ -\left[ -{10}, -9, 9, 5, -4, -3, 3, -1 -\right] -$$ -\returnType{Type: List Integer} - -This function returns $1$ if $i + j$ is even, $-1$ otherwise. -\spadcommand{ev := ( (i,j) +-> if even?(i+j) then 1 else -1)} -$$ -{\left( i, j -\right)} -\mapsto {if \ {even? -\left( -{{i+j}} -\right)} -\ {\begin{array}{l} {then \ 1} \\ -{else \ -1} -\end{array} -}} -$$ -\returnType{Type: AnonymousFunction} - -We create a four-by-four matrix containing $1$ or $-1$ depending on -whether the row plus the column index is even or not. -\spadcommand{matrix([ [ev(row,col) for row in 1..4] for col in 1..4])} -$$ -\left[ -\begin{array}{cccc} -1 & -1 & 1 & -1 \\ --1 & 1 & -1 & 1 \\ -1 & -1 & 1 & -1 \\ --1 & 1 & -1 & 1 -\end{array} -\right] -$$ -\returnType{Type: Matrix Integer} - -This function returns {\tt true} if a polynomial in $x$ has multiple -roots, {\tt false} otherwise. It is defined and applied in the same -expression. -\spadcommand{( p +-> not one?(gcd(p,D(p,x))) )(x**2+4*x+4)} -$$ -{\tt true} -$$ -\returnType{Type: Boolean} - -This and the next expression are equivalent. -\spadcommand{g(x,y,z) == cos(x + sin(y + tan(z)))} -\returnType{Type: Void} - -The one you use is a matter of taste. -\spadcommand{g == (x,y,z) +-> cos(x + sin(y + tan(z)))} -\begin{verbatim} - 1 old definition(s) deleted for function or rule g -\end{verbatim} -\returnType{Type: Void} - -\subsection{Declaring Anonymous Functions} -\label{ugUserAnonDeclare} - -If you declare any of the arguments you must declare all of them. Thus, -\begin{verbatim} -(x: INT,y): FRAC INT +-> (x + 2*y)/(y - 1) -\end{verbatim} -is not legal. - -This is an example of a fully declared anonymous function. -\index{function!declaring} \index{function!anonymous!declaring} The -output shown just indicates that the object you created is a -particular kind of map, that is, function. -\spadcommand{(x: INT,y: INT): FRAC INT +-> (x + 2*y)/(y - 1)} -$$ -\mbox{theMap(...)} -$$ -\returnType{Type: ((Integer,Integer) {\tt ->} Fraction Integer)} - -Axiom allows you to declare the arguments and not declare -the return type. -\spadcommand{(x: INT,y: INT) +-> (x + 2*y)/(y - 1)} -$$ -\mbox{theMap(...)} -$$ -\returnType{Type: ((Integer,Integer) {\tt ->} Fraction Integer)} - -The return type is computed from the types of the arguments and the -body of the function. You cannot declare the return type if you do -not declare the arguments. Therefore, -\begin{verbatim} -(x,y): FRAC INT +-> (x + 2*y)/(y - 1) -\end{verbatim} - -is not legal. This and the next expression are equivalent. -\spadcommand{h(x: INT,y: INT): FRAC INT == (x + 2*y)/(y - 1)} -\begin{verbatim} - Function declaration h : (Integer,Integer) -> Fraction Integer - has been added to workspace. -\end{verbatim} -\returnType{Type: Void} - -The one you use is a matter of taste. -\spadcommand{h == (x: INT,y: INT): FRAC INT +-> (x + 2*y)/(y - 1)} -\begin{verbatim} - Function declaration h : (Integer,Integer) -> Fraction Integer - has been added to workspace. - 1 old definition(s) deleted for function or rule h -\end{verbatim} -\returnType{Type: Void} - -When should you declare an anonymous function? -\begin{enumerate} -\item If you use an anonymous function and Axiom can't figure out what -you are trying to do, declare the function. -\item If the function has nontrivial argument types or a nontrivial -return type that Axiom may be able to determine eventually, but you -are not willing to wait that long, declare the function. -\item If the function will only be used for arguments of specific types -and it is not too much trouble to declare the function, do so. -\item If you are using the anonymous function as an argument to another -function (such as {\bf map} or {\bf sort}), consider declaring the function. -\item If you define an anonymous function inside a named function, -you {\it must} declare the anonymous function. -\end{enumerate} - -This is an example of a named function for integers that returns a -function. -\spadcommand{addx x == ((y: Integer): Integer +-> x + y)} -\returnType{Type: Void} - -We define {\bf g} to be a function that adds $10$ to its -argument. -\spadcommand{g := addx 10} -\begin{verbatim} - Compiling function addx with type - PositiveInteger -> (Integer -> Integer) -\end{verbatim} -$$ -\mbox{theMap(...)} -$$ -\returnType{Type: (Integer {\tt ->} Integer)} - -Try it out. -\spadcommand{g 3} -$$ -13 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{g(-4)} -$$ -6 -$$ -\returnType{Type: PositiveInteger} - -\index{function!anonymous!restrictions} -An anonymous function cannot be recursive: since it does not have a -name, you cannot even call it within itself! If you place an -anonymous function inside a named function, the anonymous function -must be declared. - -\section{Example: A Database} -\label{ugUserDatabase} - -This example shows how you can use Axiom to organize a database of -lineage data and then query the database for relationships. - -The database is entered as ``assertions'' that are really pieces of a -function definition. -\spadcommand{children("albert") == ["albertJr","richard","diane"]} -\returnType{Type: Void} - -Each piece $children(x) == y$ means ``the children of $x$ are $y$''. -\spadcommand{children("richard") == ["douglas","daniel","susan"]} -\returnType{Type: Void} - -This family tree thus spans four generations. -\spadcommand{children("douglas") == ["dougie","valerie"]} -\returnType{Type: Void} - -Say ``no one else has children.'' -\spadcommand{children(x) == []} -\returnType{Type: Void} - -We need some functions for computing lineage. Start with {\tt childOf}. -\spadcommand{childOf(x,y) == member?(x,children(y))} -\returnType{Type: Void} - -To find the {\tt parentOf} someone, you have to scan the database of -people applying {\tt children}. -\begin{verbatim} -parentOf(x) == - for y in people repeat - (if childOf(x,y) then return y) - "unknown" -\end{verbatim} -\returnType{Type: Void} - -And a grandparent of $x$ is just a parent of a parent of $x$. -\spadcommand{grandParentOf(x) == parentOf parentOf x} -\returnType{Type: Void} - -The grandchildren of $x$ are the people $y$ such that $x$ is a -grandparent of $y$. -\spadcommand{grandchildren(x) == [y for y in people | grandParentOf(y) = x]} -\returnType{Type: Void} - -Suppose you want to make a list of all great-grandparents. Well, a -great-grandparent is a grandparent of a person who has children. - -\begin{verbatim} -greatGrandParents == [x for x in people | - reduce(_or, - [not empty? children(y) for y in grandchildren(x)],false)] -\end{verbatim} -\returnType{Type: Void} - -Define {\tt descendants} to include the parent as well. -\begin{verbatim} -descendants(x) == - kids := children(x) - null kids => [x] - concat(x,reduce(concat,[descendants(y) - for y in kids],[])) -\end{verbatim} -\returnType{Type: Void} - -Finally, we need a list of people. Since all people are descendants -of ``albert'', let's say so. -\spadcommand{people == descendants "albert"} -\returnType{Type: Void} - -We have used ``{\tt ==}'' to define the database and some functions to -query the database. But no computation is done until we ask for some -information. Then, once and for all, the functions are analyzed and -compiled to machine code for run-time efficiency. Notice that no -types are given anywhere in this example. They are not needed. - -Who are the grandchildren of ``richard''? -\spadcommand{grandchildren "richard"} -\begin{verbatim} -Compiling function children with type String -> List String -Compiling function descendants with type String -> List String -Compiling body of rule people to compute value of type List String -Compiling function childOf with type (String,String) -> Boolean -Compiling function parentOf with type String -> String -Compiling function grandParentOf with type String -> String -Compiling function grandchildren with type String -> List String -\end{verbatim} -$$ -\left[ -\mbox{\tt "dougie"} , \mbox{\tt "valerie"} -\right] -$$ -\returnType{Type: List String} - -Who are the great-grandparents? -\spadcommand{greatGrandParents} -\begin{verbatim} -Compiling body of rule greatGrandParents to compute value of - type List String -\end{verbatim} -$$ -\left[ -\mbox{\tt "albert"} -\right] -$$ -\returnType{Type: List String} - -\section{Example: A Famous Triangle} -\label{ugUserTriangle} - -In this example we write some functions that display Pascal's -triangle. \index{Pascal's triangle} It demonstrates the use of -piece-wise definitions and some output operations you probably haven't -seen before. - -To make these output operations available, we have to {\it expose} the -domain {\tt OutputForm}. \index{OutputForm} See -\ref{ugTypesExpose} on page~\pageref{ugTypesExpose} -for more information about exposing domains and packages. -\spadcommand{)set expose add constructor OutputForm} -\begin{verbatim} - OutputForm is now explicitly exposed in frame G82322 -\end{verbatim} - -Define the values along the first row and any column $i$. -\spadcommand{pascal(1,i) == 1} -\returnType{Type: Void} - -Define the values for when the row and column index $i$ are equal. -Repeating the argument name indicates that the two index values are equal. -\spadcommand{pascal(n,n) == 1} -\returnType{Type: Void} - -\begin{verbatim} -pascal(i,j | 1 < i and i < j) == - pascal(i-1,j-1)+pascal(i,j-1) -\end{verbatim} -\returnType{Type: Void} - -Now that we have defined the coefficients in Pascal's triangle, let's -write a couple of one-liners to display it. - -First, define a function that gives the $n$-th row. -\spadcommand{pascalRow(n) == [pascal(i,n) for i in 1..n]} -\returnType{Type: Void} - -Next, we write the function {\bf displayRow} to display the row, -separating entries by blanks and centering. -\spadcommand{displayRow(n) == output center blankSeparate pascalRow(n)} -\returnType{Type: Void} - -Here we have used three output operations. Operation -\spadfunFrom{output}{OutputForm} displays the printable form of -objects on the screen, \spadfunFrom{center}{OutputForm} centers a -printable form in the width of the screen, and -\spadfunFrom{blankSeparate}{OutputForm} takes a list of nprintable -forms and inserts a blank between successive elements. - -Look at the result. -\spadcommand{for i in 1..7 repeat displayRow i} -\begin{verbatim} - Compiling function pascal with type (Integer,Integer) -> - PositiveInteger - Compiling function pascalRow with type PositiveInteger -> List - PositiveInteger - Compiling function displayRow with type PositiveInteger -> Void - - - 1 - 1 1 - 1 2 1 - 1 3 3 1 - 1 4 6 4 1 - 1 5 10 10 5 1 - 1 6 15 20 15 6 1 -\end{verbatim} -\returnType{Type: Void} - -Being purists, we find this less than satisfactory. Traditionally, -elements of Pascal's triangle are centered between the left and right -elements on the line above. - -To fix this misalignment, we go back and redefine {\bf pascalRow} to -right adjust the entries within the triangle within a width of four -characters. - -\spadcommand{pascalRow(n) == [right(pascal(i,n),4) for i in 1..n]} -\begin{verbatim} - Compiled code for pascalRow has been cleared. - Compiled code for displayRow has been cleared. - 1 old definition(s) deleted for function or rule pascalRow -\end{verbatim} -\returnType{Type: Void} - -Finally let's look at our purely reformatted triangle. -\spadcommand{for i in 1..7 repeat displayRow i} -\begin{verbatim} - Compiling function pascalRow with type PositiveInteger -> List - OutputForm - -+++ |*1;pascalRow;1;G82322| redefined - Compiling function displayRow with type PositiveInteger -> Void - -+++ |*1;displayRow;1;G82322| redefined - 1 - 1 1 - 1 2 1 - 1 3 3 1 - 1 4 6 4 1 - 1 5 10 10 5 1 - 1 6 15 20 15 6 1 -\end{verbatim} -\returnType{Type: Void} - -Unexpose {\tt OutputForm} so we don't get unexpected results later. -\spadcommand{)set expose drop constructor OutputForm} -\begin{verbatim} - OutputForm is now explicitly hidden in frame G82322 -\end{verbatim} - -\section{Example: Testing for Palindromes} -\label{ugUserPal} - -In this section we define a function {\bf pal?} that tests whether its -\index{palindrome} argument is a {\it palindrome}, that is, something -that reads the same backwards and forwards. For example, the string -``Madam I'm Adam'' is a palindrome (excluding blanks and punctuation) -and so is the number $123454321$. The definition works for any -datatype that has $n$ components that are accessed by the indices -$1\ldots n$. - -Here is the definition for {\bf pal?}. It is simply a call to an -auxiliary function called {\bf palAux?}. We are following the -convention of ending a function's name with {\tt ?} if the function -returns a {\tt Boolean} value. -\spadcommand{pal? s == palAux?(s,1,\#s)} -\returnType{Type: Void} - -Here is {\bf palAux?}. It works by comparing elements that are -equidistant from the start and end of the object. -\begin{verbatim} -palAux?(s,i,j) == - j > i => - (s.i = s.j) and palAux?(s,i+1,i-1) - true -\end{verbatim} -\returnType{Type: Void} - -Try {\bf pal?} on some examples. First, a string. -\spadcommand{pal? "Oxford"} -\begin{verbatim} - Compiling function palAux? with type (String,Integer,Integer) -> - Boolean - Compiling function pal? with type String -> Boolean -\end{verbatim} -$$ -{\tt false} -$$ -\returnType{Type: Boolean} - -A list of polynomials. -\spadcommand{pal? [4,a,x-1,0,x-1,a,4]} -\begin{verbatim} - Compiling function palAux? with type (List Polynomial Integer, - Integer,Integer) -> Boolean - Compiling function pal? with type List Polynomial Integer -> Boolean -\end{verbatim} -$$ -{\tt true} -$$ -\returnType{Type: Boolean} - -A list of integers from the example in the last section. -\spadcommand{pal? [1,6,15,20,15,6,1]} -\begin{verbatim} - Compiling function palAux? with type (List PositiveInteger,Integer, - Integer) -> Boolean - Compiling function pal? with type List PositiveInteger -> Boolean -\end{verbatim} -$$ -{\tt true} -$$ -\returnType{Type: Boolean} - -To use {\bf pal?} on an integer, first convert it to a string. -\spadcommand{pal?(1441::String)} -$$ -{\tt true} -$$ -\returnType{Type: Boolean} - -Compute an infinite stream of decimal numbers, each of which is an -obvious palindrome. -\spadcommand{ones := [reduce(+,[10**j for j in 0..i]) for i in 1..]} -$$ -\begin{array}{@{}l} -\left[ -{11}, {111}, {1111}, {11111}, {111111}, {1111111}, -\right. -\\ -\\ -\displaystyle -\left. -{11111111}, {111111111}, {1111111111}, {11111111111}, \ldots -\right] -\end{array} -$$ -\returnType{Type: Stream PositiveInteger} - -\spadcommand{)set streams calculate 9} - -How about their squares? -\spadcommand{squares := [x**2 for x in ones]} -$$ -\begin{array}{@{}l} -\left[ -{121}, {12321}, {1234321}, {123454321}, {12345654321}, {1234567654321}, -\right. -\\ -\\ -\displaystyle -{123456787654321}, {12345678987654321}, {1234567900987654321}, -\\ -\\ -\displaystyle -\left. -{123456790120987654321}, \ldots -\right] -\end{array} -$$ -\returnType{Type: Stream PositiveInteger} - -Well, let's test them all. -\spadcommand{[pal?(x::String) for x in squares]} -$$ -\left[ -{\tt true}, {\tt true}, {\tt true}, {\tt true}, {\tt true}, -{\tt true}, {\tt true}, {\tt true}, {\tt true}, {\tt true}, \ldots -\right] -$$ -\returnType{Type: Stream Boolean} - -\spadcommand{)set streams calculate 7} - -\section{Rules and Pattern Matching} -\label{ugUserRules} - -A common mathematical formula is -$$ \log(x) + \log(y) = \log(x y) \quad\forall \, x \hbox{\ and\ } y$$ -The presence of ``$\forall$'' indicates that $x$ and $y$ can stand for -arbitrary mathematical expressions in the above formula. You can use -such mathematical formulas in Axiom to specify ``rewrite rules''. -Rewrite rules are objects in Axiom that can be assigned to variables -for later use, often for the purpose of simplification. Rewrite rules -look like ordinary function definitions except that they are preceded -by the reserved word $rule$. \index{rule} For example, a rewrite rule -for the above formula is: -\begin{verbatim} -rule log(x) + log(y) == log(x * y) -\end{verbatim} - -Like function definitions, no action is taken when a rewrite rule is -issued. Think of rewrite rules as functions that take one argument. -When a rewrite rule $A = B$ is applied to an argument $f$, its meaning -is: ``rewrite every subexpression of $f$ that {\it matches} $A$ by -$B.$'' The left-hand side of a rewrite rule is called a {\it pattern}; -its right-side side is called its {\it substitution}. - -Create a rewrite rule named {\bf logrule}. The generated symbol -beginning with a ``{\tt \%}'' is a place-holder for any other terms that -might occur in the sum. -\spadcommand{logrule := rule log(x) + log(y) == log(x * y)} -$$ -{{\log -\left( -{y} -\right)}+{\log -\left( -{x} -\right)}+ - \%C} \mbox{\rm == } {{\log -\left( -{{x \ y}} -\right)}+ - \%C} -$$ -\returnType{Type: RewriteRule(Integer,Integer,Expression Integer)} - -Create an expression with logarithms. -\spadcommand{f := log sin x + log x} -$$ -{\log -\left( -{{\sin -\left( -{x} -\right)}} -\right)}+{\log -\left( -{x} -\right)} -$$ -\returnType{Type: Expression Integer} - -Apply {\bf logrule} to $f$. -\spadcommand{logrule f} -$$ -\log -\left( -{{x \ {\sin -\left( -{x} -\right)}}} -\right) -$$ -\returnType{Type: Expression Integer} - -The meaning of our example rewrite rule is: ``for all expressions $x$ -and $y$, rewrite $log(x) + log(y)$ by $log(x * y)$.'' Patterns -generally have both operation names (here, {\bf log} and ``{\tt +}'') and -variables (here, $x$ and $y$). By default, every operation name -stands for itself. Thus {\bf log} matches only ``$log$'' and not any -other operation such as {\bf sin}. On the other hand, variables do -not stand for themselves. Rather, a variable denotes a {\it pattern -variable} that is free to match any expression whatsoever. -\index{pattern!variables} - -When a rewrite rule is applied, a process called -{\it pattern matching} goes to work by systematically scanning -\index{pattern!matching} the subexpressions of the argument. When a -subexpression is found that ``matches'' the pattern, the subexpression -is replaced by the right-hand side of the rule. The details of what -happens will be covered later. - -The customary Axiom notation for patterns is actually a shorthand for -a longer, more general notation. Pattern variables can be made -explicit by using a percent ``{\tt \%}'' as the first character of the -variable name. To say that a name stands for itself, you can prefix -that name with a quote operator ``{\tt '}''. Although the current Axiom -parser does not let you quote an operation name, this more general -notation gives you an alternate way of giving the same rewrite rule: -\begin{verbatim} -rule log(%x) + log(%y) == log(x * y) -\end{verbatim} - -This longer notation gives you patterns that the standard notation -won't handle. For example, the rule -\begin{verbatim} -rule %f(c * 'x) == c*%f(x) -\end{verbatim} -means ``for all $f$ and $c$, replace $f(y)$ by $c * f(x)$ when $y$ is -the product of $c$ and the explicit variable $x$.'' - -Thus the pattern can have several adornments on the names that appear there. -Normally, all these adornments are dropped in the substitution on the -right-hand side. - -To summarize: - -\boxed{4.6in}{ -\vskip 0.1cm -To enter a single rule in Axiom, use the following syntax: \index{rule} -\begin{center} -{\tt rule {\it leftHandSide} == {\it rightHandSide}} -\end{center} - -The {\it leftHandSide} is a pattern to be matched and the {\it -rightHandSide} is its substitution. The rule is an object of type -{\tt RewriteRule} that can be assigned to a variable and applied to -expressions to transform them.\\ -} - -Rewrite rules can be collected -into rulesets so that a set of rules can be applied at once. -Here is another simplification rule for logarithms. -$$y \log(x) = \log(x^y) \quad\forall \, x \hbox{\ and\ } y$$ -If instead of giving a single rule following the reserved word $rule$ -you give a ``pile'' of rules, you create what is called a {\it -ruleset.} \index{ruleset} Like rules, rulesets are objects in Axiom -and can be assigned to variables. You will find it useful to group -commonly used rules into input files, and read them in as needed. - -Create a ruleset named $logrules$. -\begin{verbatim} -logrules := rule - log(x) + log(y) == log(x * y) - y * log x == log(x ** y) -\end{verbatim} -$$ -\left\{ -{{{\log -\left( -{y} -\right)}+{\log -\left( -{x} -\right)}+ - \%B} \mbox{\rm == } {{\log -\left( -{{x \ y}} -\right)}+ - \%B}}, {{y \ {\log -\left( -{x} -\right)}} -\mbox{\rm == } {\log -\left( -{{x \sp y}} -\right)}} -\right\} -$$ -\returnType{Type: Ruleset(Integer,Integer,Expression Integer)} - -Again, create an expression $f$ containing logarithms. -\spadcommand{f := a * log(sin x) - 2 * log x} -$$ -{a \ {\log -\left( -{{\sin -\left( -{x} -\right)}} -\right)}} --{2 \ {\log -\left( -{x} -\right)}} -$$ -\returnType{Type: Expression Integer} - -Apply the ruleset {\bf logrules} to $f$. -\spadcommand{logrules f} -$$ -\log -\left( -{{{{\sin -\left( -{x} -\right)} -\sp a} \over {x \sp 2}}} -\right) -$$ -\returnType{Type: Expression Integer} - - -We have allowed pattern variables to match arbitrary expressions in -the above examples. Often you want a variable only to match -expressions satisfying some predicate. For example, we may want to -apply the transformation -$$y \log(x) = \log(x^y)$$ -only when $y$ is an integer. - -The way to restrict a pattern variable $y$ by a predicate $f(y)$ -\index{pattern!variable!predicate} is by using a vertical bar ``{\tt |}'', -which means ``such that,'' in \index{such that} much the same way it -is used in function definitions. \index{predicate!on a pattern -variable} You do this only once, but at the earliest (meaning deepest -and leftmost) part of the pattern. - -This restricts the logarithmic rule to create integer exponents only. -\begin{verbatim} -logrules2 := rule - log(x) + log(y) == log(x * y) - (y | integer? y) * log x == log(x ** y) -\end{verbatim} -$$ -\left\{ -{{{\log -\left( -{y} -\right)}+{\log -\left( -{x} -\right)}+ - \%D} \mbox{\rm == } {{\log -\left( -{{x \ y}} -\right)}+ - \%D}}, {{y \ {\log -\left( -{x} -\right)}} -\mbox{\rm == } {\log -\left( -{{x \sp y}} -\right)}} -\right\} -$$ -\returnType{Type: Ruleset(Integer,Integer,Expression Integer)} - -Compare this with the result of applying the previous set of rules. -\spadcommand{f} -$$ -{a \ {\log -\left( -{{\sin -\left( -{x} -\right)}} -\right)}} --{2 \ {\log -\left( -{x} -\right)}} -$$ -\returnType{Type: Expression Integer} - -\spadcommand{logrules2 f} -$$ -{a \ {\log -\left( -{{\sin -\left( -{x} -\right)}} -\right)}}+{\log -\left( -{{1 \over {x \sp 2}}} -\right)} -$$ -\returnType{Type: Expression Integer} - -You should be aware that you might need to apply a function like -{\tt integer} within your predicate expression to actually apply the test -function. - -Here we use {\tt integer} because $n$ has type {\tt Expression -Integer} but {\bf even?} is an operation defined on integers. -\spadcommand{evenRule := rule cos(x)**(n | integer? n and even? integer n)==(1-sin(x)**2)**(n/2)} -$$ -{{\cos -\left( -{x} -\right)} -\sp n} \mbox{\rm == } {{\left( -{{\sin -\left( -{x} -\right)} -\sp 2}+1 -\right)} -\sp {n \over 2}} -$$ -\returnType{Type: RewriteRule(Integer,Integer,Expression Integer)} - -Here is the application of the rule. -\spadcommand{evenRule( cos(x)**2 )} -$$ --{{\sin -\left( -{x} -\right)} -\sp 2}+1 -$$ -\returnType{Type: Expression Integer} - -This is an example of some of the usual identities involving products of -sines and cosines. -\begin{verbatim} -sinCosProducts == rule - sin(x) * sin(y) == (cos(x-y) - cos(x + y))/2 - cos(x) * cos(y) == (cos(x-y) + cos(x+y))/2 - sin(x) * cos(y) == (sin(x-y) + sin(x + y))/2 -\end{verbatim} -\returnType{Type: Void} - -\spadcommand{g := sin(a)*sin(b) + cos(b)*cos(a) + sin(2*a)*cos(2*a)} -$$ -{{\sin -\left( -{a} -\right)} -\ {\sin -\left( -{b} -\right)}}+{{\cos -\left( -{{2 \ a}} -\right)} -\ {\sin -\left( -{{2 \ a}} -\right)}}+{{\cos -\left( -{a} -\right)} -\ {\cos -\left( -{b} -\right)}} -$$ -\returnType{Type: Expression Integer} - -\spadcommand{sinCosProducts g} -\begin{verbatim} - Compiling body of rule sinCosProducts to compute value of type - Ruleset(Integer,Integer,Expression Integer) -\end{verbatim} -$$ -{{\sin -\left( -{{4 \ a}} -\right)}+{2 -\ {\cos -\left( -{{b -a}} -\right)}}} -\over 2 -$$ -\returnType{Type: Expression Integer} - -Another qualification you will often want to use is to allow a pattern to -match an identity element. -Using the pattern $x + y$, for example, neither $x$ nor $y$ -matches the expression $0$. -Similarly, if a pattern contains a product $x*y$ or an exponentiation -$x**y$, then neither $x$ or $y$ matches $1$. - -If identical elements were matched, pattern matching would generally loop. -Here is an expansion rule for exponentials. -\spadcommand{exprule := rule exp(a + b) == exp(a) * exp(b)} -$$ -{e \sp {\left( b+a -\right)}} -\mbox{\rm == } {{e \sp a} \ {e \sp b}} -$$ -\returnType{Type: RewriteRule(Integer,Integer,Expression Integer)} - -This rule would cause infinite rewriting on this if either $a$ or -$b$ were allowed to match $0$. -\spadcommand{exprule exp x} -$$ -e \sp x -$$ -\returnType{Type: Expression Integer} - -There are occasions when you do want a pattern variable in a sum or -product to match $0$ or $1$. If so, prefix its name -with a ``{\tt ?}'' whenever it appears in a left-hand side of a rule. -For example, consider the following rule for the exponential integral: -$$\int \left(\frac{y+e^x}{x}\right) dx = -\int \frac{y}{x} dx + \hbox{\rm Ei}(x) \quad\forall \, x \hbox{\ and\ } y$$ -This rule is valid for $y = 0$. One solution is to create a {\tt -Ruleset} with two rules, one with and one without $y$. A better -solution is to use an ``optional'' pattern variable. - -Define rule {\tt eirule} with -a pattern variable $?y$ to indicate -that an expression may or may not occur. -\spadcommand{eirule := rule integral((?y + exp x)/x,x) == integral(y/x,x) + Ei x} -$$ -{\int \sp{\displaystyle x} {{{{e \sp \%M}+y} \over \%M} \ {d \%M}}} -\mbox{\rm == } {{{{\tt '}integral} -\left( -{{y \over x}, x} -\right)}+{{{\tt -'}Ei} -\left( -{x} -\right)}} -$$ -\returnType{Type: RewriteRule(Integer,Integer,Expression Integer)} - -Apply rule {\tt eirule} to an integral without this term. -\spadcommand{eirule integral(exp u/u, u)} -$$ -Ei -\left( -{u} -\right) -$$ -\returnType{Type: Expression Integer} - -Apply rule {\tt eirule} to an integral with this term. -\spadcommand{eirule integral(sin u + exp u/u, u)} -$$ -{\int \sp{\displaystyle u} {{\sin -\left( -{ \%M} -\right)} -\ {d \%M}}}+{Ei -\left( -{u} -\right)} -$$ -\returnType{Type: Expression Integer} - -Here is one final adornment you will find useful. When matching a -pattern of the form $x + y$ to an expression containing a long sum of -the form $a +\ldots+ b$, there is no way to predict in advance which -subset of the sum matches $x$ and which matches $y$. Aside from -efficiency, this is generally unimportant since the rule holds for any -possible combination of matches for $x$ and $y$. In some situations, -however, you may want to say which pattern variable is a sum (or -product) of several terms, and which should match only a single term. -To do this, put a prefix colon ``{\tt :}'' before the pattern variable -that you want to match multiple terms. -\index{pattern!variable!matching several terms} - -The remaining rules involve operators $u$ and $v$. \index{operator} -\spadcommand{u := operator 'u} -$$ -u -$$ -\returnType{Type: BasicOperator} - -These definitions tell Axiom that $u$ and $v$ are formal operators to -be used in expressions. -\spadcommand{v := operator 'v} -$$ -v -$$ -\returnType{Type: BasicOperator} - -First define {\tt myRule} with no restrictions on the pattern variables -$x$ and $y$. -\spadcommand{myRule := rule u(x + y) == u x + v y} -$$ -{u -\left( -{{y+x}} -\right)} -\mbox{\rm == } {{{{\tt '}v} -\left( -{y} -\right)}+{{{\tt -'}u} -\left( -{x} -\right)}} -$$ -\returnType{Type: RewriteRule(Integer,Integer,Expression Integer)} - -Apply {\tt myRule} to an expression. -\spadcommand{myRule u(a + b + c + d)} -$$ -{v -\left( -{{d+c+b}} -\right)}+{u -\left( -{a} -\right)} -$$ -\returnType{Type: Expression Integer} - -Define {\tt myOtherRule} to match several terms so that the rule gets -applied recursively. -\spadcommand{myOtherRule := rule u(:x + y) == u x + v y} -$$ -{u -\left( -{{y+x}} -\right)} -\mbox{\rm == } {{{{\tt '}v} -\left( -{y} -\right)}+{{{\tt -'}u} -\left( -{x} -\right)}} -$$ -\returnType{Type: RewriteRule(Integer,Integer,Expression Integer)} - -Apply {\tt myOtherRule} to the same expression. -\spadcommand{myOtherRule u(a + b + c + d)} -$$ -{v -\left( -{c} -\right)}+{v -\left( -{b} -\right)}+{v -\left( -{a} -\right)}+{u -\left( -{d} -\right)} -$$ -\returnType{Type: Expression Integer} - -\boxed{4.6in}{ -\vskip 0.1cm -Summary of pattern variable adornments: -\vskip .5\baselineskip -\begin{tabular}{@{}ll} -{\tt (x | predicate?(x))} & - means that the substutution $s$ for $x$\\ & - must satisfy {\tt predicate(s) = true.} \\ -{\tt ?x} & - means that $x$ can match an identity \\ & element (0 or 1). \\ -{\tt :x} & - means that $x$ can match several terms \\ & in a sum. -\end{tabular}\\ -} - -Here are some final remarks on pattern matching. Pattern matching -provides a very useful paradigm for solving certain classes of -problems, namely, those that involve transformations of one form to -another and back. However, it is important to recognize its -limitations. \index{pattern!matching!caveats} - -First, pattern matching slows down as the number of rules you have to -apply increases. Thus it is good practice to organize the sets of -rules you use optimally so that irrelevant rules are never included. - -Second, careless use of pattern matching can lead to wrong answers. -You should avoid using pattern matching to handle hidden algebraic -relationships that can go undetected by other programs. As a simple -example, a symbol such as ``J'' can easily be used to represent the -square root of $-1$ or some other important algebraic quantity. Many -algorithms branch on whether an expression is zero or not, then divide -by that expression if it is not. If you fail to simplify an -expression involving powers of $J$ to $-1,$ algorithms may incorrectly -assume an expression is non-zero, take a wrong branch, and produce a -meaningless result. - -Pattern matching should also not be used as a substitute for a domain. -In Axiom, objects of one domain are transformed to objects of other -domains using well-defined {\bf coerce} operations. Pattern matching -should be used on objects that are all the same type. Thus if your -application can be handled by type {\tt Expression} in Axiom and you -think you need pattern matching, consider this choice carefully. -\index{Expression} You may well be better served by extending an -existing domain or by building a new domain of objects for your -application. - -\setcounter{chapter}{6} - -\chapter{Graphics} -\label{ugGraph} - -\begin{figure}[htbp] -\includegraphics{ps/torusknot.ps} -\caption{Torus knot of type (15,17).} -\end{figure} - -This chapter shows how to use the Axiom graphics facilities -\index{graphics} under the X Window System. Axiom has -two-di\-men\-sion\-al and three-di\-men\-sion\-al drawing and -rendering packages that allow the drawing, coloring, transforming, -mapping, clipping, and combining of graphic output from Axiom -computations. This facility is particularly useful for investigating -problems in areas such as topology. The graphics package is capable -of plotting functions of one or more variables or plotting parametric -surfaces and curves. Various coordinate systems are also available, -such as polar and spherical. - -A graph is displayed in a viewport window and it has a -\index{viewport} control-panel that uses interactive mouse commands. -PostScript and other output forms are available so that Axiom -\index{PostScript} images can be printed or used by other programs. - -\section{Two-Dimensional Graphics} -\label{ugGraphTwoD} - -The Axiom two-di\-men\-sion\-al graphics package provides the ability -to \index{graphics!two-dimensional} display -\begin{itemize} -\item curves defined by functions of a single real variable -\item curves defined by parametric equations -\item implicit non-singular curves defined by polynomial equations -\item planar graphs generated from lists of point components. -\end{itemize} - -These graphs can be modified by specifying various options, such as -calculating points in the polar coordinate system or changing the size -of the graph viewport window. - -\subsection{Plotting Two-Dimensional Functions of One Variable} -\label{ugGraphTwoDPlot} - -\index{curve!one variable function} The first kind of -two-di\-men\-sion\-al graph is that of a curve defined by a function -$y = f(x)$ over a finite interval of the $x$ axis. - -\boxed{4.6in}{ -\vskip 0.1cm -The general format for drawing a function defined by a formula $f(x)$ is: -\begin{center} -{\tt draw(f(x), x = a..b, {\it options})} -\end{center} - -where $a..b$ defines the range of $x$, and where {\it options} -prescribes zero or more options as described in -\ref{ugGraphTwoDOptions} on page~\pageref{ugGraphTwoDOptions}. An -example of an option is $curveColor == bright red().$ An alternative -format involving functions $f$ and $g$ is also available.\\ -} - -A simple way to plot a function is to use a formula. The first -argument is the formula. For the second argument, write the name of -the independent variable (here, $x$), followed by an ``{\tt =}'', and the -range of values. - -Display this formula over the range $0 \leq x \leq 6$. -Axiom converts your formula to a compiled function so that the -results can be computed quickly and efficiently. - -\spadgraph{draw(sin(tan(x)) - tan(sin(x)),x = 0..6)} -%\begin{figure}[htbp] -%{\rm draw(sin(tan(x)) - tan(sin(x)),x = 0..6)\\} -%\vskip 0.2cm -%\includegraphics{ps/2d1vara.ps} -%\caption{$sin(tan(x)) - tan(sin(x))\ \ \ x = 0 \ldots6$} -%\end{figure} -%{\rm -%\par -%Notice that Axiom compiled the function before the graph was put -%on the screen. -% -%Here is the same graph on a different interval.\\ -% -%draw(sin(tan(x)) - tan(sin(x)),x = 10..16)\\ -%} -%\vskip 0.2cm -%\spadgraph{draw(sin(tan(x)) - tan(sin(x)),x = 10..16)} -%\begin{figure}[htbp] -%\includegraphics{ps/2d1varb.ps} -%\caption{$sin(tan(x)) - tan(sin(x))\ \ \ x = 10 \ldots16$} -%\end{figure} - -Once again the formula is converted to a compiled function before any -points were computed. If you want to graph the same function on -several intervals, it is a good idea to define the function first so -that the function has to be compiled only once. - -This time we first define the function. -\spadcommand{f(x) == (x-1)*(x-2)*(x-3) } -%\begin{figure} -%\tpd{0 0 300 300 ps/2d1vard.ps} -%\end{figure} -\returnType{Type: Void} -%\epsffile[14 14 188 199]{ps/2d1vard.ps} -%\hspace*{\baseLeftSkip}\special{psfile=ps/2dctrl.ps} - -To draw the function, the first argument is its name and the second is -just the range with no independent variable. -\spadgraph{draw(f, 0..4) } -% window was 300 x 300 -%\epsffile[0 0 295 295]{ps/2d1vard.ps} - -\subsection{Plotting Two-Dimensional Parametric Plane Curves} -\label{ugGraphTwoDPar} - -The second kind of two-di\-men\-sion\-al graph is that of -\index{parametric plane curve} curves produced by parametric -equations. \index{curve!parametric plane} Let $x = f(t)$ and -$y = g(t)$ be formulas or two functions $f$ and $g$ as the parameter $t$ -ranges over an interval $[a,b]$. The function {\bf curve} takes the -two functions $f$ and $g$ as its parameters. - -\boxed{4.6in}{ -\vskip 0.1cm -The general format for drawing a two-di\-men\-sion\-al plane curve defined by -parametric formulas $x = f(t)$ and $y = g(t)$ is: -\begin{center} -{\tt draw(curve(f(t), g(t)), t = a..b, {\it options})} -\end{center} - -where $a..b$ defines the range of the independent variable $t$, and -where {\it options} prescribes zero or more options as described in -\ref{ugGraphThreeDOptions} on page~\pageref{ugGraphThreeDOptions}. An -example of an option is $curveColor == bright red().$\\ } - -Here's an example: - -Define a parametric curve using a range involving $\%pi$, Axiom's way -of saying $\pi$. For parametric curves, Axiom -compiles two functions, one for each of the functions $f$ and $g$. -\spadgraph{draw(curve(sin(t)*sin(2*t)*sin(3*t), sin(4*t)*sin(5*t)*sin(6*t)), t = 0..2*\%pi)} -% window was 300 x 300 -%\epsffile[0 0 295 295]{ps/2dppca.ps} - -The title may be an arbitrary string and is an optional argument to -the {\bf draw} command. -\spadgraph{draw(curve(cos(t), sin(t)), t = 0..2*\%pi)} -% window was 300 x 300 -%\epsffile[0 0 295 295]{ps/2dppcb.ps} - -If you plan on plotting $x = f(t)$, $y = g(t)$ as $t$ ranges over -several intervals, you may want to define functions $f$ and $g$ first, -so that they need not be recompiled every time you create a new graph. -Here's an example: - -As before, you can first define the functions you wish to draw. -\spadcommand{f(t:DFLOAT):DFLOAT == sin(3*t/4) } -\begin{verbatim} - Function declaration f : DoubleFloat -> DoubleFloat has been - added to workspace. -\end{verbatim} -\returnType{Type: Void} - -Axiom compiles them to map {\tt DoubleFloat} values to {\tt DoubleFloat} -values. -\spadcommand{g(t:DFLOAT):DFLOAT == sin(t) } -\begin{verbatim} - Function declaration f : DoubleFloat -> DoubleFloat has been added - to workspace. -\end{verbatim} -\returnType{Type: Void} - -Give to {\tt curve} the names of the functions, then write the range -without the name of the independent variable. -\spadgraph{draw(curve(f,g),0..\%pi) } -% window was 300 x 300 -%\epsffile[0 0 295 295]{ps/2dppcc.ps} - -Here is another look at the same curve but over a different -range. Notice that $f$ and $g$ are not recompiled. Also note that -Axiom provides a default title based on the first function specified -in {\bf curve}. -\spadgraph{draw(curve(f,g),-4*\%pi..4*\%pi) } - -% window was 300 x 300 -%\epsffile[0 0 295 295]{ps/2dppce.ps} - -\subsection{Plotting Plane Algebraic Curves} -\label{ugGraphTwoDPlane} - -A third kind of two-di\-men\-sion\-al graph is a non-singular -``solution curve'' \index{curve!plane algebraic} in a rectangular -region of the plane. A solution curve is a curve defined by a -polynomial equation $p(x,y) = 0$. \index{plane algebraic curve} -Non-singular means that the curve is ``smooth'' in that it does not -cross itself or come to a point (cusp). Algebraically, this means -that for any point $(x,y)$ on the curve, that is, a point such that -$p(x,y) = 0$, the partial derivatives -${{\partial p}\over{\partial x}}(x,y)$ and -${{\partial p}\over{\partial y}}(x,y)$ are not both zero. -\index{curve!smooth} \index{curve!non-singular} \index{smooth curve} -\index{non-singular curve} - -\boxed{4.6in}{ -\vskip 0.1cm -The general format for drawing a non-singular solution curve given by -a polynomial of the form $p(x,y) = 0$ is: -\begin{center} -{\tt draw(p(x,y) = 0, x, y, range == [a..b, c..d], {\it options})} -\end{center} - -where the second and third arguments name the first and second -independent variables of $p$. A {\tt range} option is always given to -designate a bounding rectangular region of the plane -$a \leq x \leq b, c \leq y \leq d$. -Zero or more additional options as described in -\ref{ugGraphTwoDOptions} on page~\pageref{ugGraphTwoDOptions} may be given.\\ -} - -We require that the polynomial has rational or integral coefficients. -Here is an algebraic curve example (``Cartesian ovals''): -\index{Cartesian!ovals} -\spadcommand{p := ((x**2 + y**2 + 1) - 8*x)**2 - (8*(x**2 + y**2 + 1)-4*x-1) } -$$ -{y \sp 4}+{{\left( {2 \ {x \sp 2}} -{{16} \ x} -6 -\right)} -\ {y \sp 2}}+{x \sp 4} -{{16} \ {x \sp 3}}+{{58} \ {x \sp 2}} -{{12} \ x} --6 -$$ -\returnType{Type: Polynomial Integer} - -The first argument is always expressed as an equation of the form $p = 0$ -where $p$ is a polynomial. -\spadgraph{draw(p = 0, x, y, range == [-1..11, -7..7]) } - -% window was 300 x 300 -%\epsffile[0 0 295 295]{ps/2dpaca.ps} - -\subsection{Two-Dimensional Options} -\label{ugGraphTwoDOptions} - -The {\bf draw} commands take an optional list of options, such as {\tt -title} shown above. Each option is given by the syntax: -{\it name} {\tt ==} {\it value}. -Here is a list of the available options in the -order that they are described below. - -\begin{tabular}{llll} -adaptive&clip&unit\\ -clip&curveColor&range\\ -toScale&pointColor&coordinates\\ -\end{tabular} - -The $adaptive$ option turns adaptive plotting on or off. -\index{adaptive plotting} Adaptive plotting uses an algorithm that -traverses a graph and computes more points for those parts of the -graph with high curvature. The higher the curvature of a region is, -the more points the algorithm computes. -\index{graphics!2D options!adaptive} - -The {\tt adaptive} option is normally on. Here we turn it off. -\spadgraph{draw(sin(1/x),x=-2*\%pi..2*\%pi, adaptive == false)} - -% window was 300 x 300 -%\epsffile[0 0 295 295]{ps/2doptad.ps} - -The {\tt clip} option turns clipping on or off. -\index{graphics!2D options!clipping} -If on, large values are cut off according to -\spadfunFrom{clipPointsDefault}{GraphicsDefaults}. - -\spadgraph{draw(tan(x),x=-2*\%pi..2*\%pi, clip == true)} - -% window was 300 x 300 -%\epsffile[0 0 295 295]{ps/2doptcp.ps} - -Option {\tt toScale} does plotting to scale if {\tt true} or uses the -entire viewport if {\tt false}. The default can be determined using -\spadfunFrom{drawToScale}{GraphicsDefaults}. -\index{graphics!2D options!to scale} - -\spadgraph{draw(sin(x),x=-\%pi..\%pi, toScale == true, unit == [1.0,1.0])} - -% window was 300 x 300 -%\epsffile[0 0 295 295]{ps/2doptsc.ps} - -Option {\tt clip} with a range sets point clipping of a graph within -the \index{graphics!2D options!clip in a range} ranges specified in -the list $[x range,y range]$. \index{clipping} If only one range is -specified, clipping applies to the y-axis. -\spadgraph{draw(sec(x),x=-2*\%pi..2*\%pi, clip == [-2*\%pi..2*\%pi,-\%pi..\%pi], unit == [1.0,1.0])} - -% window was 300 x 300 -%\epsffile[0 0 295 295]{ps/2doptcpr.ps} - -Option {\tt curveColor} sets the color of the graph curves or lines to -be the \index{graphics!2D options!curve color} indicated palette color -\index{curve!color} (see \ref{ugGraphColor} on page~\pageref{ugGraphColor} and -\ref{ugGraphColorPalette} on page~\pageref{ugGraphColorPalette}). -\index{color!curve} - -\spadgraph{draw(sin(x),x=-\%pi..\%pi, curveColor == bright red())} - -% window was 300 x 300 -%\epsffile[0 0 295 295]{ps/2doptcvc.ps} - -Option {\tt pointColor} sets the color of the graph points to the -indicated \index{graphics!2D options!point color} palette color (see -\ref{ugGraphColor} on page~\pageref{ugGraphColor} and -\ref{ugGraphColorPalette} on page~\pageref{ugGraphColorPalette}). -\index{color!point} -\spadgraph{draw(sin(x),x=-\%pi..\%pi, pointColor == pastel yellow())} - -% window was 300 x 300 -%\epsffile[0 0 295 295]{ps/2doptptc.ps} - -Option {\tt unit} sets the intervals at which the axis units are -plotted \index{graphics!2D options!set units} according to the -indicated steps [$x$ interval, $y$ interval]. -\spadgraph{draw(curve(9*sin(3*t/4),8*sin(t)), t = -4*\%pi..4*\%pi, unit == [2.0,1.0])} - -% window was 300 x 300 -%\epsffile[0 0 295 295]{ps/2doptut.ps} - -Option {\tt range} sets the range of variables in a graph to be within -the ranges \index{graphics!2D options!range} for solving plane -algebraic curve plots. -\spadgraph{draw(y**2 + y - (x**3 - x) = 0, x, y, range == [-2..2,-2..1], unit==[1.0,1.0])} - -% window was 300 x 300 -%\epsffile[0 0 295 295]{ps/2doptrga.ps} - -A second example of a solution plot. -\spadgraph{draw(x**2 + y**2 = 1, x, y, range == [-3/2..3/2,-3/2..3/2], unit==[0.5,0.5])} - -% window was 300 x 300 -%\epsffile[0 0 295 295]{ps/2doptrgb.ps} - -Option $coordinates$ indicates the coordinate system in which the -graph \index{graphics!2D options!coordinates} is plotted. The default -is to use the Cartesian coordinate system. -\index{Cartesian!coordinate system} For more details, see -\ref{ugGraphCoord} on page~\pageref{ugGraphCoord} -{or {\tt CoordinateSystems}.} -\index{coordinate system!Cartesian} - -\spadgraph{draw(curve(sin(5*t),t),t=0..2*\%pi, coordinates == polar)} - -% window was 300 x 300 -%\epsffile[0 0 295 295]{ps/2doptplr.ps} - -\subsection{Color} -\label{ugGraphColor} - -The domain {\tt Color} \index{Color} provides operations for -manipulating \index{graphics!color} colors in two-di\-men\-sion\-al -graphs. \index{color} Colors are objects of {\tt Color}. Each color -has a {\it hue} and a {\it weight}. \index{hue} Hues are represented -by integers that range from $1$ to the -\spadfunFrom{numberOfHues()}{Color}, normally -\index{graphics!color!number of hues} $27$. \index{weight} Weights -are floats and have the value $1.0$ by default. - -\begin{description} - -\item[{\bf color}]\funArgs{integer} -creates a color of hue {\it integer} and weight $1.0$. - -\item[{\bf hue}]\funArgs{color} -returns the hue of {\it color} as an integer. -\index{graphics!color!hue function} - -\item[{\bf red}]\funArgs{} -\funSyntax{blue}{}, -\funSyntax{green}{}, and \funSyntax{yellow}{} -\index{graphics!color!primary color functions} -create colors of that hue with weight $1.0$. - -\item[$\hbox{\it color}_{1}$ {\tt +} $\hbox{\it color}_{2}$] returns the -color that results from additively combining the indicated -$\hbox{\it color}_{1}$ and $\hbox{\it color}_{2}$. -Color addition is not commutative: changing the order of the arguments -produces different results. - -\item[{\it integer} {\tt *} {\it color}] -changes the weight of {\it color} by {\it integer} -without affecting its hue. -\index{graphics!color!multiply function} -For example, -$red() + 3*yellow()$ produces a color closer to yellow than to red. -Color multiplication is not associative: changing the order of grouping -\index{color!multiplication} -produces different results. -\end{description} - -These functions can be used to change the point and curve colors -for two- and three-di\-men\-sion\-al graphs. -Use the {\tt pointColor} option for points. - -\spadgraph{draw(x**2,x=-1..1,pointColor == green())} - -% window was 300 x 300 -%\epsffile[0 0 295 295]{ps/23dcola.ps} - - -Use the {\tt curveColor} option for curves. - -\spadgraph{draw(x**2,x=-1..1,curveColor == color(13) + 2*blue())} - -% window was 300 x 300 -%\epsffile[0 0 295 295]{ps/23dcolb.ps} - -\subsection{Palette} -\label{ugGraphColorPalette} -\index{graphics!palette} - -Domain {\tt Palette} is the domain of shades of colors: -{\bf dark}, {\bf dim}, {\bf bright}, {\bf pastel}, and {\bf light}, -designated by the integers $1$ through $5$, respectively. -\index{Palette} - -Colors are normally ``bright.'' - -\spadcommand{shade red()} -$$ -3 -$$ -\returnType{Type: PositiveInteger} - -To change the shade of a color, apply the name of a shade to it. -\index{color!shade} -\index{shade} - -\spadcommand{myFavoriteColor := dark blue() } -$$ -[{ \mbox{\rm Hue: } {22} \mbox{\rm Weight: } {1.0}} \mbox{\rm ] from the } -Dark \mbox{\rm palette} -$$ -\returnType{Type: Palette} - -The expression $shade(color)$ -returns the value of a shade of $color$. - -\spadcommand{shade myFavoriteColor } -$$ -1 -$$ -\returnType{Type: PositiveInteger} - -The expression $hue(color)$ returns its hue. - -\spadcommand{hue myFavoriteColor } -$$ -\mbox{\rm Hue: } {22} \mbox{\rm Weight: } {1.0} -$$ -\returnType{Type: Color} - -Palettes can be used in specifying colors in two-di\-men\-sion\-al graphs. - -\spadgraph{draw(x**2,x=-1..1,curveColor == dark blue())} - -% window was 300 x 300 -%\epsffile[0 0 295 295]{ps/23dpal.ps} - - -\subsection{Two-Dimensional Control-Panel} -\label{ugGraphTwoDControl} - -\index{graphics!2D control-panel} -Once you have created a viewport, move your mouse to the viewport and click -with your left mouse button to display a control-panel. -The panel is displayed on the side of the viewport closest to -where you clicked. Each of the buttons which toggle on and off show the -current state of the graph. - -%\begin{texonly} -%\typeout{2D control-panel.} -\begin{figure}[htbp] -%{\epsfverbosetrue\epsfxsize=2in% -%\def\epsfsize#1#2{\epsfxsize}\hspace*{\baseLeftSkip}% -%%\epsffile[0 0 144 289]{ps/2dctrl.ps}} -%\begin{picture}(147,252)%(-143,0) -%\hspace*{\baseLeftSkip}\special{psfile=ps/2dctrl.ps} -%\end{picture} -\caption{Two-dimensional control-panel.} -\end{figure} -%\end{texonly} - -\subsubsection{Transformations} -\index{graphics!2D control-panel!transformations} - -Object transformations are executed from the control-panel by mouse-activated -potentiometer windows. -% -\begin{description} -% -\item[Scale:] To scale a graph, click on a mouse button -\index{graphics!2D control-panel!scale} -within the {\bf Scale} window in the upper left corner of the control-panel. -The axes along which the scaling is to occur are indicated by setting the -toggles above the arrow. -With {\tt X On} and {\tt Y On} appearing, both axes are selected and scaling -is uniform. -If either is not selected, for example, if {\tt X Off} appears, scaling is -non-uniform. -% -\item[Translate:] To translate a graph, click the mouse in the -\index{graphics!2D control-panel!translate} -{\bf Translate} window in the direction you wish the graph to move. -This window is located in the upper right corner of the control-panel. -Along the top of the {\bf Translate} window are two buttons for selecting -the direction of translation. -Translation along both coordinate axes results when {\tt X On} and {\tt Y -On} appear or along one axis when one is on, for example, {\tt X On} and -{\tt Y Off} appear. -\end{description} - -\subsubsection{Messages} -\index{graphics!2D control-panel!messages} - -The window directly below the transformation potentiometer windows is -used to display system messages relating to the viewport and the control-panel. -The following format is displayed: \newline -% -\begin{center} -[scaleX, scaleY] $>$graph$<$ [translateX, translateY] \newline -\end{center} -The two values to the left show the scale factor along the {\tt X} and -{\tt Y} coordinate axes. The two values to the right show the distance of -translation from the center in the {\tt X} and {\tt Y} directions. The number -in the center shows which graph in the viewport this data pertains to. -When multiple graphs exist in the same viewport, -the graph must be selected (see ``Multiple Graphs,'' below) in -order for its transformation data to be shown, otherwise the number -is 1. - -\subsubsection{Multiple Graphs} - -\index{graphics!2D control-panel!multiple graphs} -The {\bf Graphs} window contains buttons that allow the placement -of two-di\-men\-sion\-al graphs into one of nine available slots in any other -two-di\-men\-sion\-al viewport. -In the center of the window are numeral buttons from one to nine -that show whether a graph is displayed in the viewport. -Below each number button is a button showing whether a graph -that is present is selected for application of some -transformation. -When the caret symbol is displayed, then the graph in that slot -will be manipulated. -Initially, the graph for which the viewport is created occupies -the first slot, is displayed, and is selected. -% -% -\begin{description} -% -\item[Clear:] The {\bf Clear} button deselects every viewport graph slot. -\index{graphics!2D control-panel!clear} -A graph slot is reselected by selecting the button below its number. -% -\item[Query:] The {\bf Query} button is used to display the scale and -\index{graphics!2D control-panel!query} -translate data for the indicated graph. When this button is selected the -message ``Click on the graph to query'' appears. Select a slot -number button from the {\bf Graphs} window. The scaling factor and translation -offset of the graph are then displayed in the message window. -% -\item[Pick:] The {\bf Pick} button is used to select a graph -\index{graphics!2D control-panel!pick} -to be placed or dropped into the indicated viewport. When this button is -selected, the message ``Click on the graph to pick'' appears. -Click on the slot with the graph number of the desired -graph. The graph information is held waiting for -you to execute a {\bf Drop} in some other graph. -% -\item[Drop:] Once a graph has been picked up using the {\bf Pick} button, -\index{graphics!2D control-panel!drop} -the {\bf Drop} button places it into a new viewport slot. -The message ``Click on the graph to drop'' appears in the message -window when the {\bf Drop} button is selected. -By selecting one of the slot number buttons in the {\bf Graphs} -window, the graph currently being held is dropped into this slot -and displayed. -\end{description} - -\subsubsection{Buttons} -\index{graphics!2D control-panel!buttons} - -% -\begin{description} -% -\item[Axes] turns the coordinate axes on or off. -\index{graphics!2D control-panel!axes} -% -\item[Units] turns the units along the {\tt x} -and {\tt y} axis on or off. -\index{graphics!2D control-panel!units} -% -\item[Box] encloses the area of the viewport graph -in a bounding box, or removes the box if already enclosed. -\index{graphics!2D control-panel!box} -% -\item[Pts] turns on or off the display of points. -\index{graphics!2D control-panel!points} -% -\item[Lines] turns on or off the display -of lines connecting points. -\index{graphics!2D control-panel!lines} -% -\item[PS] writes the current viewport contents to -\index{graphics!2D control-panel!ps} -a file {\bf axiom2d.ps} or to a name specified in the user's {\bf -\index{graphics!.Xdefaults!PostScript file name} -.Xdefaults} file. -\index{file!.Xdefaults @{\bf .Xdefaults}} -The file is placed in the directory from which Axiom or the {\bf -viewalone} program was invoked. -\index{PostScript} -% -\item[Reset] resets the object transformation -characteristics and attributes back to their initial states. -\index{graphics!2D control-panel!reset} -% -\item[Hide] makes the control-panel disappear. -\index{graphics!2D control-panel!hide} -% -\item[Quit] queries whether the current viewport -\index{graphics!2D control-panel!quit} -session should be terminated. -\end{description} - -\subsection{Operations for Two-Dimensional Graphics} -\label{ugGraphTwoDops} - -Here is a summary of useful Axiom operations for two-di\-men\-sion\-al -graphics. -Each operation name is followed by a list of arguments. -Each argument is written as a variable informally named according -to the type of the argument (for example, {\it integer}). -If appropriate, a default value for an argument is given in -parentheses immediately following the name. - -% -\begin{description} -% -\item[{\bf adaptive}]\funArgs{\optArg{boolean\argDef{true}}} -\index{adaptive plotting} -sets or indicates whether graphs are plotted -\index{graphics!set 2D defaults!adaptive} -according to the adaptive refinement algorithm. -% -\item[{\bf axesColorDefault}]\funArgs{\optArg{color\argDef{dark blue()}}} -sets or indicates the default color of the -\index{graphics!set 2D defaults!axes color} -axes in a two-di\-men\-sion\-al graph viewport. -% -\item[{\bf clipPointsDefault}]\funArgs{\optArg{boolean\argDef{false}}} -sets or -indicates whether point clipping is -\index{graphics!set 2D defaults!clip points} -to be applied as the default for graph plots. -% -\item[{\bf drawToScale}]\funArgs{\optArg{boolean\argDef{false}}} -sets or -indicates whether the plot of a graph -\index{graphics!set 2D defaults!to scale} -is ``to scale'' or uses the entire viewport space as the default. -% -\item[{\bf lineColorDefault}]\funArgs{\optArg{color\argDef{pastel yellow()}}} -sets or indicates the default color of the -\index{graphics!set 2D defaults!line color} -lines or curves in a two-di\-men\-sion\-al graph viewport. -% -\item[{\bf maxPoints}]\funArgs{\optArg{integer\argDef{500}}} -sets or indicates -the default maximum number of -\index{graphics!set 2D defaults!max points} -possible points to be used when constructing a two-di\-men\-sion\-al graph. -% -\item[{\bf minPoints}]\funArgs{\optArg{integer\argDef{21}}} -sets or indicates the default minimum number of -\index{graphics!set 2D defaults!min points} -possible points to be used when constructing a two-di\-men\-sion\-al graph. -% -\item[{\bf pointColorDefault}]\funArgs{\optArg{color\argDef{bright red()}}} -sets or indicates the default color of the -\index{graphics!set 2D defaults!point color} -points in a two-di\-men\-sion\-al graph viewport. -% -\item[{\bf pointSizeDefault}]\funArgs{\optArg{integer\argDef{5}}} -sets or indicates the default size of the -\index{graphics!set 2D defaults!point size} -dot used to plot points in a two-di\-men\-sion\-al graph. -% -\item[{\bf screenResolution}]\funArgs{\optArg{integer\argDef{600}}} -sets or indicates the default screen -\index{graphics!set 2D defaults!screen resolution} -resolution constant used in setting the computation limit of adaptively -\index{adaptive plotting} -generated curve plots. -% -\item[{\bf unitsColorDefault}]\funArgs{\optArg{color\argDef{dim green()}}} -sets or indicates the default color of the -\index{graphics!set 2D defaults!units color} -unit labels in a two-di\-men\-sion\-al graph viewport. -% -\item[{\bf viewDefaults}]\funArgs{} -resets the default settings for the following -\index{graphics!set 2D defaults!reset viewport} -attributes: point color, line color, axes color, units color, point size, -viewport upper left-hand corner position, and the viewport size. -% -\item[{\bf viewPosDefault}]\funArgs{\optArg{list\argDef{[100,100]}}} -sets or indicates the default position of the -\index{graphics!set 2D defaults!viewport position} -upper left-hand corner of a two-di\-men\-sion\-al viewport, relative to the -display root window. -The upper left-hand corner of the display is considered to be at the -(0, 0) position. -% -\item[{\bf viewSizeDefault}]\funArgs{\optArg{list\argDef{[200,200]}}} -sets or -indicates the default size in which two -\index{graphics!set 2D defaults!viewport size} -dimensional viewport windows are shown. -It is defined by a width and then a height. -% -\item[{\bf viewWriteAvailable}] -\funArgs{\optArg{list\argDef{["pixmap","bitmap", "postscript", "image"]}}} -indicates the possible file types -\index{graphics!2D defaults!available viewport writes} -that can be created with the \spadfunFrom{write}{TwoDimensionalViewport} function. -% -\item[{\bf viewWriteDefault}]\funArgs{\optArg{list\argDef{[]}}} -sets or indicates the default types of files, in -\index{graphics!set 2D defaults!write viewport} -addition to the {\bf data} file, that are created when a -{\bf write} function is executed on a viewport. -% -\item[{\bf units}]\funArgs{viewport, integer\argDef{1}, string\argDef{"off"}} -turns the units on or off for the graph with index {\it integer}. -% -\item[{\bf axes}]\funArgs{viewport, integer\argDef{1}, string\argDef{"on"}} -turns the axes on -\index{graphics!2D commands!axes} -or off for the graph with index {\it integer}. -% -\item[{\bf close}]\funArgs{viewport} -closes {\it viewport}. -\index{graphics!2D commands!close} -% -\item[{\bf connect}]\funArgs{viewport, integer\argDef{1}, string\argDef{"on"}} -declares whether lines -\index{graphics!2D commands!connect} -connecting the points are displayed or not. -% -\item[{\bf controlPanel}]\funArgs{viewport, string\argDef{"off"}} -declares -whether the two-di\-men\-sion\-al control-panel is automatically displayed -or not. -% -\item[{\bf graphs}]\funArgs{viewport} -returns a list -\index{graphics!2D commands!graphs} -describing the state of each graph. -If the graph state is not being used this is shown by {\tt "undefined"}, -otherwise a description of the graph's contents is shown. -% -\item[{\bf graphStates}]\funArgs{viewport} -displays -\index{graphics!2D commands!state of graphs} -a list of all the graph states available for {\it viewport}, giving the -values for every property. -% -\item[{\bf key}]\funArgs{viewport} -returns the process -\index{graphics!2D commands!key} -ID number for {\it viewport}. -% -\item[{\bf move}]\funArgs{viewport, -$integer_{x}$(viewPosDefault), -$integer_{y}$(viewPosDefault)} -moves {\it viewport} on the screen so that the -\index{graphics!2D commands!move} -upper left-hand corner of {\it viewport} is at the position {\it (x,y)}. -% -\item[{\bf options}]\funArgs{\it viewport} -returns a list -\index{graphics!2D commands!options} -of all the {\tt DrawOption}s used by {\it viewport}. -% -\item[{\bf points}]\funArgs{viewport, integer\argDef{1}, string\argDef{"on"}} -specifies whether the graph points for graph {\it integer} are -\index{graphics!2D commands!points} -to be displayed or not. -% -\item[{\bf region}]\funArgs{viewport, integer\argDef{1}, string\argDef{"off"}} -declares whether graph {\it integer} is or is not to be displayed -with a bounding rectangle. -% -\item[{\bf reset}]\funArgs{viewport} -resets all the properties of {\it viewport}. -% -\item[{\bf resize}]\funArgs{viewport, -$integer_{width}$,$integer_{height}$} -\index{graphics!2D commands!resize} -resizes {\it viewport} with a new {\it width} and {\it height}. -% -\item[{\bf scale}]\funArgs{viewport, $integer_{n}$\argDef{1}, -$integer_{x}$\argDef{0.9}, $integer_{y}$\argDef{0.9}} -scales values for the -\index{graphics!2D commands!scale} -{\it x} and {\it y} coordinates of graph {\it n}. -% -\item[{\bf show}]\funArgs{viewport, $integer_{n}$\argDef{1}, -string\argDef{"on"}} -indicates if graph {\it n} is shown or not. -% -\item[{\bf title}]\funArgs{viewport, string\argDef{"Axiom 2D"}} -designates the title for {\it viewport}. -% -\item[{\bf translate}]\funArgs{viewport, -$integer_{n}$\argDef{1}, -$float_{x}$\argDef{0.0}, $float_{y}$\argDef{0.0}} -\index{graphics!2D commands!translate} -causes graph {\it n} to be moved {\it x} and {\it y} units in the respective directions. -% -\item[{\bf write}]\funArgs{viewport, $string_{directory}$, -\optArg{strings}} -if no third argument is given, writes the {\bf data} file onto the directory -with extension {\bf data}. -The third argument can be a single string or a list of strings with some or -all the entries {\tt "pixmap"}, {\tt "bitmap"}, {\tt "postscript"}, and -{\tt "image"}. -\end{description} - -\subsection{Addendum: Building Two-Dimensional Graphs} -\label{ugGraphTwoDbuild} - -In this section we demonstrate how to create two-di\-men\-sion\-al graphs from -lists of points and give an example showing how to read the lists -of points from a file. - -\subsubsection{Creating a Two-Dimensional Viewport from a List of Points} - -Axiom creates lists of points in a two-di\-men\-sion\-al viewport by utilizing -the {\tt GraphImage} and {\tt TwoDimensionalViewport} domains. -In this example, the \spadfunFrom{makeGraphImage}{GraphImage} -function takes a list of lists of points parameter, a list of colors for -each point in the graph, a list of colors for each line in the graph, and -a list of sizes for each point in the graph. -% - -The following expressions create a list of lists of points which will be read -by Axiom and made into a two-di\-men\-sion\-al viewport. - -\spadcommand{p1 := point [1,1]\$(Point DFLOAT) } -$$ -\left[ -{1.0}, {1.0} -\right] -$$ -\returnType{Type: Point DoubleFloat} - -\spadcommand{p2 := point [0,1]\$(Point DFLOAT) } -$$ -\left[ -{0.0}, {1.0} -\right] -$$ -\returnType{Type: Point DoubleFloat} - -\spadcommand{p3 := point [0,0]\$(Point DFLOAT) } -$$ -\left[ -{0.0}, {0.0} -\right] -$$ -\returnType{Type: Point DoubleFloat} - -\spadcommand{p4 := point [1,0]\$(Point DFLOAT) } -$$ -\left[ -{1.0}, {0.0} -\right] -$$ -\returnType{Type: Point DoubleFloat} - -\spadcommand{p5 := point [1,.5]\$(Point DFLOAT) } -$$ -\left[ -{1.0}, {0.5} -\right] -$$ -\returnType{Type: Point DoubleFloat} - -\spadcommand{p6 := point [.5,0]\$(Point DFLOAT) } -$$ -\left[ -{0.5}, {0.0} -\right] -$$ -\returnType{Type: Point DoubleFloat} - -\spadcommand{p7 := point [0,0.5]\$(Point DFLOAT) } -$$ -\left[ -{0.0}, {0.5} -\right] -$$ -\returnType{Type: Point DoubleFloat} - -\spadcommand{p8 := point [.5,1]\$(Point DFLOAT) } -$$ -\left[ -{0.5}, {1.0} -\right] -$$ -\returnType{Type: Point DoubleFloat} - -\spadcommand{p9 := point [.25,.25]\$(Point DFLOAT) } -$$ -\left[ -{0.25}, {0.25} -\right] -$$ -\returnType{Type: Point DoubleFloat} - -\spadcommand{p10 := point [.25,.75]\$(Point DFLOAT) } -$$ -\left[ -{0.25}, {0.75} -\right] -$$ -\returnType{Type: Point DoubleFloat} - -\spadcommand{p11 := point [.75,.75]\$(Point DFLOAT) } -$$ -\left[ -{0.75}, {0.75} -\right] -$$ -\returnType{Type: Point DoubleFloat} - -\spadcommand{p12 := point [.75,.25]\$(Point DFLOAT) } -$$ -\left[ -{0.75}, {0.25} -\right] -$$ -\returnType{Type: Point DoubleFloat} - -Finally, here is the list. - -\spadcommand{llp := [ [p1,p2], [p2,p3], [p3,p4], [p4,p1], [p5,p6], [p6,p7], [p7,p8], [p8,p5], [p9,p10], [p10,p11], [p11,p12], [p12,p9] ] } -$$ -\left[ -{\left[ {\left[ {1.0}, {1.0} -\right]}, - {\left[ {0.0}, {1.0} -\right]} -\right]}, - {\left[ {\left[ {0.0}, {1.0} -\right]}, - {\left[ {0.0}, {0.0} -\right]} -\right]}, - {\left[ {\left[ {0.0}, {0.0} -\right]}, - {\left[ {1.0}, {0.0} -\right]} -\right]}, - {\left[ {\left[ {1.0}, {0.0} -\right]}, - {\left[ {1.0}, {1.0} -\right]} -\right]}, - {\left[ {\left[ {1.0}, {0.5} -\right]}, - {\left[ {0.5}, {0.0} -\right]} -\right]}, - {\left[ {\left[ {0.5}, {0.0} -\right]}, - {\left[ {0.0}, {0.5} -\right]} -\right]}, - {\left[ {\left[ {0.0}, {0.5} -\right]}, - {\left[ {0.5}, {1.0} -\right]} -\right]}, - {\left[ {\left[ {0.5}, {1.0} -\right]}, - {\left[ {1.0}, {0.5} -\right]} -\right]}, - {\left[ {\left[ {0.25}, {0.25} -\right]}, - {\left[ {0.25}, {0.75} -\right]} -\right]}, - {\left[ {\left[ {0.25}, {0.75} -\right]}, - {\left[ {0.75}, {0.75} -\right]} -\right]}, - {\left[ {\left[ {0.75}, {0.75} -\right]}, - {\left[ {0.75}, {0.25} -\right]} -\right]}, - {\left[ {\left[ {0.75}, {0.25} -\right]}, - {\left[ {0.25}, {0.25} -\right]} -\right]} -\right] -$$ -\returnType{Type: List List Point DoubleFloat} - -Now we set the point sizes for all components of the graph. - -\spadcommand{size1 := 6::PositiveInteger } -$$ -6 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{size2 := 8::PositiveInteger } -$$ -8 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{size3 := 10::PositiveInteger } - -\spadcommand{lsize := [size1, size1, size1, size1, size2, size2, size2, size2, size3, size3, size3, size3] } -$$ -\left[ -6, 6, 6, 6, 8, 8, 8, 8, size3, size3, size3, -size3 -\right] -$$ -\returnType{Type: List Polynomial Integer} - -Here are the colors for the points. - -\spadcommand{pc1 := pastel red() } -$$ -[{ \mbox{\rm Hue: } 1 \mbox{\rm Weight: } {1.0}} \mbox{\rm ] from the } -Pastel \mbox{\rm palette} -$$ -\returnType{Type: Palette} - -\spadcommand{pc2 := dim green() } -$$ -[{ \mbox{\rm Hue: } {14} \mbox{\rm Weight: } {1.0}} \mbox{\rm ] from the } -Dim \mbox{\rm palette} -$$ -\returnType{Type: Palette} - -\spadcommand{pc3 := pastel yellow() } -$$ -[{ \mbox{\rm Hue: } {11} \mbox{\rm Weight: } {1.0}} \mbox{\rm ] from the } -Pastel \mbox{\rm palette} -$$ -\returnType{Type: Palette} - -\spadcommand{lpc := [pc1, pc1, pc1, pc1, pc2, pc2, pc2, pc2, pc3, pc3, pc3, pc3] } -$$ -\left[ -{[{ \mbox{\rm Hue: } 1 \mbox{\rm Weight: } {1.0}} \mbox{\rm ] from the } -Pastel \mbox{\rm palette} }, {[{ \mbox{\rm Hue: } 1 \mbox{\rm Weight: } -{1.0}} \mbox{\rm ] from the } Pastel \mbox{\rm palette} }, {[{ \mbox{\rm -Hue: } 1 \mbox{\rm Weight: } {1.0}} \mbox{\rm ] from the } Pastel \mbox{\rm -palette} }, {[{ \mbox{\rm Hue: } 1 \mbox{\rm Weight: } {1.0}} \mbox{\rm ] -from the } Pastel \mbox{\rm palette} }, {[{ \mbox{\rm Hue: } {14} -\mbox{\rm Weight: } {1.0}} \mbox{\rm ] from the } Dim \mbox{\rm palette} }, - {[{ \mbox{\rm Hue: } {14} \mbox{\rm Weight: } {1.0}} \mbox{\rm ] from the -} Dim \mbox{\rm palette} }, {[{ \mbox{\rm Hue: } {14} \mbox{\rm Weight: } -{1.0}} \mbox{\rm ] from the } Dim \mbox{\rm palette} }, {[{ \mbox{\rm Hue: -} {14} \mbox{\rm Weight: } {1.0}} \mbox{\rm ] from the } Dim \mbox{\rm -palette} }, {[{ \mbox{\rm Hue: } {11} \mbox{\rm Weight: } {1.0}} \mbox{\rm -] from the } Pastel \mbox{\rm palette} }, {[{ \mbox{\rm Hue: } {11} -\mbox{\rm Weight: } {1.0}} \mbox{\rm ] from the } Pastel \mbox{\rm palette} -}, {[{ \mbox{\rm Hue: } {11} \mbox{\rm Weight: } {1.0}} \mbox{\rm ] from -the } Pastel \mbox{\rm palette} }, {[{ \mbox{\rm Hue: } {11} \mbox{\rm -Weight: } {1.0}} \mbox{\rm ] from the } Pastel \mbox{\rm palette} } -\right] -$$ -\returnType{Type: List Palette} - -Here are the colors for the lines. - -\spadcommand{lc := [pastel blue(), light yellow(), dim green(), bright red(), light green(), dim yellow(), bright blue(), dark red(), pastel red(), light blue(), dim green(), light yellow()] } -$$ -\left[ -{[{ \mbox{\rm Hue: } {22} \mbox{\rm Weight: } {1.0}} \mbox{\rm ] from the } -Pastel \mbox{\rm palette} }, {[{ \mbox{\rm Hue: } {11} \mbox{\rm Weight: } -{1.0}} \mbox{\rm ] from the } Light \mbox{\rm palette} }, {[{ \mbox{\rm -Hue: } {14} \mbox{\rm Weight: } {1.0}} \mbox{\rm ] from the } Dim \mbox{\rm -palette} }, {[{ \mbox{\rm Hue: } 1 \mbox{\rm Weight: } {1.0}} \mbox{\rm ] -from the } Bright \mbox{\rm palette} }, {[{ \mbox{\rm Hue: } {14} -\mbox{\rm Weight: } {1.0}} \mbox{\rm ] from the } Light \mbox{\rm palette} }, - {[{ \mbox{\rm Hue: } {11} \mbox{\rm Weight: } {1.0}} \mbox{\rm ] from the -} Dim \mbox{\rm palette} }, {[{ \mbox{\rm Hue: } {22} \mbox{\rm Weight: } -{1.0}} \mbox{\rm ] from the } Bright \mbox{\rm palette} }, {[{ \mbox{\rm -Hue: } 1 \mbox{\rm Weight: } {1.0}} \mbox{\rm ] from the } Dark \mbox{\rm -palette} }, {[{ \mbox{\rm Hue: } 1 \mbox{\rm Weight: } {1.0}} \mbox{\rm ] -from the } Pastel \mbox{\rm palette} }, {[{ \mbox{\rm Hue: } {22} -\mbox{\rm Weight: } {1.0}} \mbox{\rm ] from the } Light \mbox{\rm palette} }, - {[{ \mbox{\rm Hue: } {14} \mbox{\rm Weight: } {1.0}} \mbox{\rm ] from the -} Dim \mbox{\rm palette} }, {[{ \mbox{\rm Hue: } {11} \mbox{\rm Weight: } -{1.0}} \mbox{\rm ] from the } Light \mbox{\rm palette} } -\right] -$$ -\returnType{Type: List Palette} - -Now the {\tt GraphImage} is created according to the component -specifications indicated above. - -\spadcommand{g := makeGraphImage(llp,lpc,lc,lsize)\$GRIMAGE } - -The \spadfunFrom{makeViewport2D}{TwoDimensionalViewport} function now -creates a {\tt TwoDimensionalViewport} for this graph according to the -list of options specified within the brackets. - -\spadgraph{makeViewport2D(g,[title("Lines")])\$VIEW2D } - -%See Figure #.#. - -This example demonstrates the use of the {\tt GraphImage} functions -\spadfunFrom{component}{GraphImage} and \spadfunFrom{appendPoint}{GraphImage} -in adding points to an empty {\tt GraphImage}. - -\spadcommand{)clear all } - -\spadcommand{g := graphImage()\$GRIMAGE } -$$ -\mbox{\rm Graph with } 0 \mbox{\rm point lists} -$$ -\returnType{Type: GraphImage} - -\spadcommand{p1 := point [0,0]\$(Point DFLOAT) } -$$ -\left[ -{0.0}, {0.0} -\right] -$$ -\returnType{Type: Point DoubleFloat} - -\spadcommand{p2 := point [.25,.25]\$(Point DFLOAT) } -$$ -\left[ -{0.25}, {0.25} -\right] -$$ -\returnType{Type: Point DoubleFloat} - -\spadcommand{p3 := point [.5,.5]\$(Point DFLOAT) } -$$ -\left[ -{0.5}, {0.5} -\right] -$$ -\returnType{Type: Point DoubleFloat} - -\spadcommand{p4 := point [.75,.75]\$(Point DFLOAT) } -$$ -\left[ -{0.75}, {0.75} -\right] -$$ -\returnType{Type: Point DoubleFloat} - -\spadcommand{p5 := point [1,1]\$(Point DFLOAT) } -$$ -\left[ -{1.0}, {1.0} -\right] -$$ -\returnType{Type: Point DoubleFloat} - -\spadcommand{component(g,p1)\$GRIMAGE} -\returnType{Type: Void} - -\spadcommand{component(g,p2)\$GRIMAGE} -\returnType{Type: Void} - -\spadcommand{appendPoint(g,p3)\$GRIMAGE} -\returnType{Type: Void} - -\spadcommand{appendPoint(g,p4)\$GRIMAGE} -\returnType{Type: Void} - -\spadcommand{appendPoint(g,p5)\$GRIMAGE} -\returnType{Type: Void} - -\spadcommand{g1 := makeGraphImage(g)\$GRIMAGE } - -Here is the graph. - -\spadgraph{makeViewport2D(g1,[title("Graph Points")])\$VIEW2D } - -% -%See Figure #.#. -% - -A list of points can also be made into a {\tt GraphImage} by using -the operation \spadfunFrom{coerce}{GraphImage}. It is equivalent to adding -each point to $g2$ using \spadfunFrom{component}{GraphImage}. - -\spadcommand{g2 := coerce([ [p1],[p2],[p3],[p4],[p5] ])\$GRIMAGE } - -Now, create an empty {\tt TwoDimensionalViewport}. - -\spadcommand{v := viewport2D()\$VIEW2D } - -\spadcommand{options(v,[title("Just Points")])\$VIEW2D } - -Place the graph into the viewport. - -\spadcommand{putGraph(v,g2,1)\$VIEW2D } - -Take a look. - -\spadgraph{makeViewport2D(v)\$VIEW2D } - -%See Figure #.#. - -\subsubsection{Creating a Two-Dimensional Viewport of a List of Points from a File} - -The following three functions read a list of points from a -file and then draw the points and the connecting lines. The -points are stored in the file in readable form as floating point numbers -(specifically, {\tt DoubleFloat} values) as an alternating -stream of $x$- and $y$-values. For example, -\begin{verbatim} -0.0 0.0 1.0 1.0 2.0 4.0 -3.0 9.0 4.0 16.0 5.0 25.0 -\end{verbatim} - -\begin{verbatim} -drawPoints(lp:List Point DoubleFloat):VIEW2D == - g := graphImage()$GRIMAGE - for p in lp repeat - component(g,p,pointColorDefault(),lineColorDefault(), - pointSizeDefault()) - gi := makeGraphImage(g)$GRIMAGE - makeViewport2D(gi,[title("Points")])$VIEW2D - -drawLines(lp:List Point DoubleFloat):VIEW2D == - g := graphImage()$GRIMAGE - component(g, lp, pointColorDefault(), lineColorDefault(), - pointSizeDefault())$GRIMAGE - gi := makeGraphImage(g)$GRIMAGE - makeViewport2D(gi,[title("Points")])$VIEW2D - -plotData2D(name, title) == - f:File(DFLOAT) := open(name,"input") - lp:LIST(Point DFLOAT) := empty() - while ((x := readIfCan!(f)) case DFLOAT) repeat - y : DFLOAT := read!(f) - lp := cons(point [x,y]$(Point DFLOAT), lp) - lp - close!(f) - drawPoints(lp) - drawLines(lp) -\end{verbatim} -% -This command will actually create the viewport and the graph if -the point data is in the file $"file.data"$. -\begin{verbatim} -plotData2D("file.data", "2D Data Plot") -\end{verbatim} - -\subsection{Addendum: Appending a Graph to a Viewport Window Containing a Graph} -\label{ugGraphTwoDappend} - -This section demonstrates how to append a two-di\-men\-sion\-al graph to a viewport -already containing other graphs. -The default {\bf draw} command places a graph into the first -{\tt GraphImage} slot position of the {\tt TwoDimensionalViewport}. - -This graph is in the first slot in its viewport. - -\spadcommand{v1 := draw(sin(x),x=0..2*\%pi) } - -So is this graph. - -\spadcommand{v2 := draw(cos(x),x=0..2*\%pi, curveColor==light red()) } - -The operation \spadfunFrom{getGraph}{TwoDimensionalViewport} -retrieves the {\tt GraphImage} $g1$ from the first slot position -in the viewport $v1$. - -\spadcommand{g1 := getGraph(v1,1) } - -Now \spadfunFrom{putGraph}{TwoDimensionalViewport} -places $g1$ into the the second slot position of $v2$. - -\spadcommand{putGraph(v2,g1,2) } - -Display the new {\tt TwoDimensionalViewport} containing both graphs. - -\spadgraph{makeViewport2D(v2) } - -% -%See Figure #.#. -% - -\section{Three-Dimensional Graphics} -\label{ugGraphThreeD} - -% -The Axiom three-di\-men\-sion\-al graphics package provides the ability to -\index{graphics!three-dimensional} -% -\begin{itemize} -% -\item generate surfaces defined by a function of two real variables -% -\item generate space curves and tubes defined by parametric equations -% -\item generate surfaces defined by parametric equations -\end{itemize} -These graphs can be modified by using various options, such as calculating -points in the spherical coordinate system or changing the polygon grid size -of a surface. - -\subsection{Plotting Three-Dimensional Functions of Two Variables} -\label{ugGraphThreeDPlot} - -\index{surface!two variable function} -The simplest three-di\-men\-sion\-al graph is that of a surface defined by a function -of two variables, $z = f(x,y)$. - -\boxed{4.6in}{ -\vskip 0.1cm -The general format for drawing a surface defined by a formula $f(x,y)$ -of two variables $x$ and $y$ is: -% -\begin{center} -{\tt draw(f(x,y), x = a..b, y = c..d, {\it options})} -\end{center} -where $a..b$ and $c..d$ define the range of $x$ -and $y$, and where {\it options} prescribes zero or more -options as described in \ref{ugGraphThreeDOptions} -on page~\pageref{ugGraphThreeDOptions}. -An example of an option is $title == "Title of Graph".$ -An alternative format involving a function $f$ is also -available.\\ -} - -The simplest way to plot a function of two variables is to use a formula. -With formulas you always precede the range specifications with -the variable name and an {\tt =} sign. - -\spadgraph{draw(cos(x*y),x=-3..3,y=-3..3)} - -% window was 300 x 300 -%\epsffile[0 0 295 295]{ps/3d2vara.ps} - -If you intend to use a function more than once, -or it is long and complex, then first -give its definition to Axiom. - -\spadcommand{f(x,y) == sin(x)*cos(y) } -\returnType{Type: Void} - -To draw the function, just give its name and drop the variables -from the range specifications. -Axiom compiles your function for efficient computation -of data for the graph. -Notice that Axiom uses the text of your function as a -default title. - -\spadgraph{draw(f,-\%pi..\%pi,-\%pi..\%pi) } - -% window was 300 x 300 -%\epsffile[0 0 295 295]{ps/3d2varb.ps} - -\subsection{Plotting Three-Dimensional Parametric Space Curves} -\label{ugGraphThreeDParm} - - -A second kind of three-di\-men\-sion\-al graph is a three-di\-men\-sion\-al space curve -\index{curve!parametric space} -defined by the parametric equations for $x(t)$, $y(t)$, -\index{parametric space curve} -and $z(t)$ as a function of an independent variable $t$. - -\boxed{4.6in}{ -\vskip 0.1cm -The general format for drawing a three-di\-men\-sion\-al space curve defined by -parametric formulas $x = f(t)$, $y = g(t)$, and -$z = h(t)$ is: -% -\begin{center} -{\tt draw(curve(f(t),g(t),h(t)), t = a..b, {\it options})} -\end{center} -where $a..b$ defines the range of the independent variable -$t$, and where {\it options} prescribes zero or more options -as described in \ref{ugGraphThreeDOptions} -on page~\pageref{ugGraphThreeDOptions}. -An example of an option is $title == "Title of Graph".$ -An alternative format involving functions $f$, $g$ and -$h$ is also available.\\ -} - -If you use explicit formulas to draw a space curve, always precede -the range specification with the variable name and an -{\tt =} sign. - -\spadgraph{draw(curve(5*cos(t), 5*sin(t),t), t=-12..12)} - -% window was 300 x 300 -%\epsffile[0 0 295 295]{ps/3dpsca.ps} - -Alternatively, you can draw space curves by referring to functions. - -\spadcommand{i1(t:DFLOAT):DFLOAT == sin(t)*cos(3*t/5) } -\begin{verbatim} - Function declaration i1 : DoubleFloat -> DoubleFloat has been added - to workspace. -\end{verbatim} -\returnType{Type: Void} - -This is useful if the functions are to be used more than once \ldots - -\spadcommand{i2(t:DFLOAT):DFLOAT == cos(t)*cos(3*t/5) } -\begin{verbatim} - Function declaration i2 : DoubleFloat -> DoubleFloat has been added - to workspace. -\end{verbatim} -\returnType{Type: Void} - -or if the functions are long and complex. - -\spadcommand{i3(t:DFLOAT):DFLOAT == cos(t)*sin(3*t/5) } -\begin{verbatim} - Function declaration i3 : DoubleFloat -> DoubleFloat has been added - to workspace. -\end{verbatim} -\returnType{Type: Void} - -Give the names of the functions and -drop the variable name specification in the second argument. -Again, Axiom supplies a default title. - -\spadgraph{draw(curve(i1,i2,i3),0..15*\%pi) } - -% window was 300 x 300 -%\epsffile[0 0 295 295]{ps/3dpscb.ps} - -\subsection{Plotting Three-Dimensional Parametric Surfaces} -\label{ugGraphThreeDPar} - -\index{surface!parametric} -A third kind of three-di\-men\-sion\-al graph is a surface defined by -\index{parametric surface} -parametric equations for $x(u,v)$, $y(u,v)$, and -$z(u,v)$ of two independent variables $u$ and $v$. - - -\boxed{4.6in}{ -\vskip 0.1cm -The general format for drawing a three-di\-men\-sion\-al graph defined by -parametric formulas $x = f(u,v)$, $y = g(u,v)$, -and $z = h(u,v)$ is: -% -\begin{center} -{\tt draw(surface(f(u,v),g(u,v),h(u,v)), u = a..b, v = c..d, {\it options})} -\end{center} -where $a..b$ and $c..d$ define the range of the -independent variables $u$ and $v$, and where -{\it options} prescribes zero or more options as described in -\ref{ugGraphThreeDOptions} on page~\pageref{ugGraphThreeDOptions}. -An example of an option is $title == "Title of Graph".$ -An alternative format involving functions $f$, $g$ and -$h$ is also available.\\ -} - -This example draws a graph of a surface plotted using the -parabolic cylindrical coordinate system option. -\index{coordinate system!parabolic cylindrical} -The values of the functions supplied to {\bf surface} are -\index{parabolic cylindrical coordinate system} -interpreted in coordinates as given by a {\tt coordinates} option, -here as parabolic cylindrical coordinates (see -\ref{ugGraphCoord} on page~\pageref{ugGraphCoord}). - -\spadgraph{draw(surface(u*cos(v), u*sin(v), v*cos(u)), u=-4..4, v=0..\%pi, coordinates== parabolicCylindrical)} - -% window was 300 x 300 -%\epsffile[0 0 295 295]{ps/3dpsa.ps} - -Again, you can graph these parametric surfaces using functions, -if the functions are long and complex. - -Here we declare the types of arguments and values to be of type -{\tt DoubleFloat}. - -\spadcommand{n1(u:DFLOAT,v:DFLOAT):DFLOAT == u*cos(v) } -\begin{verbatim} - Function declaration n1 : DoubleFloat -> DoubleFloat has been added - to workspace. -\end{verbatim} -\returnType{Type: Void} - -As shown by previous examples, these declarations are necessary. - -\spadcommand{n2(u:DFLOAT,v:DFLOAT):DFLOAT == u*sin(v) } -\begin{verbatim} - Function declaration n2 : DoubleFloat -> DoubleFloat has been added - to workspace. -\end{verbatim} -\returnType{Type: Void} - -In either case, Axiom compiles the functions -when needed to graph a result. - -\spadcommand{n3(u:DFLOAT,v:DFLOAT):DFLOAT == u } -\begin{verbatim} - Function declaration n3 : DoubleFloat -> DoubleFloat has been added - to workspace. -\end{verbatim} -\returnType{Type: Void} - -Without these declarations, you have to suffix floats -with $@DFLOAT$ to get a {\tt DoubleFloat} result. -However, a call here with an unadorned float produces a {\tt DoubleFloat}. - -\spadcommand{n3(0.5,1.0)} -\begin{verbatim} - Compiling function n3 with type (DoubleFloat,DoubleFloat) -> - DoubleFloat -\end{verbatim} -\returnType{Type: DoubleFloat} - -Draw the surface by referencing the function names, this time -choosing the toroidal coordinate system. -\index{coordinate system!toroidal} -\index{toroidal coordinate system} - -\spadgraph{draw(surface(n1,n2,n3), 1..4, 1..2*\%pi, coordinates == toroidal(1\$DFLOAT)) } - -% window was 300 x 300 -%\epsffile[0 0 295 295]{ps/3dpsb.ps} - -\subsection{Three-Dimensional Options} -\label{ugGraphThreeDOptions} - -\index{graphics!3D options} -The {\bf draw} commands optionally take an optional list of options such -as {\tt coordinates} as shown in the last example. -Each option is given by the syntax: $name$ {\tt ==} $value$. -Here is a list of the available options in the order that they are -described below: - -\begin{tabular}{llll} -title&coordinates&var1Steps\\ -style&tubeRadius&var2Steps\\ -colorFunction&tubePoints&space\\ -\end{tabular} - -The option $title$ gives your graph a title. -\index{graphics!3D options!title} - -\spadgraph{draw(cos(x*y),x=0..2*\%pi,y=0..\%pi,title == "Title of Graph") } - -% window was 300 x 300 -%\epsffile[0 0 295 295]{ps/3doptttl.ps} - -The $style$ determines which of four rendering algorithms is used for -\index{rendering} -the graph. -The choices are -{\tt "wireMesh"}, {\tt "solid"}, {\tt "shade"}, and {\tt "smooth"}. - -\spadgraph{draw(cos(x*y),x=-3..3,y=-3..3, style=="smooth", title=="Smooth Option")} - -% window was 300 x 300 -%\epsffile[0 0 295 295]{ps/3doptsty.ps} - -In all but the wire-mesh style, polygons in a surface or tube plot -are normally colored in a graph according to their -$z$-coordinate value. Space curves are colored according to their -parametric variable value. -\index{graphics!3D options!color function} -To change this, you can give a coloring function. -\index{function!coloring} -The coloring function is sampled across the range of its arguments, then -normalized onto the standard Axiom colormap. - -A function of one variable makes the color depend on the -value of the parametric variable specified for a tube plot. - -\spadcommand{color1(t) == t } -\returnType{Type: Void} - -\spadgraph{draw(curve(sin(t), cos(t),0), t=0..2*\%pi, tubeRadius == .3, colorFunction == color1) } - -% window was 300 x 300 -%\epsffile[0 0 295 295]{ps/3doptcf1.ps} - -A function of two variables makes the color depend on the -values of the independent variables. - -\spadcommand{color2(u,v) == u**2 - v**2 } -\returnType{Type: Void} - -Use the option {\tt colorFunction} for special coloring. - -\spadgraph{draw(cos(u*v), u=-3..3, v=-3..3, colorFunction == color2) } - -% window was 300 x 300 -%\epsffile[0 0 295 295]{ps/3doptcf2.ps} - -With a three variable function, the -color also depends on the value of the function. - -\spadcommand{color3(x,y,fxy) == sin(x*fxy) + cos(y*fxy) } -\returnType{Type: Void} - -\spadgraph{draw(cos(x*y), x=-3..3, y=-3..3, colorFunction == color3) } - -% window was 300 x 300 -%\epsffile[0 0 295 295]{ps/3doptcf3.ps} - -Normally the Cartesian coordinate system is used. -\index{Cartesian!coordinate system} -To change this, use the {\tt coordinates} option. -\index{coordinate system!Cartesian} -For details, see \ref{ugGraphCoord} on page~\pageref{ugGraphCoord}. - -\spadcommand{m(u:DFLOAT,v:DFLOAT):DFLOAT == 1 } -\begin{verbatim} - Function declaration m : (DoubleFloat,DoubleFloat) -> DoubleFloat - has been added to workspace. -\end{verbatim} -\returnType{Type: Void} - -Use the spherical -\index{spherical coordinate system} -coordinate system. -\index{coordinate system!spherical} - -\spadgraph{draw(m, 0..2*\%pi,0..\%pi, coordinates == spherical, style=="shade") } - -% window was 300 x 300 -%\epsffile[0 0 295 295]{ps/3doptcrd.ps} - -Space curves may be displayed as tubes with polygonal cross sections. -\index{tube} -Two options, {\tt tubeRadius} and {\tt tubePoints}, control the size and -shape of this cross section. -% - -The {\tt tubeRadius} option specifies the radius of the tube that -\index{tube!radius} -encircles the specified space curve. - -\spadgraph{draw(curve(sin(t),cos(t),0),t=0..2*\%pi, style=="shade", tubeRadius == .3)} - -% window was 300 x 300 -%\epsffile[0 0 295 295]{ps/3doptrad.ps} - -The {\tt tubePoints} option specifies the number of vertices -\index{tube!points in polygon} -defining the polygon that is used to create a tube around the -specified space curve. -The larger this number is, the more cylindrical the tube becomes. - -\spadgraph{draw(curve(sin(t), cos(t), 0), t=0..2*\%pi, style=="shade", tubeRadius == .25, tubePoints == 3)} - -% window was 300 x 300 -%\epsffile[0 0 295 295]{ps/3doptpts.ps} - -\index{graphics!3D options!variable steps} -% - -Options \spadfunFrom{var1Steps}{DrawOption} and -\spadfunFrom{var2Steps}{DrawOption} specify the number of intervals into -which the grid defining a surface plot is subdivided with respect to the -first and second parameters of the surface function(s). - -\spadgraph{draw(cos(x*y),x=-3..3,y=-3..3, style=="shade", var1Steps == 30, var2Steps == 30)} - -% window was 300 x 300 -%\epsffile[0 0 295 295]{ps/3doptvb.ps} - -The {\tt space} option -of a {\bf draw} command lets you build multiple graphs in three space. -To use this option, first create an empty three-space object, -then use the {\tt space} option thereafter. -There is no restriction as to the number or kinds -of graphs that can be combined this way. - -Create an empty three-space object. - -\spadcommand{s := create3Space()\$(ThreeSpace DFLOAT) } -$$ -{3-Space with }0 \mbox{\rm components} -$$ -\returnType{Type: ThreeSpace DoubleFloat} - -\spadcommand{m(u:DFLOAT,v:DFLOAT):DFLOAT == 1 } -\begin{verbatim} - Function declaration m : (DoubleFloat,DoubleFloat) -> DoubleFloat - has been added to workspace. -\end{verbatim} -\returnType{Type: Void} - -Add a graph to this three-space object. -The new graph destructively inserts the graph -into $s$. - -\spadgraph{draw(m,0..\%pi,0..2*\%pi, coordinates == spherical, space == s) } - -% window was 300 x 300 -%\epsffile[0 0 295 295]{ps/3dmult1a.ps} - -Add a second graph to $s$. - -\spadgraph{v := draw(curve(1.5*sin(t), 1.5*cos(t),0), t=0..2*\%pi, tubeRadius == .25, space == s) } - -% window was 300 x 300 -%\epsffile[0 0 295 295]{ps/3dmult1b.ps} - -A three-space object can also be obtained from an existing three-di\-men\-sion\-al viewport -using the \spadfunFrom{subspace}{ThreeSpace} command. -You can then use {\bf makeViewport3D} to create a viewport window. - -Assign to $subsp$ the three-space object in viewport $v$. - -\spadcommand{subsp := subspace v } - -Reset the space component of $v$ to the value of $subsp$. - -\spadcommand{subspace(v, subsp) } - -Create a viewport window from a three-space object. - -\spadgraph{makeViewport3D(subsp,"Graphs") } - -\subsection{The makeObject Command} -\label{ugGraphMakeObject} - -An alternate way to create multiple graphs is to use -{\bf makeObject}. -The {\bf makeObject} command is similar to the {\bf draw} -command, except that it returns a three-space object rather than a -{\tt ThreeDimensionalViewport}. -In fact, {\bf makeObject} is called by the {\bf draw} -command to create the {\tt ThreeSpace} then -\spadfunFrom{makeViewport3D}{ThreeDimensionalViewport} to create a -viewport window. - -\spadcommand{m(u:DFLOAT,v:DFLOAT):DFLOAT == 1 } -\begin{verbatim} - Function declaration m : (DoubleFloat,DoubleFloat) -> DoubleFloat - has been added to workspace. -\end{verbatim} -\returnType{Type: Void} - -Do the last example a new way. -First use {\bf makeObject} to -create a three-space object $sph$. - -\spadcommand{sph := makeObject(m, 0..\%pi, 0..2*\%pi, coordinates==spherical)} -\begin{verbatim} - Compiling function m with type (DoubleFloat,DoubleFloat) -> - DoubleFloat -\end{verbatim} -$$ -{3-Space with }1 \mbox{\rm component} -$$ -\returnType{Type: ThreeSpace DoubleFloat} - -Add a second object to $sph$. - -\spadcommand{makeObject(curve(1.5*sin(t), 1.5*cos(t), 0), t=0..2*\%pi, space == sph, tubeRadius == .25) } -\begin{verbatim} - Compiling function %D with type DoubleFloat -> DoubleFloat - Compiling function %F with type DoubleFloat -> DoubleFloat - Compiling function %H with type DoubleFloat -> DoubleFloat -\end{verbatim} -$$ -{3-Space with }2 \mbox{\rm components} -$$ -\returnType{Type: ThreeSpace DoubleFloat} - -Create and display a viewport -containing $sph$. - -\spadgraph{makeViewport3D(sph,"Multiple Objects") } - -Note that an undefined {\tt ThreeSpace} parameter declared in a -{\bf makeObject} or {\bf draw} command results in an error. -Use the \spadfunFrom{create3Space}{ThreeSpace} function to define a -{\tt ThreeSpace}, or obtain a {\tt ThreeSpace} that has been -previously generated before including it in a command line. - -\subsection{Building Three-Dimensional Objects From Primitives} -\label{ugGraphThreeDBuild} - -Rather than using the {\bf draw} and {\bf makeObject} commands, -\index{graphics!advanced!build 3D objects} -you can create three-di\-men\-sion\-al graphs from primitives. -Operation \spadfunFrom{create3Space}{ThreeSpace} creates a -three-space object to which points, curves and polygons -can be added using the operations from the {\tt ThreeSpace} -domain. -The resulting object can then be displayed in a viewport using -\spadfunFrom{makeViewport3D}{ThreeDimensionalViewport}. - -Create the empty three-space object $space$. - -\spadcommand{space := create3Space()\$(ThreeSpace DFLOAT) } -$$ -{3-Space with }0 \mbox{\rm components} -$$ -\returnType{Type: ThreeSpace DoubleFloat} - -Objects can be sent to this $space$ using the operations -exported by the {\tt ThreeSpace} domain. -\index{ThreeSpace} -The following examples place curves into $space$. - -Add these eight curves to the space. - -\spadcommand{closedCurve(space,[ [0,30,20], [0,30,30], [0,40,30], [0,40,100], [0,30,100],[0,30,110], [0,60,110], [0,60,100], [0,50,100], [0,50,30], [0,60,30], [0,60,20] ]) } -$$ -{3-Space with }1 \mbox{\rm component} -$$ -\returnType{Type: ThreeSpace DoubleFloat} - -\spadcommand{closedCurve(space,[ [80,0,30], [80,0,100], [70,0,110], [40,0,110], [30,0,100], [30,0,90], [40,0,90], [40,0,95], [45,0,100], [65,0,100], [70,0,95], [70,0,35] ]) } -$$ -{3-Space with }2 \mbox{\rm components} -$$ -\returnType{Type: ThreeSpace DoubleFloat} - -\spadcommand{closedCurve(space,[ [70,0,35], [65,0,30], [45,0,30], [40,0,35], [40,0,60], [50,0,60], [50,0,70], [30,0,70], [30,0,30], [40,0,20], [70,0,20], [80,0,30] ]) } -$$ -{3-Space with }3 \mbox{\rm components} -$$ -\returnType{Type: ThreeSpace DoubleFloat} - -\spadcommand{closedCurve(space,[ [0,70,20], [0,70,110], [0,110,110], [0,120,100], [0,120,70], [0,115,65], [0,120,60], [0,120,30], [0,110,20], [0,80,20], [0,80,30], [0,80,20] ]) } -$$ -{3-Space with }4 \mbox{\rm components} -$$ -\returnType{Type: ThreeSpace DoubleFloat} - -\spadcommand{closedCurve(space,[ [0,105,30], [0,110,35], [0,110,55], [0,105,60], [0,80,60], [0,80,70], [0,105,70], [0,110,75], [0,110,95], [0,105,100], [0,80,100], [0,80,20], [0,80,30] ]) } -$$ -{3-Space with }5 \mbox{\rm components} -$$ -\returnType{Type: ThreeSpace DoubleFloat} - -\spadcommand{closedCurve(space,[ [140,0,20], [140,0,110], [130,0,110], [90,0,20], [101,0,20],[114,0,50], [130,0,50], [130,0,60], [119,0,60], [130,0,85], [130,0,20] ]) } -$$ -{3-Space with }6 \mbox{\rm components} -$$ -\returnType{Type: ThreeSpace DoubleFloat} - -\spadcommand{closedCurve(space,[ [0,140,20], [0,140,110], [0,150,110], [0,170,50], [0,190,110], [0,200,110], [0,200,20], [0,190,20], [0,190,75], [0,175,35], [0,165,35],[0,150,75], [0,150,20] ]) } -$$ -{3-Space with }7 \mbox{\rm components} -$$ -\returnType{Type: ThreeSpace DoubleFloat} - -\spadcommand{closedCurve(space,[ [200,0,20], [200,0,110], [189,0,110], [160,0,45], [160,0,110], [150,0,110], [150,0,20], [161,0,20], [190,0,85], [190,0,20] ]) } -$$ -{3-Space with }8 \mbox{\rm components} -$$ -\returnType{Type: ThreeSpace DoubleFloat} - -Create and display the viewport using {\bf makeViewport3D}. -Options may also be given but here are displayed as a list with values -enclosed in parentheses. - -\spadgraph{makeViewport3D(space, title == "Letters") } - -% window was 300 x 300 -%\epsffile[0 0 295 295]{ps/3dbuilda.ps} - -\subsubsection{Cube Example} - -As a second example of the use of primitives, we generate a cube using a -polygon mesh. -It is important to use a consistent orientation of the polygons for -correct generation of three-di\-men\-sion\-al objects. - -Again start with an empty three-space object. - -\spadcommand{spaceC := create3Space()\$(ThreeSpace DFLOAT) } -$$ -{3-Space with }0 \mbox{\rm components} -$$ -\returnType{Type: ThreeSpace DoubleFloat} - -For convenience, -give {\tt DoubleFloat} values $+1$ and $-1$ names. - -\spadcommand{x: DFLOAT := 1 } -$$ -1.0 -$$ -\returnType{Type: DoubleFloat} - -\spadcommand{y: DFLOAT := -1 } -$$ --{1.0} -$$ -\returnType{Type: DoubleFloat} - -Define the vertices of the cube. - -\spadcommand{a := point [x,x,y,1::DFLOAT]\$(Point DFLOAT) } -$$ -\left[ -{1.0}, {1.0}, -{1.0}, {1.0} -\right] -$$ -\returnType{Type: Point DoubleFloat} - -\spadcommand{b := point [y,x,y,4::DFLOAT]\$(Point DFLOAT) } -$$ -\left[ --{1.0}, {1.0}, -{1.0}, {4.0} -\right] -$$ -\returnType{Type: Point DoubleFloat} - -\spadcommand{c := point [y,x,x,8::DFLOAT]\$(Point DFLOAT) } -$$ -\left[ --{1.0}, {1.0}, {1.0}, {8.0} -\right] -$$ -\returnType{Type: Point DoubleFloat} - -\spadcommand{d := point [x,x,x,12::DFLOAT]\$(Point DFLOAT) } -$$ -\left[ -{1.0}, {1.0}, {1.0}, {12.0} -\right] -$$ -\returnType{Type: Point DoubleFloat} - -\spadcommand{e := point [x,y,y,16::DFLOAT]\$(Point DFLOAT) } -$$ -\left[ -{1.0}, -{1.0}, -{1.0}, {16.0} -\right] -$$ -\returnType{Type: Point DoubleFloat} - -\spadcommand{f := point [y,y,y,20::DFLOAT]\$(Point DFLOAT) } -$$ -\left[ --{1.0}, -{1.0}, -{1.0}, {20.0} -\right] -$$ -\returnType{Type: Point DoubleFloat} - -\spadcommand{g := point [y,y,x,24::DFLOAT]\$(Point DFLOAT) } -$$ -\left[ --{1.0}, -{1.0}, {1.0}, {24.0} -\right] -$$ -\returnType{Type: Point DoubleFloat} - -\spadcommand{h := point [x,y,x,27::DFLOAT]\$(Point DFLOAT) } -$$ -\left[ -{1.0}, -{1.0}, {1.0}, {27.0} -\right] -$$ -\returnType{Type: Point DoubleFloat} - -Add the faces of the cube as polygons to the space using a -consistent orientation. - -\spadcommand{polygon(spaceC,[d,c,g,h]) } -$$ -{3-Space with }1 \mbox{\rm component} -$$ -\returnType{Type: ThreeSpace DoubleFloat} - -\spadcommand{polygon(spaceC,[d,h,e,a]) } -$$ -{3-Space with }2 \mbox{\rm components} -$$ -\returnType{Type: ThreeSpace DoubleFloat} - -\spadcommand{polygon(spaceC,[c,d,a,b]) } -$$ -{3-Space with }3 \mbox{\rm components} -$$ -\returnType{Type: ThreeSpace DoubleFloat} - -\spadcommand{polygon(spaceC,[g,c,b,f]) } -$$ -{3-Space with }4 \mbox{\rm components} -$$ -\returnType{Type: ThreeSpace DoubleFloat} - -\spadcommand{polygon(spaceC,[h,g,f,e]) } -$$ -{3-Space with }5 \mbox{\rm components} -$$ -\returnType{Type: ThreeSpace DoubleFloat} - -\spadcommand{polygon(spaceC,[e,f,b,a]) } -$$ -{3-Space with }6 \mbox{\rm components} -$$ -\returnType{Type: ThreeSpace DoubleFloat} - -Create and display the viewport. - -\spadgraph{makeViewport3D(spaceC, title == "Cube") } - -% window was 300 x 300 -%\epsffile[0 0 295 295]{ps/3dbuildb.ps} - -\subsection{Coordinate System Transformations} -\label{ugGraphCoord} -\index{graphics!advanced!coordinate systems} - -The {\tt CoordinateSystems} package provides coordinate transformation -functions that map a given data point from the coordinate system specified -into the Cartesian coordinate system. -\index{CoordinateSystems} -The default coordinate system, given a triplet $(f(u,v), u, v)$, assumes -that $z = f(u, v)$, $x = u$ and $y = v$, -that is, reads the coordinates in $(z, x, y)$ order. - -\spadcommand{m(u:DFLOAT,v:DFLOAT):DFLOAT == u**2 } -\begin{verbatim} - Function declaration m : (DoubleFloat,DoubleFloat) -> DoubleFloat - has been added to workspace. -\end{verbatim} -\returnType{Type: Void} - -Graph plotted in default coordinate system. - -\spadgraph{draw(m,0..3,0..5) } - -% window was 300 x 300 -%\epsffile[0 0 295 295]{ps/defcoord.ps} - -The $z$ coordinate comes first since the first argument of -the {\bf draw} command gives its values. -In general, the coordinate systems Axiom provides, or any -that you make up, must provide a map to an $(x, y, z)$ triplet in -order to be compatible with the -\spadfunFrom{coordinates}{DrawOption} {\tt DrawOption}. -\index{DrawOption} -Here is an example. - -Define the identity function. - -\spadcommand{cartesian(point:Point DFLOAT):Point DFLOAT == point } -\begin{verbatim} - Function declaration cartesian : Point DoubleFloat -> Point - DoubleFloat has been added to workspace. -\end{verbatim} -\returnType{Type: Void} - -Pass $cartesian$ as the \spadfunFrom{coordinates}{DrawOption} -parameter to the {\bf draw} command. - -\spadgraph{draw(m,0..3,0..5,coordinates==cartesian) } - -% window was 300 x 300 -%\epsffile[0 0 295 295]{ps/cartcoord.ps} - -What happened? The option {\tt coordinates == cartesian} directs -Axiom to treat the dependent variable $m$ defined by $m=u^2$ as the -$x$ coordinate. Thus the triplet of values $(m, u, v)$ is transformed -to coordinates $(x, y, z)$ and so we get the graph of $x=y^2$. - -Here is another example. -The \spadfunFrom{cylindrical}{CoordinateSystems} transform takes -\index{coordinate system!cylindrical} -input of the form $(w,u,v)$, interprets it in the order -\index{cylindrical coordinate system} -($r$,$\theta$,$z$) -and maps it to the Cartesian coordinates -$x=r\cos(\theta)$, $y=r\sin(\theta)$, $z=z$ -in which -$r$ is the radius, -$\theta$ is the angle and -$z$ is the z-coordinate. - -An example using the \spadfunFrom{cylindrical}{CoordinateSystems} -coordinates for the constant $r = 3$. - -\spadcommand{f(u:DFLOAT,v:DFLOAT):DFLOAT == 3 } -\begin{verbatim} - Function declaration f : (DoubleFloat,DoubleFloat) -> DoubleFloat - has been added to workspace. -\end{verbatim} -\returnType{Type: Void} - -Graph plotted in cylindrical coordinates. - -\spadgraph{draw(f,0..\%pi,0..6,coordinates==cylindrical) } - -% window was 300 x 300 -%\epsffile[0 0 295 295]{ps/cylcoord.ps} - -Suppose you would like to specify $z$ as a function of -$r$ and $\theta$ instead of just $r$? -Well, you still can use the {\bf cylindrical} Axiom -transformation but we have to reorder the triplet before -passing it to the transformation. - -First, let's create a point to -work with and call it $pt$ with some color $col$. - -\spadcommand{col := 5 } -$$ -5 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{pt := point[1,2,3,col]\$(Point DFLOAT) } -$$ -\left[ -{1.0}, {2.0}, {3.0}, {5.0} -\right] -$$ -\returnType{Type: Point DoubleFloat} - -The reordering you want is -$(z,r, \theta)$ to -$(r, \theta,z)$ -so that the first element is moved to the third element, while the second -and third elements move forward and the color element does not change. - -Define a function {\bf reorder} to reorder the point elements. - -\spadcommand{reorder(p:Point DFLOAT):Point DFLOAT == point[p.2, p.3, p.1, p.4] } -\begin{verbatim} - Function declaration reorder : Point DoubleFloat -> Point - DoubleFloat has been added to workspace. -\end{verbatim} -\returnType{Type: Void} - -The function moves the second and third elements -forward but the color does not change. - -\spadcommand{reorder pt } -$$ -\left[ -{2.0}, {3.0}, {1.0}, {5.0} -\right] -$$ -\returnType{Type: Point DoubleFloat} - -The function {\bf newmap} converts our reordered version of -the cylindrical coordinate system to the standard -$(x,y,z)$ Cartesian system. - -\spadcommand{newmap(pt:Point DFLOAT):Point DFLOAT == cylindrical(reorder pt) } -\begin{verbatim} - Function declaration newmap : Point DoubleFloat -> Point DoubleFloat - has been added to workspace. -\end{verbatim} -\returnType{Type: Void} - -\spadcommand{newmap pt } -$$ -\left[ --{1.9799849932008908}, {0.28224001611973443}, {1.0}, {5.0} -\right] -$$ -\returnType{Type: Point DoubleFloat} - -Graph the same function $f$ using the coordinate mapping of the function -$newmap$, so it is now interpreted as -$z=3$: - -\spadgraph{draw(f,0..3,0..2*\%pi,coordinates==newmap) } - -% window was 300 x 300 -%\epsffile[0 0 295 295]{ps/newmap.ps} - -% I think this is good to say here: it shows a lot of depth. RSS -%{\sloppy -The {\tt CoordinateSystems} package exports the following -\index{coordinate system} -operations: -{\bf bipolar}, -{\bf bipolarCylindrical}, -{\bf cartesian}, -{\bf conical}, -{\bf cylindrical}, -{\bf elliptic}, -{\bf ellipticCylindrical}, -{\bf oblateSpheroidal}, -{\bf parabolic}, -{\bf parabolicCylindrical}, -{\bf paraboloidal}, -{\bf polar}, -{\bf prolateSpheroidal}, -{\bf spherical}, and -{\bf toroidal}. -Use Browse or the {\tt )show} system command -\index{show} -to get more information. - -\subsection{Three-Dimensional Clipping} -\label{ugGraphClip} - -A three-di\-men\-sion\-al graph can be explicitly clipped within the {\bf draw} -\index{graphics!advanced!clip} -command by indicating a minimum and maximum threshold for the -\index{clipping} -given function definition. -These thresholds can be defined using the Axiom {\bf min} -and {\bf max} functions. - -\begin{verbatim} -gamma(x,y) == - g := Gamma complex(x,y) - point [x, y, max( min(real g, 4), -4), argument g] -\end{verbatim} - -Here is an example that clips -the gamma function in order to eliminate the extreme divergence it creates. - -\spadgraph{draw(gamma,-\%pi..\%pi,-\%pi..\%pi,var1Steps==50,var2Steps==50) } - -% window was 300 x 300 -%\epsffile[0 0 295 295]{ps/clipgamma.ps} - -\subsection{Three-Dimensional Control-Panel} -\label{ugGraphThreeDControl} - -\index{graphics!3D control-panel} -Once you have created a viewport, move your mouse to the viewport -and click with your left mouse button. -This displays a control-panel on the side of the viewport -that is closest to where you clicked. - -\begin{figure}[htbp] -%{\epsfverbosetrue\epsfxsize=2in% -%\def\epsfsize#1#2{\epsfxsize}\hspace*{\baseLeftSkip}% -%%\epsffile[0 0 144 289]{ps/3dctrl.ps}} -\begin{picture}(183,252)%(-125,0) -\hspace*{\baseLeftSkip}\special{psfile=ps/3dctrl.ps} -\end{picture} -\caption{Three-dimensional control-panel.} -\end{figure} - -\subsubsection{Transformations} - -We recommend you first select the {\bf Bounds} button while -\index{graphics!3D control-panel!transformations} -executing transformations since the bounding box displayed -indicates the object's position as it changes. -% -\begin{description} -% -\item[Rotate:] A rotation transformation occurs by clicking the mouse -\index{graphics!3D control-panel!rotate} -within the {\bf Rotate} window in the upper left corner of the -control-panel. -The rotation is computed in spherical coordinates, using the -horizontal mouse position to increment or decrement the value of -the longitudinal angle $\theta$ within the -range of 0 to 2$\pi$ and the vertical mouse position -to increment or decrement the value of the latitudinal angle -$\phi$ within the range of -$\pi$ -to $\pi$. -The active mode of rotation is displayed in green on a color -monitor or in clear text on a black and white monitor, while the -inactive mode is displayed in red for color display or a mottled -pattern for black and white. -% -\begin{description} -% -\item[origin:] The {\bf origin} button indicates that the -rotation is to occur with respect to the origin of the viewing space, that is -indicated by the axes. -% -\item[object:] The {\bf object} button indicates that the -rotation is to occur with respect to the center of volume of the object, -independent of the axes' origin position. -\end{description} -% -\item[Scale:] A scaling transformation occurs by clicking the mouse -\index{graphics!3D control-panel!scale} -within the {\bf Scale} window in the upper center of the -control-panel, containing a zoom arrow. -The axes along which the scaling is to occur are indicated by -selecting the appropriate button above the zoom arrow window. -The selected axes are displayed in green on a color monitor or in -clear text on a black and white monitor, while the unselected axes -are displayed in red for a color display or a mottled pattern for -black and white. -% -\begin{description} -% -\item[uniform:] Uniform scaling along the {\tt x}, {\tt y} -and {\tt z} axes occurs when all the axes buttons are selected. -% -\item[non-uniform:] If any of the axes buttons are -not selected, non-uniform scaling occurs, that is, scaling occurs only in the -direction of the axes that are selected. -\end{description} -% -\item[Translate:] Translation occurs by indicating with the mouse in the -\index{graphics!3D control-panel!translate} -{\bf Translate} window the direction you want the graph to move. -This window is located in the upper right corner of the -control-panel and contains a potentiometer with crossed arrows -pointing up, down, left and right. -Along the top of the {\bf Translate} window are three buttons -({\bf XY}, -{\bf XZ}, and {\bf YZ}) indicating the three orthographic projection planes. -Each orientates the group as a view into that plane. -Any translation of the graph occurs only along this plane. -\end{description} - -\subsubsection{Messages} - -\index{graphics!3D control-panel!messages} - -The window directly below the potentiometer windows for transformations is -used to display system messages relating to the viewport, the control-panel -and the current graph displaying status. - -\subsubsection{Colormap} - -\index{graphics!3D control-panel!color map} - -Directly below the message window is the colormap range indicator -window. -\index{colormap} -The Axiom Colormap shows a sampling of the spectrum from -which hues can be drawn to represent the colors of a surface. -The Colormap is composed of five shades for each of the hues along -this spectrum. -By moving the markers above and below the Colormap, the range of -hues that are used to color the existing surface are set. -The bottom marker shows the hue for the low end of the color range -and the top marker shows the hue for the upper end of the range. -Setting the bottom and top markers at the same hue results in -monochromatic smooth shading of the graph when {\bf Smooth} mode is selected. -At each end of the Colormap are {\bf +} and {\bf -} buttons. -When clicked on, these increment or decrement the top or bottom -marker. - -\subsubsection{Buttons} -\index{graphics!3D control-panel!buttons} - -Below the Colormap window and to the left are located various -buttons that determine the characteristics of a graph. -The buttons along the bottom and right hand side all have special -meanings; the remaining buttons in the first row indicate the mode -or style used to display the graph. -The second row are toggles that turn on or off a property of the -graph. -On a color monitor, the property is on if green (clear text, on a -monochrome monitor) and off if red (mottled pattern, on a -monochrome monitor). -Here is a list of their functions. -% -\begin{description} -% -\item[Wire] displays surface and tube plots as a -\index{graphics!3D control-panel!wire} -wireframe image in a single color (blue) with no hidden surfaces removed, -or displays space curve plots in colors based upon their parametric variables. -This is the fastest mode for displaying a graph. -This is very useful when you -want to find a good orientation of your graph. -% -\item[Solid] displays the graph with hidden -\index{graphics!3D control-panel!solid} -surfaces removed, drawing each polygon beginning with the furthest -from the viewer. -The edges of the polygons are displayed in the hues specified by -the range in the Colormap window. -% -\item[Shade] displays the graph with hidden -\index{graphics!3D control-panel!shade} -surfaces removed and with the polygons shaded, drawing each -polygon beginning with the furthest from the viewer. -Polygons are shaded in the hues specified by the range in the -Colormap window using the Phong illumination model. -\index{Phong!illumination model} -% -\item[Smooth] displays the graph using a -\index{graphics!3D control-panel!smooth} -renderer that computes the graph one line at a time. -The location and color of the graph at each visible point on the -screen are determined and displayed using the Phong illumination -\index{Phong!illumination model} -model. -Smooth shading is done in one of two ways, depending on the range -selected in the colormap window and the number of colors available -from the hardware and/or window manager. -When the top and bottom markers of the colormap range are set to -different hues, the graph is rendered by dithering between the -\index{dithering} -transitions in color hue. -When the top and bottom markers of the colormap range are set to -the same hue, the graph is rendered using the Phong smooth shading -model. -\index{Phong!smooth shading model} -However, if enough colors cannot be allocated for this purpose, -the renderer reverts to the color dithering method until a -sufficient color supply is available. -For this reason, it may not be possible to render multiple Phong -smooth shaded graphs at the same time on some systems. -% -\item[Bounds] encloses the entire volume of the -viewgraph within a bounding box, or removes the box if previously selected. -\index{graphics!3D control-panel!bounds} -The region that encloses the entire volume of the viewport graph is displayed. -% -\item[Axes] displays Cartesian -\index{graphics!3D control-panel!axes} -coordinate axes of the space, or turns them off if previously selected. -% -\item[Outline] causes -\index{graphics!3D control-panel!outline} -quadrilateral polygons forming the graph surface to be outlined in black when -the graph is displayed in {\bf Shade} mode. -% -\item[BW] converts a color viewport to black and white, or vice-versa. -\index{graphics!3D control-panel!bw} -When this button is selected the -control-panel and viewport switch to an immutable colormap composed of a range -of grey scale patterns or tiles that are used wherever shading is necessary. -% -\item[Light] takes you to a control-panel described below. -% -\item[ViewVolume] takes you to another control-panel as described below. -\index{graphics!3D control-panel!save} -% -\item[Save] creates a menu of the possible file types that can -be written using the control-panel. -The {\bf Exit} button leaves the save menu. -The {\bf Pixmap} button writes an Axiom pixmap of -\index{graphics!3D control-panel!pixmap} -the current viewport contents. The file is called {\bf axiom3D.pixmap} and is -located in the directory from which Axiom or {\bf viewalone} was -started. -The {\bf PS} button writes the current viewport contents to -\index{graphics!3D control-panel!ps} -PostScript output rather than to the viewport window. -By default the file is called {\bf axiom3D.ps}; however, if a file -\index{file!.Xdefaults @{\bf .Xdefaults}} -name is specified in the user's {\bf .Xdefaults} file it is -\index{graphics!.Xdefaults!PostScript file name} -used. -The file is placed in the directory from which the Axiom or -{\bf viewalone} session was begun. -See also the \spadfunFrom{write}{ThreeDimensionalViewport} -function. -\index{PostScript} -% -\item[Reset] returns the object transformation -\index{graphics!3D control-panel!reset} -characteristics back to their initial states. -% -\item[Hide] causes the control-panel for the -\index{graphics!3D control-panel!hide} -corresponding viewport to disappear from the screen. -% -\item[Quit] queries whether the current viewport -\index{graphics!3D control-panel!quit} -session should be terminated. -\end{description} - -\subsubsection{Light} - -\index{graphics!3D control-panel!light} - -%>>>\begin{figure}[htbp] -%>>>\begin{picture}(183,252)(-125,0) -%>>>\special{psfile=ps/3dlight.ps} -%>>>\end{picture} -%>>>\caption{Three-Dimensional Lighting Panel.} -%>>>\end{figure} - -The {\bf Light} button changes the control-panel into the -{\bf Lighting Control-Panel}. At the top of this panel, the three axes -are shown with the same orientation as the object. A light vector from -the origin of the axes shows the current position of the light source -relative to the object. At the bottom of the panel is an {\bf Abort} -button that cancels any changes to the lighting that were made, and a -{\bf Return} button that carries out the current set of lighting changes -on the graph. -% -\begin{description} -% -\item[XY:] The {\bf XY} lighting axes window is below the -\index{graphics!3D control-panel!move xy} -{\bf Lighting Control-Panel} title and to the left. -This changes the light vector within the {\bf XY} view plane. -% -\item[Z:] The {\bf Z} lighting axis window is below the -\index{graphics!3D control-panel!move z} -{\bf Lighting Control-Panel} title and in the center. This -changes the {\bf Z} -location of the light vector. -% -\item[Intensity:] -Below the {\bf Lighting Control-Panel} title -\index{graphics!3D control-panel!intensity} -and to the right is the light intensity meter. -Moving the intensity indicator down decreases the amount of -light emitted from the light source. -When the indicator is at the top of the meter the light source is -emitting at 100\% intensity. -At the bottom of the meter the light source is emitting at a level -slightly above ambient lighting. -\end{description} - -\subsubsection{View Volume} - -\index{graphics!3D control-panel!view volume} - -The {\bf View Volume} button changes the control-panel into -the {\bf Viewing Volume Panel}. -At the bottom of the viewing panel is an {\bf Abort} button that -cancels any changes to the viewing volume that were made and a -{\it Return} button that carries out the current set of -viewing changes to the graph. -% -%>>>\begin{figure}[htbp] -%>>>\begin{picture}(183,252)(-125,0) -%>>>\special{psfile=ps/3dvolume.ps} -%>>>\end{picture} -%>>>\caption{Three-Dimensional Volume Panel.} -%>>>\end{figure} - -\begin{description} - -\item[Eye Reference:] At the top of this panel is the -\index{graphics!3D control-panel!eye reference} -{\bf Eye Reference} window. -It shows a planar projection of the viewing pyramid from the eye -of the viewer relative to the location of the object. -This has a bounding region represented by the rectangle on the -left. -Below the object rectangle is the {\bf Hither} window. -By moving the slider in this window the hither clipping plane sets -\index{hither clipping plane} -the front of the view volume. -As a result of this depth clipping all points of the object closer -to the eye than this hither plane are not shown. -The {\bf Eye Distance} slider to the right of the {\bf Hither} -slider is used to change the degree of perspective in the image. -% -\item[Clip Volume:] The {\bf Clip Volume} window is at the -\index{graphics!3D control-panel!clip volume} -bottom of the {\bf Viewing Volume Panel}. -On the right is a {\bf Settings} menu. -In this menu are buttons to select viewing attributes. -Selecting the {\bf Perspective} button computes the image using -perspective projection. -\index{graphics!3D control-panel!perspective} -The {\bf Show Region} button indicates whether the clipping region -of the -\index{graphics!3D control-panel!show clip region} -volume is to be drawn in the viewport and the {\bf Clipping On} -button shows whether the view volume clipping is to be in effect -when the image -\index{graphics!3D control-panel!clipping on} -is drawn. -The left side of the {\bf Clip Volume} window shows the clipping -\index{graphics!3D control-panel!clip volume} -boundary of the graph. -Moving the knobs along the {\bf X}, {\bf Y}, and {\bf Z} sliders -adjusts the volume of the clipping region accordingly. -\end{description} - -\subsection{Operations for Three-Dimensional Graphics} -\label{ugGraphThreeDops} - - -Here is a summary of useful Axiom operations for three-di\-men\-sion\-al -graphics. -Each operation name is followed by a list of arguments. -Each argument is written as a variable informally named according -to the type of the argument (for example, {\it integer}). -If appropriate, a default value for an argument is given in -parentheses immediately following the name. - -% -\bgroup\hbadness = 10001\sloppy -\begin{description} -% -\item[{\bf adaptive3D?}]\funArgs{} -tests whether space curves are to be plotted -\index{graphics!plot3d defaults!adaptive} -according to the -\index{adaptive plotting} -adaptive refinement algorithm. - -% -\item[{\bf axes}]\funArgs{viewport, string\argDef{"on"}} -turns the axes on and off. -\index{graphics!3D commands!axes} - -% -\item[{\bf close}]\funArgs{viewport} -closes the viewport. -\index{graphics!3D commands!close} - -% -\item[{\bf colorDef}]\funArgs{viewport, -$\hbox{\it color}_{1}$\argDef{1}, $\hbox{\it color}_{2}$\argDef{27}} -sets the colormap -\index{graphics!3D commands!define color} -range to be from -$\hbox{\it color}_{1}$ to $\hbox{\it color}_{2}$. - -% -\item[{\bf controlPanel}]\funArgs{viewport, string\argDef{"off"}} -declares whether the -\index{graphics!3D commands!control-panel} -control-panel for the viewport is to be displayed or not. - -% -\item[{\bf diagonals}]\funArgs{viewport, string\argDef{"off"}} -declares whether the -\index{graphics!3D commands!diagonals} -polygon outline includes the diagonals or not. - -% -\item[{\bf drawStyle}]\funArgs{viewport, style} -selects which of four drawing styles -\index{graphics!3D commands!drawing style} -are used: {\tt "wireMesh", "solid", "shade",} or {\tt "smooth".} - -% -\item[{\bf eyeDistance}]\funArgs{viewport,float\argDef{500}} -sets the distance of the eye from the origin of the object -\index{graphics!3D commands!eye distance} -for use in the \spadfunFrom{perspective}{ThreeDimensionalViewport}. - -% -\item[{\bf key}]\funArgs{viewport} -returns the operating -\index{graphics!3D commands!key} -system process ID number for the viewport. - -% -\item[{\bf lighting}]\funArgs{viewport, -$float_{x}$\argDef{-0.5}, -$float_{y}$\argDef{0.5}, $float_{z}$\argDef{0.5}} -sets the Cartesian -\index{graphics!3D commands!lighting} -coordinates of the light source. - -% -\item[{\bf modifyPointData}]\funArgs{viewport,integer,point} -replaces the coordinates of the point with -\index{graphics!3D commands!modify point data} -the index {\it integer} with {\it point}. - -% -\item[{\bf move}]\funArgs{viewport, -$integer_{x}$\argDef{viewPosDefault}, -$integer_{y}$\argDef{viewPosDefault}} -moves the upper -\index{graphics!3D commands!move} -left-hand corner of the viewport to screen position -\allowbreak -({\small $integer_{x}$, $integer_{y}$}). - -% -\item[{\bf options}]\funArgs{viewport} -returns a list of all current draw options. - -% -\item[{\bf outlineRender}]\funArgs{viewport, string\argDef{"off"}} -turns polygon outlining -\index{graphics!3D commands!outline} -off or on when drawing in {\tt "shade"} mode. - -% -\item[{\bf perspective}]\funArgs{viewport, string\argDef{"on"}} -turns perspective -\index{graphics!3D commands!perspective} -viewing on and off. - -% -\item[{\bf reset}]\funArgs{viewport} -resets the attributes of a viewport to their -\index{graphics!3D commands!reset} -initial settings. - -% -\item[{\bf resize}]\funArgs{viewport, -$integer_{width}$ \argDef{viewSizeDefault}, -$integer_{height}$ \argDef{viewSizeDefault}} -resets the width and height -\index{graphics!3D commands!resize} -values for a viewport. - -% -\item[{\bf rotate}]\funArgs{viewport, -$number_{\theta}$\argDef{viewThetaDefapult}, -$number_{\phi}$\argDef{viewPhiDefault}} -rotates the viewport by rotation angles for longitude -({\it $\theta$}) and -latitude ({\it $\phi$}). -Angles designate radians if given as floats, or degrees if given -\index{graphics!3D commands!rotate} -as integers. - -% -\item[{\bf setAdaptive3D}]\funArgs{boolean\argDef{true}} -sets whether space curves are to be plotted -\index{graphics!plot3d defaults!set adaptive} -according to the adaptive -\index{adaptive plotting} -refinement algorithm. - -% -\item[{\bf setMaxPoints3D}]\funArgs{integer\argDef{1000}} - sets the default maximum number of possible -\index{graphics!plot3d defaults!set max points} -points to be used when constructing a three-di\-men\-sion\-al space curve. - -% -\item[{\bf setMinPoints3D}]\funArgs{integer\argDef{49}} -sets the default minimum number of possible -\index{graphics!plot3d defaults!set min points} -points to be used when constructing a three-di\-men\-sion\-al space curve. - -% -\item[{\bf setScreenResolution3D}]\funArgs{integer\argDef{49}} -sets the default screen resolution constant -\index{graphics!plot3d defaults!set screen resolution} -used in setting the computation limit of adaptively -\index{adaptive plotting} -generated three-di\-men\-sion\-al space curve plots. - -% -\item[{\bf showRegion}]\funArgs{viewport, string\argDef{"off"}} -declares whether the bounding -\index{graphics!3D commands!showRegion} -box of a graph is shown or not. -% -\item[{\bf subspace}]\funArgs{viewport} -returns the space component. -% -\item[{\bf subspace}]\funArgs{viewport, subspace} -resets the space component -\index{graphics!3D commands!subspace} -to {\it subspace}. - -% -\item[{\bf title}]\funArgs{viewport, string} -gives the viewport the -\index{graphics!3D commands!title} -title {\it string}. - -% -\item[{\bf translate}]\funArgs{viewport, -$float_{x}$\argDef{viewDeltaXDefault}, -$float_{y}$\argDef{viewDeltaYDefault}} -translates -\index{graphics!3D commands!translate} -the object horizontally and vertically relative to the center of the viewport. - -% -\item[{\bf intensity}]\funArgs{viewport,float\argDef{1.0}} -resets the intensity {\it I} of the light source, -\index{graphics!3D commands!intensity} -$0 \le I \le 1.$ - -% -\item[{\bf tubePointsDefault}]\funArgs{\optArg{integer\argDef{6}}} -sets or indicates the default number of -\index{graphics!3D defaults!tube points} -vertices defining the polygon that is used to create a tube around -a space curve. - -% -\item[{\bf tubeRadiusDefault}]\funArgs{\optArg{float\argDef{0.5}}} -sets or indicates the default radius of -\index{graphics!3D defaults!tube radius} -the tube that encircles a space curve. - -% -\item[{\bf var1StepsDefault}]\funArgs{\optArg{integer\argDef{27}}} -sets or indicates the default number of -\index{graphics!3D defaults!var1 steps} -increments into which the grid defining a surface plot is subdivided with -respect to the first parameter declared in the surface function. - -% -\item[{\bf var2StepsDefault}]\funArgs{\optArg{integer\argDef{27}}} -sets or indicates the default number of -\index{graphics!3D defaults!var2 steps} -increments into which the grid defining a surface plot is subdivided with -respect to the second parameter declared in the surface function. - -% -\item[{\bf viewDefaults}]\funArgs{{\tt [}$integer_{point}$, -$integer_{line}$, $integer_{axes}$, -$integer_{units}$, $float_{point}$, -\allowbreak$list_{position}$, -$list_{size}${\tt ]}} -resets the default settings for the -\index{graphics!3D defaults!reset viewport defaults} -point color, line color, axes color, units color, point size, -viewport upper left-hand corner position, and the viewport size. - -% -\item[{\bf viewDeltaXDefault}]\funArgs{\optArg{float\argDef{0}}} -resets the default horizontal offset -\index{graphics!3D commands!deltaX default} -from the center of the viewport, or returns the current default offset if no argument is given. - -% -\item[{\bf viewDeltaYDefault}]\funArgs{\optArg{float\argDef{0}}} -resets the default vertical offset -\index{graphics!3D commands!deltaY default} -from the center of the viewport, or returns the current default offset if no argument is given. - -% -\item[{\bf viewPhiDefault}]\funArgs{\optArg{float\argDef{-$\pi$/4}}} -resets the default latitudinal view angle, -or returns the current default angle if no argument is given. -\index{graphics!3D commands!phi default} -$\phi$ is set to this value. - -% -\item[{\bf viewpoint}]\funArgs{viewport, $float_{x}$, -$float_{y}$, $float_{z}$} -sets the viewing position in Cartesian coordinates. - -% -\item[{\bf viewpoint}]\funArgs{viewport, -$float_{\theta}$, -$Float_{\phi}$} -sets the viewing position in spherical coordinates. - -% -\item[{\bf viewpoint}]\funArgs{viewport, -$Float_{\theta}$, -$Float_{\phi}$, -$Float_{scaleFactor}$, -$Float_{xOffset}$, $Float_{yOffset}$} -sets the viewing position in spherical coordinates, -the scale factor, and offsets. -\index{graphics!3D commands!viewpoint} -$\theta$ (longitude) and -$\phi$ (latitude) are in radians. - -% -\item[{\bf viewPosDefault}]\funArgs{\optArg{list\argDef{[0,0]}}} -sets or indicates the position of the upper -\index{graphics!3D defaults!viewport position} -left-hand corner of a two-di\-men\-sion\-al viewport, relative to the display root -window (the upper left-hand corner of the display is $[0, 0]$). - -% -\item[{\bf viewSizeDefault}]\funArgs{\optArg{list\argDef{[400,400]}}} -sets or indicates the width and height dimensions -\index{graphics!3D defaults!viewport size} -of a viewport. - -% -\item[{\bf viewThetaDefault}]\funArgs{\optArg{float\argDef{$\pi$/4}}} -resets the default longitudinal view angle, -or returns the current default angle if no argument is given. -\index{graphics!3D commands!theta default} -When a parameter is specified, the default longitudinal view angle -$\theta$ is set to this value. - -% -\item[{\bf viewWriteAvailable}]\funArgs{\optArg{list\argDef{["pixmap", -"bitmap", "postscript", "image"]}}} -indicates the possible file types -\index{graphics!3D defaults!available viewport writes} -that can be created with the \spadfunFrom{write}{ThreeDimensionalViewport} function. - -% -\item[{\bf viewWriteDefault}]\funArgs{\optArg{list\argDef{[]}}} -sets or indicates the default types of files -that are created in addition to the {\bf data} file when a -\spadfunFrom{write}{ThreeDimensionalViewport} command -\index{graphics!3D defaults!viewport writes} -is executed on a viewport. - -% -\item[{\bf viewScaleDefault}]\funArgs{\optArg{float}} -sets the default scaling factor, or returns -\index{graphics!3D commands!scale default} -the current factor if no argument is given. - -% -\item[{\bf write}]\funArgs{viewport, directory, \optArg{option}} -writes the file {\bf data} for {\it viewport} -in the directory {\it directory}. -An optional third argument specifies a file type (one of {\tt -pixmap}, {\tt bitmap}, {\tt postscript}, or {\tt image}), or a -list of file types. -An additional file is written for each file type listed. - -% -\item[{\bf scale}]\funArgs{viewport, float\argDef{2.5}} -specifies the scaling factor. -\index{graphics!3D commands!scale} -\index{scaling graphs} -\end{description} -\egroup - -\subsection{Customization using .Xdefaults} -\label{ugXdefaults} - -\index{graphics!.Xdefaults} - -Both the two-di\-men\-sion\-al and three-di\-men\-sion\-al drawing facilities consult -the {\bf .Xdefaults} file for various defaults. -\index{file!.Xdefaults @{\bf .Xdefaults}} -The list of defaults that are recognized by the graphing routines -is discussed in this section. -These defaults are preceded by {\tt Axiom.3D.} -for three-di\-men\-sion\-al viewport defaults, {\tt Axiom.2D.} -for two-di\-men\-sion\-al viewport defaults, or {\tt Axiom*} (no dot) for -those defaults that are acceptable to either viewport type. - -% -\begin{description} -% -\item[{\tt Axiom*buttonFont:\ \it font}] \ \newline -This indicates which -\index{graphics!.Xdefaults!button font} -font type is used for the button text on the control-panel. -{\bf Rom11} -% -\item[{\tt Axiom.2D.graphFont:\ \it font}] \quad (2D only) \newline -This indicates -\index{graphics!.Xdefaults!graph number font} -which font type is used for displaying the graph numbers and -slots in the {\bf Graphs} section of the two-di\-men\-sion\-al control-panel. -{\bf Rom22} -% -\item[{\tt Axiom.3D.headerFont:\ \it font}] \ \newline -This indicates which -\index{graphics!.Xdefaults!graph label font} -font type is used for the axes labels and potentiometer -header names on three-di\-men\-sion\-al viewport windows. -This is also used for two-di\-men\-sion\-al control-panels for indicating -which font type is used for potentionmeter header names and -multiple graph title headers. -%for example, {\tt Axiom.2D.headerFont: 8x13}. -{\bf Itl14} -% -\item[{\tt Axiom*inverse:\ \it switch}] \ \newline -This indicates whether the -\index{graphics!.Xdefaults!inverting background} -background color is to be inverted from white to black. -If {\tt on}, the graph viewports use black as the background -color. -If {\tt off} or no declaration is made, the graph viewports use a -white background. -{\bf off} -% -\item[{\tt Axiom.3D.lightingFont:\ \it font}] \quad (3D only) \newline -This indicates which font type is used for the {\bf x}, -\index{graphics!.Xdefaults!lighting font} -{\bf y}, and {\bf z} labels of the two lighting axes potentiometers, and for -the {\bf Intensity} title on the lighting control-panel. -{\bf Rom10} -% -\item[{\tt Axiom.2D.messageFont, Axiom.3D.messageFont:\ \it font}] \ \newline -These indicate the font type -\index{graphics!.Xdefaults!message font} -to be used for the text in the control-panel message window. -{\bf Rom14} -% -\item[{\tt Axiom*monochrome:\ \it switch}] \ \newline -This indicates whether the -\index{graphics!.Xdefaults!monochrome} -graph viewports are to be displayed as if the monitor is black and -white, that is, a 1 bit plane. -If {\tt on} is specified, the viewport display is black and white. -If {\tt off} is specified, or no declaration for this default is -given, the viewports are displayed in the normal fashion for the -monitor in use. -{\bf off} -% -\item[{\tt Axiom.2D.postScript:\ \it filename}] \ \newline -This specifies -\index{graphics!.Xdefaults!PostScript file name} -the name of the file that is generated when a 2D PostScript graph -\index{PostScript} -is saved. -{\bf axiom2d.ps} -% -\item[{\tt Axiom.3D.postScript:\ \it filename}] \ \newline -This specifies -\index{graphics!.Xdefaults!PostScript file name} -the name of the file that is generated when a 3D PostScript graph -\index{PostScript} -is saved. -{\bf axiom3D.ps} -% -\item[{\tt Axiom*titleFont \it font}] \ \newline -This -\index{graphics!.Xdefaults!title font} -indicates which font type is used -for the title text and, for three-di\-men\-sion\-al graphs, -in the lighting and viewing-volume control-panel windows. -\index{graphics!Xdefaults!2d} -{\bf Rom14} -% -\item[{\tt Axiom.2D.unitFont:\ \it font}] \quad (2D only) \newline -This indicates -\index{graphics!.Xdefaults!unit label font} -which font type is used for displaying the unit labels on -two-di\-men\-sion\-al viewport graphs. -{\bf 6x10} -% -\item[{\tt Axiom.3D.volumeFont:\ \it font}] \quad (3D only) \newline -This indicates which font type is used for the {\bf x}, -\index{graphics!.Xdefaults!volume label font} -{\bf y}, and {\bf z} labels of the clipping region sliders; for the -{\bf Perspective}, {\bf Show Region}, and {\bf Clipping On} buttons under -{\bf Settings}, and above the windows for the {\bf Hither} and -{\bf Eye Distance} sliders in the {\bf Viewing Volume Panel} of the -three-di\-men\-sion\-al control-panel. -{\bf Rom8} -\end{description} - -\setcounter{chapter}{7} % Chapter 8 -% viewSizeDefault [300,300] - -\chapter{Advanced Problem Solving} -\label{ugProblem} - -In this chapter we describe techniques useful in solving advanced problems -with Axiom. - -\section{Numeric Functions} -\label{ugProblemNumeric} - -% -Axiom provides two basic floating-point types: {\tt Float} and -{\tt DoubleFloat}. This section describes how to use numerical -\index{function!numeric} -operations defined on these types and the related complex types. -\index{numeric operations} -% -As we mentioned in Chapter -\ref{ugIntro} on page~\pageref{ugIntro}, the {\tt Float} type is a software -implementation of floating-point numbers in which the exponent and the -\index{floating-point number} -significand may have any number of digits. -\index{number!floating-point} -See -\ref{FloatXmpPage} on page~\pageref{FloatXmpPage} -for detailed information about this domain. -The {\tt DoubleFloat} (see \ref{DoubleFloatXmpPage} on -page~\pageref{DoubleFloatXmpPage}) is usually a hardware implementation -of floating point numbers, corresponding to machine double -precision. -The types {\tt Complex Float} and {\tt Complex DoubleFloat} are -\index{floating-point number!complex} -the corresponding software implementations of complex floating-point numbers. -\index{complex!floating-point number} -In this section the term {\it floating-point type} means any of these -\index{number!complex floating-point} -four types. -% -The floating-point types implement the basic elementary functions. -These include (where {\tt \$} means -{\tt DoubleFloat}, -{\tt Float}, -{\tt Complex DoubleFloat}, or -{\tt Complex Float}): - -\noindent -{\bf exp}, {\bf log}: $\$ -> \$$ \newline -{\bf sin}, {\bf cos}, {\bf tan}, {\bf cot}, {\bf sec}, {\bf csc}: $\$ -> \$$ \newline -{\bf sin}, {\bf cos}, {\bf tan}, {\bf cot}, {\bf sec}, {\bf csc}: $\$ -> \$$ \newline -{\bf asin}, {\bf acos}, {\bf atan}, {\bf acot}, {\bf asec}, {\bf acsc}: $\$ -> \$$ \newline -{\bf sinh}, {\bf cosh}, {\bf tanh}, {\bf coth}, {\bf sech}, {\bf csch}: $\$ -> \$$ \newline -{\bf asinh}, {\bf acosh}, {\bf atanh}, {\bf acoth}, {\bf asech}, {\bf acsch}: $\$ -> \$$ \newline -{\bf pi}: $() -> \$$ \newline -{\bf sqrt}: $\$ -> \$$ \newline -{\bf nthRoot}: $(\$, Integer) -> \$$ \newline -\spadfunFrom{**}{Float}: $(\$, Fraction Integer) -> \$$ \newline -\spadfunFrom{**}{Float}: $(\$,\$) -> \$$ \newline - -The handling of roots depends on whether the floating-point type -\index{root!numeric approximation} -is real or complex: for the real floating-point types, -{\tt DoubleFloat} and {\tt Float}, if a real root exists -the one with the same sign as the radicand is returned; for the -complex floating-point types, the principal value is returned. -\index{principal value} -Also, for real floating-point types the inverse functions -produce errors if the results are not real. -This includes cases such as $asin(1.2)$, $log(-3.2)$, -$sqrt(-1.1)$. -% - -The default floating-point type is {\tt Float} so to evaluate -functions using {\tt Float} or {\tt Complex Float}, just use -normal decimal notation. - -\spadcommand{exp(3.1)} -$$ -22.1979512814 41633405 -$$ -\returnType{Type: Float} - -\spadcommand{exp(3.1 + 4.5 * \%i)} -$$ --{4.6792348860 969899118} -{{21.6991659280 71731864} \ i} -$$ -\returnType{Type: Complex Float} - -To evaluate functions using {\tt DoubleFloat} -or {\tt Complex DoubleFloat}, -a declaration or conversion is required. - -\spadcommand{r: DFLOAT := 3.1; t: DFLOAT := 4.5; exp(r + t*\%i)} -$$ --{4.6792348860969906} -{{21.699165928071732} \ i} -$$ -\returnType{Type: Complex DoubleFloat} - -\spadcommand{exp(3.1::DFLOAT + 4.5::DFLOAT * \%i)} -$$ --{4.6792348860969906} -{{21.699165928071732} \ i} -$$ -\returnType{Type: Complex DoubleFloat} - -A number of special functions are provided by the package -{\tt DoubleFloatSpecialFunctions} for the machine-precision -\index{special functions} -floating-point types. -\index{DoubleFloatSpecialFunctions} -The special functions provided are listed below, where $F$ stands for -the types {\tt DoubleFloat} and {\tt Complex DoubleFloat}. -The real versions of the functions yield an error if the result is not real. -\index{function!special} - -\noindent -{\bf Gamma}: $F -> F$\hfill\newline -$Gamma(z)$ is the Euler gamma function, -\index{function!Gamma} - $\Gamma(z)$, - defined by -\index{Euler!gamma function} - $$\Gamma(z) = \int_{0}^{\infty} t^{z-1} e^{-t} dt.$$ - - -\noindent -{\bf Beta}: $F -> F$\hfill\newline - $Beta(u, v)$ is the Euler Beta function, -\index{function!Euler Beta} - $Beta(u,v)$, defined by -\index{Euler!Beta function} - $$Beta(u,v) = \int_{0}^{1} t^{u-1} (1-t)^{v-1} dt.$$ - - This is related to $\Gamma(z)$ by - $$Beta(u,v) = \frac{\Gamma(u) \Gamma(v)}{\Gamma(u + v)}.$$ - -\noindent -{\bf logGamma}: $F -> F$\hfill\newline - $logGamma(z)$ is the natural logarithm of -$\Gamma(z)$. - This can often be computed even if $\Gamma(z)$ -cannot. -% - -\noindent -{\bf digamma}: $F -> F$\hfill\newline - $digamma(z)$, also called $psi(z)$, -\index{psi @ $\psi$} -is the function $\psi(z)$, -\index{function!digamma} - defined by $$\psi(z) = \Gamma'(z)/\Gamma(z).$$ - -\noindent -{\bf polygamma}: $(NonNegativeInteger, F) -> F$\hfill\newline - $polygamma(n, z)$ is the $n$-th derivative of -\index{function!polygamma} - $\psi(z)$, written $\psi^{(n)}(z)$. - -\noindent -{\bf E1}: $(DoubleFloat) -> OnePointCompletion DoubleFloat$\hfill\newline - E1(x) is the Exponential Integral function - The current implementation is a piecewise approximation - involving one poly from $-4..4$ and a second poly for $x > 4$ -\index{function!E1} - -\noindent -{\bf En}: $(PI, DFLOAT) -> OnePointCompletion DoubleFloat$\hfill\newline - En(PI,R) is the nth Exponential Integral -\index{function!En} - -\noindent -{\bf Ei}: $(OnePointCompletion DFLOAT) -> OnePointCompletion DFLOAT$ -\hfill\newline - Ei is the Exponential Integral function - This is computed using a 6 part piecewise approximation. - DoubleFloat can only preserve about 16 digits but the - Chebyshev approximation used can give 30 digits. -\index{function!Ei} - -\noindent -{\bf Ei1}: $(DoubleFloat) -> DoubleFloat$\hfill\newline - Ei1 is the first approximation of Ei where the result is - $x*e^-x*Ei(x)$ from -infinity to -10 (preserves digits) -\index{function!Ei1} - -\noindent -{\bf Ei2}: $(DoubleFloat) -> DoubleFloat$\hfill\newline - Ei2 is the first approximation of Ei where the result is - $x*e^-x*Ei(x)$ from -10 to -4 (preserves digits) -\index{function!Ei2} - -\noindent -{\bf Ei3}: $(DoubleFloat) -> DoubleFloat$\hfill\newline - Ei3 is the first approximation of Ei where the result is - $(Ei(x)-log |x| - gamma)/x$ from -4 to 4 (preserves digits) -\index{function!Ei3} - -\noindent -{\bf Ei4}: $(DoubleFloat) -> DoubleFloat$\hfill\newline - Ei4 is the first approximation of Ei where the result is - $x*e^-x*Ei(x)$ from 4 to 12 (preserves digits) -\index{function!Ei4} - -\noindent -{\bf Ei5}: $(DoubleFloat) -> DoubleFloat$\hfill\newline - Ei5 is the first approximation of Ei where the result is - $x*e^-x*Ei(x)$ from 12 to 32 (preserves digits) -\index{function!Ei5} - -\noindent -{\bf Ei6}: $(DoubleFloat) -> DoubleFloat$\hfill\newline - Ei6 is the first approximation of Ei where the result is - $x*e^-x*Ei(x)$ from 32 to infinity (preserves digits) -\index{function!Ei6} - -\noindent -{\bf besselJ}: $(F,F) -> F$\hfill\newline - $besselJ(v,z)$ is the Bessel function of the first kind, -\index{function!Bessel} - $J_\nu (z)$. - This function satisfies the differential equation - $$z^2 w''(z) + z w'(z) + (z^2-\nu^2)w(z) = 0.$$ - -\noindent -{\bf besselY}: $(F,F) -> F$\hfill\newline - $besselY(v,z)$ is the Bessel function of the second kind, -\index{function!Bessel} - $Y_\nu (z)$. - This function satisfies the same differential equation as - {\bf besselJ}. - The implementation simply uses the relation - $$Y_\nu (z) = \frac{J_\nu (z) \cos(\nu \pi) - J_{-\nu} (z)}{\sin(\nu \pi)}.$$ - -\noindent -{\bf besselI}: $(F,F) -> F$\hfill\newline - $besselI(v,z)$ is the modified Bessel function of the first kind, -\index{function!Bessel} - $I_\nu (z)$. - This function satisfies the differential equation - $$z^2 w''(z) + z w'(z) - (z^2+\nu^2)w(z) = 0.$$ - -\noindent -{\bf besselK}: $(F,F) -> F$\hfill\newline - $besselK(v,z)$ is the modified Bessel function of the second kind, -\index{function!Bessel} - $K_\nu (z)$. - This function satisfies the same differential equation as {\bf besselI}. -\index{Bessel function} - The implementation simply uses the relation - $$K_\nu (z) = \pi \frac{I_{-\nu} (z) - I_{\nu} (z)}{2 \sin(\nu \pi)}.$$ - - -\noindent -{\bf airyAi}: $F -> F$\hfill\newline - $airyAi(z)$ is the Airy function $Ai(z)$. -\index{function!Airy Ai} - This function satisfies the differential equation - $w''(z) - z w(z) = 0.$ - The implementation simply uses the relation - $$Ai(-z) = \frac{1}{3}\sqrt{z} ( J_{-1/3} (\frac{2}{3}z^{3/2}) + J_{1/3} (\frac{2}{3}z^{3/2}) ).$$ - -\noindent -{\bf airyBi}: $F -> F$\hfill\newline - $airyBi(z)$ is the Airy function $Bi(z)$. -\index{function!Airy Bi} - This function satisfies the same differential equation as {\bf airyAi}. -\index{Airy function} - The implementation simply uses the relation - $$Bi(-z) = \frac{1}{3}\sqrt{3 z} ( J_{-1/3} (\frac{2}{3}z^{3/2}) - J_{1/3} (\frac{2}{3}z^{3/2}) ).$$ - -\noindent -{\bf hypergeometric0F1}: $(F,F) -> F$\hfill\newline - $hypergeometric0F1(c,z)$ is the hypergeometric function -\index{function!hypergeometric} - ${}_0 F_1 ( ; c; z)$. - -The above special functions are defined only for small floating-point types. -If you give {\tt Float} arguments, they are converted to -{\tt DoubleFloat} by Axiom. - -\spadcommand{Gamma(0.5)**2} -$$ -3.14159265358979 -$$ -\returnType{Type: DoubleFloat} - -\spadcommand{a := 2.1; b := 1.1; besselI(a + \%i*b, b*a + 1)} -$$ -{2.489481690673867} -{{2.365846713181643} \ i} -$$ -\returnType{Type: Complex DoubleFloat} - -A number of additional operations may be used to compute numerical values. -These are special polynomial functions that can be evaluated for values in -any commutative ring $R$, and in particular for values in any -floating-point type. -The following operations are provided by the package -{\tt OrthogonalPolynomialFunctions}: -\index{OrthogonalPolynomialFunctions} - -\noindent -{\bf chebyshevT}: $(NonNegativeInteger, R) -> R$\hbox{}\hfill\newline - $chebyshevT(n,z)$ is the $n$-th Chebyshev polynomial of the first - kind, $T_n (z)$. These are defined by - $$\frac{1-t z}{1-2 t z+t^2} = \sum_{n=0}^{\infty} T_n (z) t^n.$$ - -\noindent -{\bf chebyshevU}: $(NonNegativeInteger, R) -> R$\hbox{}\hfill\newline - $chebyshevU(n,z)$ is the $n$-th Chebyshev polynomial of the second - kind, $U_n (z)$. These are defined by - $$\frac{1}{1-2 t z+t^2} = \sum_{n=0}^{\infty} U_n (z) t^n.$$ - -\noindent -{\bf hermiteH}: $(NonNegativeInteger, R) -> R$\hbox{}\hfill\newline - $hermiteH(n,z)$ is the $n$-th Hermite polynomial, - $H_n (z)$. - These are defined by - $$e^{2 t z - t^2} = \sum_{n=0}^{\infty} H_n (z) \frac{t^n}{n!}.$$ - -\noindent -{\bf laguerreL}: $(NonNegativeInteger, R) -> R$\hbox{}\hfill\newline - $laguerreL(n,z)$ is the $n$-th Laguerre polynomial, - $L_n (z)$. - These are defined by - $$\frac{e^{-\frac{t z}{1-t}}}{1-t} = \sum_{n=0}^{\infty} L_n (z) \frac{t^n}{n!}.$$ - -\noindent -{\bf laguerreL}: $(NonNegativeInteger, NonNegativeInteger, R) -> R$\hbox{}\hfill\newline - $laguerreL(m,n,z)$ is the associated Laguerre polynomial, - $L^m_n (z)$. - This is the $m$-th derivative of $L_n (z)$. - -\noindent -{\bf legendreP}: $(NonNegativeInteger, R) -> R$\hbox{}\hfill\newline - $legendreP(n,z)$ is the $n$-th Legendre polynomial, - $P_n (z)$. These are defined by - $$\frac{1}{\sqrt{1-2 t z+t^2}} = \sum_{n=0}^{\infty} P_n (z) t^n.$$ - -These operations require non-negative integers for the indices, but otherwise -the argument can be given as desired. - -\spadcommand{[chebyshevT(i, z) for i in 0..5]} -$$ -\left[ -1, z, {{2 \ {z \sp 2}} -1}, {{4 \ {z \sp 3}} -{3 \ z}}, {{8 \ -{z \sp 4}} -{8 \ {z \sp 2}}+1}, {{{16} \ {z \sp 5}} -{{20} \ {z \sp -3}}+{5 \ z}} -\right] -$$ -\returnType{Type: List Polynomial Integer} - -The expression $chebyshevT(n,z)$ evaluates to the $n$-th Chebyshev -\index{polynomial!Chebyshev!of the first kind} -polynomial of the first kind. - -\spadcommand{chebyshevT(3, 5.0 + 6.0*\%i)} -$$ --{1675.0}+{{918.0} \ i} -$$ -\returnType{Type: Complex Float} - -\spadcommand{chebyshevT(3, 5.0::DoubleFloat)} -$$ -485.0 -$$ -\returnType{Type: DoubleFloat} - -The expression $chebyshevU(n,z)$ evaluates to the $n$-th Chebyshev -\index{polynomial!Chebyshev!of the second kind} -polynomial of the second kind. - -\spadcommand{[chebyshevU(i, z) for i in 0..5]} -$$ -\left[ -1, {2 \ z}, {{4 \ {z \sp 2}} -1}, {{8 \ {z \sp 3}} -{4 \ z}}, -{{{16} \ {z \sp 4}} -{{12} \ {z \sp 2}}+1}, {{{32} \ {z \sp 5}} -{{32} -\ {z \sp 3}}+{6 \ z}} -\right] -$$ -\returnType{Type: List Polynomial Integer} - -\spadcommand{chebyshevU(3, 0.2)} -$$ --{0.736} -$$ -\returnType{Type: Float} - -The expression $hermiteH(n,z)$ evaluates to the $n$-th Hermite -\index{polynomial!Hermite} -polynomial. - -\spadcommand{[hermiteH(i, z) for i in 0..5]} -$$ -\left[ -1, {2 \ z}, {{4 \ {z \sp 2}} -2}, {{8 \ {z \sp 3}} -{{12} \ z}}, - {{{16} \ {z \sp 4}} -{{48} \ {z \sp 2}}+{12}}, {{{32} \ {z \sp 5}} --{{160} \ {z \sp 3}}+{{120} \ z}} -\right] -$$ -\returnType{Type: List Polynomial Integer} - -\spadcommand{hermiteH(100, 1.0)} -$$ --{0.1448706729 337934088 E 93} -$$ -\returnType{Type: Float} - -The expression $laguerreL(n,z)$ evaluates to the $n$-th Laguerre -\index{polynomial!Laguerre} -polynomial. - -\spadcommand{[laguerreL(i, z) for i in 0..4]} -$$ -\left[ -1, {-z+1}, {{z \sp 2} -{4 \ z}+2}, {-{z \sp 3}+{9 \ {z \sp 2}} --{{18} \ z}+6}, {{z \sp 4} -{{16} \ {z \sp 3}}+{{72} \ {z \sp 2}} --{{96} \ z}+{24}} -\right] -$$ -\returnType{Type: List Polynomial Integer} - -\spadcommand{laguerreL(4, 1.2)} -$$ --{13.0944} -$$ -\returnType{Type: Float} - -\spadcommand{[laguerreL(j, 3, z) for j in 0..4]} -$$ -\left[ -{-{z \sp 3}+{9 \ {z \sp 2}} -{{18} \ z}+6}, {-{3 \ {z \sp 2}}+{{18} \ -z} -{18}}, {-{6 \ z}+{18}}, -6, 0 -\right] -$$ -\returnType{Type: List Polynomial Integer} - -\spadcommand{laguerreL(1, 3, 2.1)} -$$ -6.57 -$$ -\returnType{Type: Float} - -The expression -\index{polynomial!Legendre} -$legendreP(n,z)$ evaluates to the $n$-th Legendre polynomial, - -\spadcommand{[legendreP(i,z) for i in 0..5]} -$$ -\left[ -1, z, {{{3 \over 2} \ {z \sp 2}} -{1 \over 2}}, {{{5 \over 2} \ {z -\sp 3}} -{{3 \over 2} \ z}}, {{{{35} \over 8} \ {z \sp 4}} -{{{15} \over -4} \ {z \sp 2}}+{3 \over 8}}, {{{{63} \over 8} \ {z \sp 5}} -{{{35} -\over 4} \ {z \sp 3}}+{{{15} \over 8} \ z}} -\right] -$$ -\returnType{Type: List Polynomial Fraction Integer} - -\spadcommand{legendreP(3, 3.0*\%i)} -$$ --{{72.0} \ i} -$$ -\returnType{Type: Complex Float} - -Finally, three number-theoretic polynomial operations may be evaluated. -\index{number theory} -The following operations are provided by the package -{\tt NumberTheoreticPolynomialFunctions}. -\index{NumberTheoreticPolynomialFunctions}. - -\noindent -{\bf bernoulliB}: $(NonNegativeInteger, R) -> R$ \hbox{}\hfill\newline - $bernoulliB(n,z)$ is the $n$-th Bernoulli polynomial, -\index{polynomial!Bernoulli} - $B_n (z)$. These are defined by - $$\frac{t e^{z t}}{e^t - 1} = \sum_{n=0}^{\infty} B_n (z) \frac{t^n}{n!}.$$ - -\noindent -{\bf eulerE}: $(NonNegativeInteger, R) -> R$ \hbox{}\hfill\newline - $eulerE(n,z)$ is the $n$-th Euler polynomial, -\index{Euler!polynomial} - $E_n (z)$. These are defined by -\index{polynomial!Euler} - $$\frac{2 e^{z t}}{e^t + 1} = \sum_{n=0}^{\infty} E_n (z) \frac{t^n}{n!}.$$ - -\noindent -{\bf cyclotomic}: $(NonNegativeInteger, R) -> R$\hbox{}\hfill\newline - $cyclotomic(n,z)$ is the $n$-th cyclotomic polynomial - $\Phi_n (z)$. This is the polynomial whose - roots are precisely the primitive $n$-th roots of unity. -\index{Euler!totient function} - This polynomial has degree given by the Euler totient function -\index{function!totient} - $\phi(n)$. - -The expression $bernoulliB(n,z)$ evaluates to the $n$-th Bernoulli -\index{polynomial!Bernouilli} -polynomial. - -\spadcommand{bernoulliB(3, z)} -$$ -{z \sp 3} -{{3 \over 2} \ {z \sp 2}}+{{1 \over 2} \ z} -$$ -\returnType{Type: Polynomial Fraction Integer} - -\spadcommand{bernoulliB(3, 0.7 + 0.4 * \%i)} -$$ --{0.138} -{{0.116} \ i} -$$ -\returnType{Type: Complex Float} - -The expression -\index{polynomial!Euler} -$eulerE(n,z)$ evaluates to the $n$-th Euler polynomial. - -\spadcommand{eulerE(3, z)} -$$ -{z \sp 3} -{{3 \over 2} \ {z \sp 2}}+{1 \over 4} -$$ -\returnType{Type: Polynomial Fraction Integer} - -\spadcommand{eulerE(3, 0.7 + 0.4 * \%i)} -$$ --{0.238} -{{0.316} \ i} -$$ -\returnType{Type: Complex Float} - -The expression -\index{polynomial!cyclotomic} -$cyclotomic(n,z)$ evaluates to the $n$-th cyclotomic polynomial. -\index{cyclotomic polynomial} - -\spadcommand{cyclotomic(3, z)} -$$ -{z \sp 2}+z+1 -$$ -\returnType{Type: Polynomial Integer} - -\spadcommand{cyclotomic(3, (-1.0 + 0.0 * \%i)**(2/3))} -$$ -0.0 -$$ -\returnType{Type: Complex Float} - -Drawing complex functions in Axiom is presently somewhat -awkward compared to drawing real functions. -It is necessary to use the {\bf draw} operations that operate -on functions rather than expressions. - -This is the complex exponential function (rotated interactively). -\index{function!complex exponential} -When this is displayed in color, the height is the value of the real part of -the function and the color is the imaginary part. -Red indicates large negative imaginary values, green indicates imaginary -values near zero and blue/violet indicates large positive imaginary values. - -\spadgraph{draw((x,y)+-> real exp complex(x,y), -2..2, -2*\%pi..2*\%pi, colorFunction == (x, y) +-> imag exp complex(x,y), title=="exp(x+\%i*y)", style=="smooth")} - -%\epsffile[0 0 295 295]{ps/compexp.ps} - -This is the complex arctangent function. -\index{function!complex arctangent} -Again, the height is the real part of the function value but here -the color indicates the function value's phase. -The position of the branch cuts are clearly visible and one can -see that the function is real only for a real argument. - -\spadgraph{vp := draw((x,y) +-> real atan complex(x,y), -\%pi..\%pi, -\%pi..\%pi, colorFunction==(x,y) +->argument atan complex(x,y), title=="atan(x+\%i*y)", style=="shade"); rotate(vp,-160,-45); vp} - -%\epsffile[0 0 295 295]{ps/compatan.ps} - -This is the complex Gamma function. - -\spadgraph{draw((x,y) +-> max(min(real Gamma complex(x,y),4),-4), -\%pi..\%pi, -\%pi..\%pi, style=="shade", colorFunction == (x,y) +-> argument Gamma complex(x,y), title == "Gamma(x+\%i*y)", var1Steps == 50, var2Steps== 50)} - -%\epsffile[0 0 295 295]{ps/compgamm.ps} - -This shows the real Beta function near the origin. - -\spadgraph{draw(Beta(x,y)/100, x=-1.6..1.7, y = -1.6..1.7, style=="shade", title=="Beta(x,y)", var1Steps==40, var2Steps==40)} - -%\epsffile[0 0 295 295]{ps/realbeta.ps} - -This is the Bessel function $J_\alpha (x)$ -for index $\alpha$ in the range $-6..4$ and -argument $x$ in the range $2..14$. - -\spadgraph{draw((alpha,x) +-> min(max(besselJ(alpha, x+8), -6), 6), -6..4, -6..6, title=="besselJ(alpha,x)", style=="shade", var1Steps==40, var2Steps==40)} - -%\epsffile[0 0 295 295]{ps/bessel.ps} - -This is the modified Bessel function -$I_\alpha (x)$ -evaluated for various real values of the index $\alpha$ -and fixed argument $x = 5$. - -\spadgraph{draw(besselI(alpha, 5), alpha = -12..12, unit==[5,20])} - -%\epsffile[0 0 295 295]{ps/modbess.ps} - -This is similar to the last example -except the index $\alpha$ -takes on complex values in a $6 x 6$ rectangle centered on the origin. - -\spadgraph{draw((x,y) +-> real besselI(complex(x/20, y/20),5), -60..60, -60..60, colorFunction == (x,y)+-> argument besselI(complex(x/20,y/20),5), title=="besselI(x+i*y,5)", style=="shade")} - -%\epsffile[0 0 295 295]{ps/modbessc.ps} - -\section{Polynomial Factorization} -\label{ugProblemFactor} - -% -The Axiom polynomial factorization -\index{polynomial!factorization} -facilities are available for all polynomial types and a wide variety of -coefficient domains. -\index{factorization} -Here are some examples. - -\subsection{Integer and Rational Number Coefficients} -\label{ugProblemFactorIntRat} - -Polynomials with integer -\index{polynomial!factorization!integer coefficients} -coefficients can be be factored. - -\spadcommand{v := (4*x**3+2*y**2+1)*(12*x**5-x**3*y+12) } -$$ --{2 \ {x \sp 3} \ {y \sp 3}}+{{\left( {{24} \ {x \sp 5}}+{24} -\right)} -\ {y \sp 2}}+{{\left( -{4 \ {x \sp 6}} -{x \sp 3} -\right)} -\ y}+{{48} \ {x \sp 8}}+{{12} \ {x \sp 5}}+{{48} \ {x \sp 3}}+{12} -$$ -\returnType{Type: Polynomial Integer} - -\spadcommand{factor v } -$$ --{{\left( {{x \sp 3} \ y} -{{12} \ {x \sp 5}} -{12} -\right)} -\ {\left( {2 \ {y \sp 2}}+{4 \ {x \sp 3}}+1 -\right)}} -$$ -\returnType{Type: Factored Polynomial Integer} - -Also, Axiom can factor polynomials with -\index{polynomial!factorization!rational number coefficients} -rational number coefficients. - -\spadcommand{w := (4*x**3+(2/3)*x**2+1)*(12*x**5-(1/2)*x**3+12) } -$$ -{{48} \ {x \sp 8}}+{8 \ {x \sp 7}} -{2 \ {x \sp 6}}+{{{35} \over 3} \ {x -\sp 5}}+{{{95} \over 2} \ {x \sp 3}}+{8 \ {x \sp 2}}+{12} -$$ -\returnType{Type: Polynomial Fraction Integer} - -\spadcommand{factor w } -$$ -{48} \ {\left( {x \sp 3}+{{1 \over 6} \ {x \sp 2}}+{1 \over 4} -\right)} -\ {\left( {x \sp 5} -{{1 \over {24}} \ {x \sp 3}}+1 -\right)} -$$ -\returnType{Type: Factored Polynomial Fraction Integer} - -\subsection{Finite Field Coefficients} -\label{ugProblemFactorFF} - -Polynomials with coefficients in a finite field -\index{polynomial!factorization!finite field coefficients} -can be also be factored. -\index{finite field!factoring polynomial with coefficients in} - -\spadcommand{u : POLY(PF(19)) :=3*x**4+2*x**2+15*x+18 } -$$ -{3 \ {x \sp 4}}+{2 \ {x \sp 2}}+{{15} \ x}+{18} -$$ -\returnType{Type: Polynomial PrimeField 19} - -These include the integers mod $p$, where $p$ is prime, and -extensions of these fields. - -\spadcommand{factor u } -$$ -3 \ {\left( x+{18} -\right)} -\ {\left( {x \sp 3}+{x \sp 2}+{8 \ x}+{13} -\right)} -$$ -\returnType{Type: Factored Polynomial PrimeField 19} - -Convert this to have coefficients in the finite -field with $19^3$ elements. -See \ref{ugProblemFinite} on page~\pageref{ugProblemFinite} -for more information about finite fields. - -\spadcommand{factor(u :: POLY FFX(PF 19,3)) } -$$ -3 \ {\left( x+{18} -\right)} -\ {\left( x+{5 \ { \%I \sp 2}}+{3 \ \%I}+{13} -\right)} -\ {\left( x+{{16} \ { \%I \sp 2}}+{{14} \ \%I}+{13} -\right)} -\ {\left( x+{{17} \ { \%I \sp 2}}+{2 \ \%I}+{13} -\right)} -$$ -\returnType{Type: Factored Polynomial FiniteFieldExtension(PrimeField 19,3)} - -\subsection{Simple Algebraic Extension Field Coefficients} -\label{ugProblemFactorAlg} - -Polynomials with coefficients in simple algebraic extensions -\index{polynomial!factorization!algebraic extension field coefficients} -of the rational numbers can be factored. -\index{algebraic number} -\index{number!algebraic} - -Here, $aa$ and $bb$ are symbolic roots of polynomials. - -\spadcommand{aa := rootOf(aa**2+aa+1) } -$$ -aa -$$ -\returnType{Type: AlgebraicNumber} - -\spadcommand{p:=(x**3+aa**2*x+y)*(aa*x**2+aa*x+aa*y**2)**2 } -$$ -\begin{array}{@{}l} -{{\left( -aa -1 \right)} \ {y \sp 5}}+ -{{\left( {{\left( -aa -1 \right)}\ {x \sp 3}}+ -{aa \ x} \right)}\ {y \sp 4}}+ -\\ -\\ -\displaystyle -{{\left( {{\left( -{2 \ aa} -2 \right)}\ {x \sp 2}}+ -{{\left( -{2 \ aa} -2 \right)}\ x} \right)}\ {y \sp 3}}+ -\\ -\\ -\displaystyle -{{\left( {{\left( -{2 \ aa} -2 \right)}\ {x \sp 5}}+ -{{\left( -{2 \ aa} -2 \right)}\ {x \sp 4}}+ -{2 \ aa \ {x \sp 3}}+{2 \ aa \ {x \sp 2}} \right)}\ {y \sp 2}}+ -\\ -\\ -\displaystyle -{{\left( {{\left( -aa -1 \right)}\ {x \sp 4}}+ -{{\left( -{2 \ aa} -2 \right)}\ {x \sp 3}}+ -{{\left( -aa -1 \right)}\ {x \sp 2}} \right)}\ y}+ -\\ -\\ -\displaystyle -{{\left( -aa -1 \right)}\ {x \sp 7}}+ -{{\left( -{2 \ aa} -2 \right)}\ {x \sp 6}} - -{x \sp 5}+ -{2 \ aa \ {x \sp 4}}+ -{aa \ {x \sp 3}} -\end{array} -$$ -\returnType{Type: Polynomial AlgebraicNumber} - -Note that the second argument to factor can be a list of -algebraic extensions to factor over. - -\spadcommand{factor(p,[aa]) } -%Note: this answer differs from the book but is equivalent. -$$ -{\left( -aa -1 -\right)} -\ {\left( y+{x \sp 3}+{{\left( -aa -1 -\right)} -\ x} -\right)} -\ {{\left( {y \sp 2}+{x \sp 2}+x -\right)} -\sp 2} -$$ -\returnType{Type: Factored Polynomial AlgebraicNumber} - -This factors $x**2+3$ over the integers. - -\spadcommand{factor(x**2+3)} -$$ -{x \sp 2}+3 -$$ -\returnType{Type: Factored Polynomial Integer} - -Factor the same polynomial over the field obtained by adjoining -$aa$ to the rational numbers. - -\spadcommand{factor(x**2+3,[aa]) } -$$ -{\left( x -{2 \ aa} -1 -\right)} -\ {\left( x+{2 \ aa}+1 -\right)} -$$ -\returnType{Type: Factored Polynomial AlgebraicNumber} - -Factor $x**6+108$ over the same field. - -\spadcommand{factor(x**6+108,[aa]) } -$$ -{\left( {x \sp 3} -{{12} \ aa} -6 -\right)} -\ {\left( {x \sp 3}+{{12} \ aa}+6 -\right)} -$$ -\returnType{Type: Factored Polynomial AlgebraicNumber} - -\spadcommand{bb:=rootOf(bb**3-2) } -$$ -bb -$$ -\returnType{Type: AlgebraicNumber} - -\spadcommand{factor(x**6+108,[bb]) } -$$ -{\left( {x \sp 2} -{3 \ bb \ x}+{3 \ {bb \sp 2}} -\right)} -\ {\left( {x \sp 2}+{3 \ {bb \sp 2}} -\right)} -\ {\left( {x \sp 2}+{3 \ bb \ x}+{3 \ {bb \sp 2}} -\right)} -$$ -\returnType{Type: Factored Polynomial AlgebraicNumber} - -Factor again over the field obtained by adjoining both $aa$ -and $bb$ to the rational numbers. - -\spadcommand{factor(x**6+108,[aa,bb]) } -$$ -\begin{array}{@{}l} -{\left( x+{{\left( -{2 \ aa} -1 \right)}\ bb} \right)} -\ {\left( x+{{\left( -aa -2 \right)}\ bb} \right)} -\ {\left( x+{{\left( -aa+1 \right)}\ bb} \right)} -\\ -\\ -\displaystyle -\ {\left( x+{{\left( aa -1 \right)}\ bb} \right)} -\ {\left( x+{{\left( aa+2 \right)}\ bb} \right)} -\ {\left( x+{{\left( {2 \ aa}+1 \right)}\ bb} \right)} -\end{array} -$$ -\returnType{Type: Factored Polynomial AlgebraicNumber} - -\subsection{Factoring Rational Functions} -\label{ugProblemFactorRatFun} - -Since fractions of polynomials form a field, every element (other than zero) -\index{rational function!factoring} -divides any other, so there is no useful notion of irreducible factors. -Thus the {\bf factor} operation is not very useful for fractions -of polynomials. - -There is, instead, a specific operation {\bf factorFraction} -that separately factors the numerator and denominator and returns -a fraction of the factored results. - -\spadcommand{factorFraction((x**2-4)/(y**2-4))} -$$ -{{\left( x -2 -\right)} -\ {\left( x+2 -\right)}} -\over {{\left( y -2 -\right)} -\ {\left( y+2 -\right)}} -$$ -\returnType{Type: Fraction Factored Polynomial Integer} - -You can also use {\bf map}. This expression -applies the {\bf factor} operation -to the numerator and denominator. - -\spadcommand{map(factor,(x**2-4)/(y**2-4))} -$$ -{{\left( x -2 -\right)} -\ {\left( x+2 -\right)}} -\over {{\left( y -2 -\right)} -\ {\left( y+2 -\right)}} -$$ -\returnType{Type: Fraction Factored Polynomial Integer} - -\section{Manipulating Symbolic Roots of a Polynomial} -\label{ugProblemSymRoot} - -% -In this section we show you how to work with one root or all roots -\index{root!symbolic} -of a polynomial. -These roots are represented symbolically (as opposed to being -numeric approximations). -See \ref{ugxProblemOnePol} on page~\pageref{ugxProblemOnePol} and -\ref{ugxProblemPolSys} on page~\pageref{ugxProblemPolSys} for -information about solving for the roots of one or more -polynomials. - -\subsection{Using a Single Root of a Polynomial} -\label{ugxProblemSymRootOne} - -Use {\bf rootOf} to get a symbolic root of a polynomial: -$rootOf(p, x)$ returns a root of $p(x)$. - -This creates an algebraic number $a$. -\index{algebraic number} -\index{number!algebraic} - -\spadcommand{a := rootOf(a**4+1,a) } -$$ -a -$$ -\returnType{Type: Expression Integer} - -To find the algebraic relation that defines $a$, -use {\bf definingPolynomial}. - -\spadcommand{definingPolynomial a } -$$ -{a \sp 4}+1 -$$ -\returnType{Type: Expression Integer} - -You can use $a$ in any further expression, -including a nested {\bf rootOf}. - -\spadcommand{b := rootOf(b**2-a-1,b) } -$$ -b -$$ -\returnType{Type: Expression Integer} - -Higher powers of the roots are automatically reduced during -calculations. - -\spadcommand{a + b } -$$ -b+a -$$ -\returnType{Type: Expression Integer} - -\spadcommand{\% ** 5 } -$$ -{{\left( {{10} \ {a \sp 3}}+{{11} \ {a \sp 2}}+{2 \ a} -4 -\right)} -\ b}+{{15} \ {a \sp 3}}+{{10} \ {a \sp 2}}+{4 \ a} -{10} -$$ -\returnType{Type: Expression Integer} - -The operation {\bf zeroOf} is similar to {\bf rootOf}, -except that it may express the root using radicals in some cases. -\index{radical} - -\spadcommand{rootOf(c**2+c+1,c)} -$$ -c -$$ -\returnType{Type: Expression Integer} - -\spadcommand{zeroOf(d**2+d+1,d)} -$$ -{{\sqrt {-3}} -1} \over 2 -$$ -\returnType{Type: Expression Integer} - -\spadcommand{rootOf(e**5-2,e)} -$$ -e -$$ -\returnType{Type: Expression Integer} - -\spadcommand{zeroOf(f**5-2,f)} -$$ -\root {5} \of {2} -$$ -\returnType{Type: Expression Integer} - -\subsection{Using All Roots of a Polynomial} -\label{ugxProblemSymRootAll} - -Use {\bf rootsOf} to get all symbolic roots of a polynomial: -$rootsOf(p, x)$ returns a -list of all the roots of $p(x)$. -If $p(x)$ has a multiple root of order $n$, then that root -\index{root!multiple} -appears $n$ times in the list. -\typeout{Make sure these variables are x0 etc} - -Compute all the roots of $x**4 + 1$. - -\spadcommand{l := rootsOf(x**4+1,x) } -$$ -\left[ - \%x0, { \%x0 \ \%x1}, - \%x0, -{ \%x0 \ \%x1} -\right] -$$ -\returnType{Type: List Expression Integer} - -As a side effect, the variables $\%x0, \%x1$ and $\%x2$ are bound -to the first three roots of $x**4+1$. - -\spadcommand{\%x0**5 } -$$ -- \%x0 -$$ -\returnType{Type: Expression Integer} - -Although they all satisfy $x**4 + 1 = 0, \%x0, \%x1,$ -and $\%x2$ are different algebraic numbers. -To find the algebraic relation that defines each of them, -use {\bf definingPolynomial}. - -\spadcommand{definingPolynomial \%x0 } -$$ -{ \%x0 \sp 4}+1 -$$ -\returnType{Type: Expression Integer} - -\spadcommand{definingPolynomial \%x1 } -$$ -{ \%x1 \sp 2}+1 -$$ -\returnType{Type: Expression Integer} - -\spadcommand{definingPolynomial \%x2 } -$$ -- \%x2+ \%\%var -$$ -\returnType{Type: Expression Integer} - -We can check that the sum and product of the roots of $x**4+1$ are -its trace and norm. - -\spadcommand{x3 := last l } -$$ --{ \%x0 \ \%x1} -$$ -\returnType{Type: Expression Integer} - -\spadcommand{\%x0 + \%x1 + \%x2 + x3 } -$$ -{{\left( - \%x0+1 -\right)} -\ \%x1}+ \%x0+ \%x2 -$$ -\returnType{Type: Expression Integer} - -\spadcommand{\%x0 * \%x1 * \%x2 * x3 } -$$ - \%x2 \ { \%x0 \sp 2} -$$ -\returnType{Type: Expression Integer} - -Corresponding to the pair of operations -{\bf rootOf}/{\bf zeroOf} in -\ref{ugxProblemOnePol} on page~\pageref{ugxProblemOnePol}, there is -an operation {\bf zerosOf} that, like {\bf rootsOf}, -computes all the roots -of a given polynomial, but which expresses some of them in terms of -radicals. - -\spadcommand{zerosOf(y**4+1,y) } -$$ -\left[ -{{{\sqrt {-1}}+1} \over {\sqrt {2}}}, {{{\sqrt {-1}} -1} \over {\sqrt -{2}}}, {{-{\sqrt {-1}} -1} \over {\sqrt {2}}}, {{-{\sqrt {-1}}+1} \over -{\sqrt {2}}} -\right] -$$ -\returnType{Type: List Expression Integer} - -As you see, only one implicit algebraic number was created -($\%y1$), and its defining equation is this. -The other three roots are expressed in radicals. - -\spadcommand{definingPolynomial \%y1 } -$$ -{ \%\%var \sp 2}+1 -$$ -\returnType{Type: Expression Integer} - -\section{Computation of Eigenvalues and Eigenvectors} -\label{ugProblemEigen} -% -In this section we show you -some of Axiom's facilities for computing and -\index{eigenvalue} -manipulating eigenvalues and eigenvectors, also called -\index{eigenvector} -characteristic values and characteristic vectors, -\index{characteristic!value} -respectively. -\index{characteristic!vector} - -\vskip 4pc - -Let's first create a matrix with integer entries. - -\spadcommand{m1 := matrix [ [1,2,1],[2,1,-2],[1,-2,4] ] } -$$ -\left[ -\begin{array}{ccc} -1 & 2 & 1 \\ -2 & 1 & -2 \\ -1 & -2 & 4 -\end{array} -\right] -$$ -\returnType{Type: Matrix Integer} - -To get a list of the {\it rational} eigenvalues, -use the operation {\bf eigenvalues}. - -\spadcommand{leig := eigenvalues(m1) } -$$ -\left[ -5, {\left( \%K \mid {{ \%K \sp 2} - \%K -5} -\right)} -\right] -$$ -\returnType{Type: List Union(Fraction Polynomial Integer,SuchThat(Symbol,Polynomial Integer))} - -Given an explicit eigenvalue, {\bf eigenvector} computes the eigenvectors -corresponding to it. - -\spadcommand{eigenvector(first(leig),m1) } -$$ -\left[ -{\left[ -\begin{array}{c} -0 \\ --{1 \over 2} \\ -1 -\end{array} -\right]} -\right] -$$ -\returnType{Type: List Matrix Fraction Polynomial Fraction Integer} - -The operation {\bf eigenvectors} returns a list of pairs of values and -vectors. When an eigenvalue is rational, Axiom gives you -the value explicitly; otherwise, its minimal polynomial is given, -(the polynomial of lowest degree with the eigenvalues as roots), -together with a parametric representation of the eigenvector using the -eigenvalue. -This means that if you ask Axiom to {\bf solve} -the minimal polynomial, then you can substitute these roots -\index{polynomial!minimal} -into the parametric form of the corresponding eigenvectors. -\index{minimal polynomial} - -You must be aware that unless an exact eigenvalue has been computed, -the eigenvector may be badly in error. - -\spadcommand{eigenvectors(m1) } -$$ -\begin{array}{@{}l} -\left[ -{\left[ {eigval=5}, {eigmult=1}, {eigvec= -{\left[ -{\left[ -\begin{array}{c} -0 \\ --{1 \over 2} \\ -1 -\end{array} -\right]} -\right]}} -\right]}, -\right. -\\ -\\ -\displaystyle -\left. -{\left[ -{eigval={\left( \%L \mid {{ \%L \sp 2} - \%L -5} \right)}}, -{eigmult=1}, {eigvec= -{\left[ -{\left[ -\begin{array}{c} - \%L \\ -2 \\ -1 -\end{array} -\right]} -\right]}} -\right]} -\right] -\end{array} -$$ -\returnType{Type: List Record(eigval: Union(Fraction Polynomial Integer,SuchThat(Symbol,Polynomial Integer)),eigmult: NonNegativeInteger,eigvec: List Matrix Fraction Polynomial Integer)} - -Another possibility is to use the operation -{\bf radicalEigenvectors} -tries to compute explicitly the eigenvectors -in terms of radicals. -\index{radical} - -\spadcommand{radicalEigenvectors(m1) } -$$ -\begin{array}{@{}l} -\left[ -{\left[ {radval={{{\sqrt {{21}}}+1} \over 2}}, {radmult=1}, -{radvect={\left[ {\left[ -\begin{array}{c} -{{{\sqrt {{21}}}+1} \over 2} \\ -2 \\ -1 -\end{array} -\right]} -\right]}} -\right]}, -\right. -\\ -\\ -\displaystyle - \left[ {radval={{-{\sqrt {{21}}}+1} \over 2}}, {radmult=1}, -{radvect={\left[ {\left[ -\begin{array}{c} -{{-{\sqrt {{21}}}+1} \over 2} \\ -2 \\ -1 -\end{array} -\right]} -\right]}} -\right], -\\ -\\ -\displaystyle -\left. - \left[ {radval=5}, {radmult=1}, -{radvect={\left[ {\left[ -\begin{array}{c} -0 \\ --{1 \over 2} \\ -1 -\end{array} -\right]} -\right]}} -\right] -\right] -\end{array} -$$ -\returnType{Type: List Record(radval: Expression Integer,radmult: Integer,radvect: List Matrix Expression Integer)} - -Alternatively, Axiom can compute real or complex approximations to the -\index{approximation} -eigenvectors and eigenvalues using the operations {\bf realEigenvectors} -or {\bf complexEigenvectors}. -They each take an additional argument $\epsilon$ -to specify the ``precision'' required. -\index{precision} -In the real case, this means that each approximation will be within -$\pm\epsilon$ of the actual -result. -In the complex case, this means that each approximation will be within -$\pm\epsilon$ of the actual result -in each of the real and imaginary parts. - -The precision can be specified as a {\tt Float} if the results are -desired in floating-point notation, or as {\tt Fraction Integer} if the -results are to be expressed using rational (or complex rational) numbers. - -\spadcommand{realEigenvectors(m1,1/1000) } -$$ -\begin{array}{@{}l} -\left[ -{\left[ {outval=5}, {outmult=1}, {outvect={\left[ {\left[ -\begin{array}{c} -0 \\ --{1 \over 2} \\ -1 -\end{array} -\right]} -\right]}} -\right]}, -\right. -\\ -\\ -\displaystyle - {\left[ {outval={{5717} \over {2048}}}, {outmult=1}, -{outvect={\left[ {\left[ -\begin{array}{c} -{{5717} \over {2048}} \\ -2 \\ -1 -\end{array} -\right]} -\right]}} -\right]}, -\\ -\\ -\displaystyle -\left. - {\left[ {outval=-{{3669} \over {2048}}}, {outmult=1}, -{outvect={\left[ {\left[ -\begin{array}{c} --{{3669} \over {2048}} \\ -2 \\ -1 -\end{array} -\right]} -\right]}} -\right]} -\right] -\end{array} -$$ -\returnType{Type: List Record(outval: Fraction Integer,outmult: Integer,outvect: List Matrix Fraction Integer)} - -If an $n$ by $n$ matrix has $n$ distinct eigenvalues (and -therefore $n$ eigenvectors) the operation {\bf eigenMatrix} -gives you a matrix of the eigenvectors. - -\spadcommand{eigenMatrix(m1) } -$$ -\left[ -\begin{array}{ccc} -{{{\sqrt {{21}}}+1} \over 2} & {{-{\sqrt {{21}}}+1} \over 2} & 0 \\ -2 & 2 & -{1 \over 2} \\ -1 & 1 & 1 -\end{array} -\right] -$$ -\returnType{Type: Union(Matrix Expression Integer,...)} - -\spadcommand{m2 := matrix [ [-5,-2],[18,7] ] } -$$ -\left[ -\begin{array}{cc} --5 & -2 \\ -{18} & 7 -\end{array} -\right] -$$ -\returnType{Type: Matrix Integer} - -\spadcommand{eigenMatrix(m2) } -$$ -\mbox{\tt "failed"} -$$ -\returnType{Type: Union("failed",...)} - -If a symmetric matrix -\index{matrix!symmetric} -has a basis of orthonormal eigenvectors, then -\index{basis!orthonormal} -{\bf orthonormalBasis} computes a list of these vectors. -\index{orthonormal basis} - -\spadcommand{m3 := matrix [ [1,2],[2,1] ] } -$$ -\left[ -\begin{array}{cc} -1 & 2 \\ -2 & 1 -\end{array} -\right] -$$ -\returnType{Type: Matrix Integer} - -\spadcommand{orthonormalBasis(m3) } -$$ -\left[ -{\left[ -\begin{array}{c} --{1 \over {\sqrt {2}}} \\ -{1 \over {\sqrt {2}}} -\end{array} -\right]}, - {\left[ -\begin{array}{c} -{1 \over {\sqrt {2}}} \\ -{1 \over {\sqrt {2}}} -\end{array} -\right]} -\right] -$$ -\returnType{Type: List Matrix Expression Integer} - -\section{Solution of Linear and Polynomial Equations} -\label{ugProblemLinPolEqn} -% -In this section we discuss the Axiom facilities for solving -systems of linear equations, finding the roots of polynomials and -\index{linear equation} -solving systems of polynomial equations. -For a discussion of the solution of differential equations, see -\ref{ugProblemDEQ} on page~\pageref{ugProblemDEQ}. - -\subsection{Solution of Systems of Linear Equations} -\label{ugxProblemLinSys} - -You can use the operation {\bf solve} to solve systems of linear equations. -\index{equation!linear!solving} - -The operation {\bf solve} takes two arguments, the list of equations and the -list of the unknowns to be solved for. -A system of linear equations need not have a unique solution. - -To solve the linear system: -$$ -\begin{array}{rcrcrcr} - x &+& y &+& z &=& 8 \\ -3 x &-& 2 y &+& z &=& 0 \\ - x &+& 2 y &+& 2 z &=& 17 -\end{array} -$$ -evaluate this expression. - -\spadcommand{solve([x+y+z=8,3*x-2*y+z=0,x+2*y+2*z=17],[x,y,z])} -$$ -\left[ -{\left[ {x=-1}, {y=2}, {z=7} -\right]} -\right] -$$ -\returnType{Type: List List Equation Fraction Polynomial Integer} - -Parameters are given as new variables starting with a percent sign and -{\tt \%} and -the variables are expressed in terms of the parameters. -If the system has no solutions then the empty list is returned. - -When you solve the linear system -$$ -\begin{array}{rcrcrcr} - x&+&2 y&+&3 z&=&2 \\ -2 x&+&3 y&+&4 z&=&2 \\ -3 x&+&4 y&+&5 z&=&2 -\end{array} -$$ -with this expression -you get a solution involving a parameter. - -\spadcommand{solve([x+2*y+3*z=2,2*x+3*y+4*z=2,3*x+4*y+5*z=2],[x,y,z])} -$$ -\left[ -{\left[ {x={ \%Q -2}}, {y={-{2 \ \%Q}+2}}, {z= \%Q} -\right]} -\right] -$$ -\returnType{Type: List List Equation Fraction Polynomial Integer} - -The system can also be presented as a matrix and a vector. -The matrix contains the coefficients of the linear equations and -the vector contains the numbers appearing on the right-hand sides -of the equations. -You may input the matrix as a list of rows and the vector as a -list of its elements. - -To solve the system: -$$ -\begin{array}{rcrcrcr} - x&+& y&+& z&=&8 \\ -3 x&-&2 y&+& z&=&0 \\ - x&+&2 y&+&2 z&=&17 -\end{array} -$$ -in matrix form you would evaluate this expression. - -\spadcommand{solve([ [1,1,1],[3,-2,1],[1,2,2] ],[8,0,17])} -$$ -\left[ -{particular={\left[ -1, 2, 7\right]}}, -{basis={\left[ {\left[ 0, 0, 0\right]}\right]}} -\right] -$$ -\returnType{Type: Record(particular: Union(Vector Fraction Integer,"failed"), -basis: List Vector Fraction Integer)} - -The solutions are presented as a {\tt Record} with two -components: the component {\it particular} contains a particular solution of the given system or -the item {\tt "failed"} if there are no solutions, the component -{\it basis} contains a list of vectors that -are a basis for the space of solutions of the corresponding -homogeneous system. -If the system of linear equations does not have a unique solution, -then the {\it basis} component contains -non-trivial vectors. - -This happens when you solve the linear system -$$ -\begin{array}{rcrcrcr} - x&+&2 y&+&3 z&=&2 \\ -2 x&+&3 y&+&4 z&=&2 \\ -3 x&+&4 y&+&5 z&=&2 -\end{array} -$$ -with this command. - -\spadcommand{solve([ [1,2,3],[2,3,4],[3,4,5] ],[2,2,2])} -$$ -\left[ -{particular={\left[ -2, 2, 0 \right]}}, -{basis={\left[ {\left[ 1, -2, 1 \right]}\right]}} -\right] -$$ -\returnType{Type: Record(particular: Union(Vector Fraction Integer,"failed"), -basis: List Vector Fraction Integer)} - - -All solutions of this system are obtained by adding the particular -solution with a linear combination of the {\it basis} vectors. - -When no solution exists then {\tt "failed"} is returned as the -{\it particular} component, as follows: - -\spadcommand{solve([ [1,2,3],[2,3,4],[3,4,5] ],[2,3,2])} -$$ -\left[ -{particular= \mbox{\tt "failed"} }, -{basis={\left[ {\left[ 1, -2, 1\right]}\right]}} -\right] -$$ -\returnType{Type: Record(particular: Union(Vector Fraction Integer,"failed"), -basis: List Vector Fraction Integer)} - -When you want to solve a system of homogeneous equations (that is, -a system where the numbers on the right-hand sides of the -\index{nullspace} -equations are all zero) in the matrix form you can omit the second -argument and use the {\bf nullSpace} operation. - -This computes the solutions of the following system of equations: -$$ -\begin{array}{rcrcrcr} - x&+&2 y&+&3 z&=&0 \\ -2 x&+&3 y&+&4 z&=&0 \\ -3 x&+&4 y&+&5 z&=&0 -\end{array} -$$ -The result is given as a list of vectors and -these vectors form a basis for the solution space. - -\spadcommand{nullSpace([ [1,2,3],[2,3,4],[3,4,5] ])} -$$ -\left[ -{\left[ 1, -2, 1 \right]} -\right] -$$ -\returnType{Type: List Vector Integer} - -\subsection{Solution of a Single Polynomial Equation} -\label{ugxProblemOnePol} - -Axiom can solve polynomial equations producing either approximate -\index{polynomial!root finding} -or exact solutions. -\index{equation!polynomial!solving} -Exact solutions are either members of the ground -field or can be presented symbolically as roots of irreducible polynomials. - -This returns the one rational root along with an irreducible -polynomial describing the other solutions. - -\spadcommand{solve(x**3 = 8,x)} -$$ -\left[ -{x=2}, {{{x \sp 2}+{2 \ x}+4}=0} -\right] -$$ -\returnType{Type: List Equation Fraction Polynomial Integer} - -If you want solutions expressed in terms of radicals you would use this -instead. -\index{radical} - -\spadcommand{radicalSolve(x**3 = 8,x)} -$$ -\left[ -{x={-{\sqrt {-3}} -1}}, {x={{\sqrt {-3}} -1}}, {x=2} -\right] -$$ -\returnType{Type: List Equation Expression Integer} - -The {\bf solve} command always returns a value but -{\bf radicalSolve} returns only the solutions that it is -able to express in terms of radicals. -\index{radical} - -If the polynomial equation has rational coefficients -you can ask for approximations to its real roots by calling -solve with a second argument that specifies the ``precision'' -\index{precision} -$\epsilon$. -This means that each approximation will be within -$\pm\epsilon$ of the actual -result. - -Notice that the type of second argument controls the type of the result. - -\spadcommand{solve(x**4 - 10*x**3 + 35*x**2 - 50*x + 25,.0001)} -$$ -\left[ -{x={3.6180114746 09375}}, {x={1.3819885253 90625}} -\right] -$$ -\returnType{Type: List Equation Polynomial Float} - -If you give a floating-point precision you get a floating-point result; -if you give the precision as a rational number you get a rational result. - -\spadcommand{solve(x**3-2,1/1000)} -$$ -\left[ -{x={{2581} \over {2048}}} -\right] -$$ -\returnType{Type: List Equation Polynomial Fraction Integer} - -If you want approximate complex results you should use the -\index{approximation} -command {\bf complexSolve} that takes the same precision argument -$\epsilon$. - -\spadcommand{complexSolve(x**3-2,.0001)} -$$ -\begin{array}{@{}l} -\left[ -{x={1.2599182128 90625}}, -\right. -\\ -\\ -\displaystyle -{x={-{0.6298943279 5395613131} -{{1.0910949707 03125} \ i}}}, -\\ -\\ -\displaystyle -\left. -{x={-{0.6298943279 5395613131}+{{1.0910949707 03125} \ -i}}} -\right] -\end{array} -$$ -\returnType{Type: List Equation Polynomial Complex Float} - -Each approximation will be within -$\pm\epsilon$ of the actual result -in each of the real and imaginary parts. - -\spadcommand{complexSolve(x**2-2*\%i+1,1/100)} -$$ -\left[ -{x={-{{13028925} \over {16777216}} -{{{325} \over {256}} \ i}}}, -{x={{{13028925} \over {16777216}}+{{{325} \over {256}} \ i}}} -\right] -$$ -\returnType{Type: List Equation Polynomial Complex Fraction Integer} - -Note that if you omit the {\tt =} from the first argument -Axiom generates an equation by equating the first argument to zero. -Also, when only one variable is present in the equation, you -do not need to specify the variable to be solved for, that is, -you can omit the second argument. - -Axiom can also solve equations involving rational functions. -Solutions where the denominator vanishes are discarded. - -\spadcommand{radicalSolve(1/x**3 + 1/x**2 + 1/x = 0,x)} -$$ -\left[ -{x={{-{\sqrt {-3}} -1} \over 2}}, {x={{{\sqrt {-3}} -1} \over 2}} -\right] -$$ -\returnType{Type: List Equation Expression Integer} - -\subsection{Solution of Systems of Polynomial Equations} -\label{ugxProblemPolSys} - -Given a system of equations of rational functions with exact coefficients: -\index{equation!polynomial!solving} -\vskip 0.1cm -$$ -\begin{array}{c} -p_1(x_1, \ldots, x_n) \\ \vdots \\ p_m(x_1,\ldots,x_n) -\end{array} -$$ - -Axiom can find -numeric or symbolic solutions. -The system is first split into irreducible components, then for -each component, a triangular system of equations is found that reduces -the problem to sequential solution of univariate polynomials resulting -from substitution of partial solutions from the previous stage. -$$ -\begin{array}{c} -q_1(x_1, \ldots, x_n) \\ \vdots \\ q_m(x_n) -\end{array} -$$ - -Symbolic solutions can be presented using ``implicit'' algebraic numbers -defined as roots of irreducible polynomials or in terms of radicals. -Axiom can also find approximations to the real or complex roots -of a system of polynomial equations to any user-specified accuracy. - -The operation {\bf solve} for systems is used in a way similar -to {\bf solve} for single equations. -Instead of a polynomial equation, one has to give a list of -equations and instead of a single variable to solve for, a list of -variables. -For solutions of single equations see -\ref{ugxProblemOnePol} on page~\pageref{ugxProblemOnePol}. - -Use the operation {\bf solve} if you want implicitly presented -solutions. - -\spadcommand{solve([3*x**3 + y + 1,y**2 -4],[x,y])} -$$ -\left[ -{\left[ {x=-1}, {y=2} \right]}, -{\left[ {{{x \sp 2} -x+1}=0}, {y=2} \right]}, -{\left[ {{{3 \ {x \sp 3}} -1}=0}, {y=-2} \right]} -\right] -$$ -\returnType{Type: List List Equation Fraction Polynomial Integer} - -\spadcommand{solve([x = y**2-19,y = z**2+x+3,z = 3*x],[x,y,z])} -$$ -\left[ -{\left[ {x={z \over 3}}, -{y={{{3 \ {z \sp 2}}+z+9} \over 3}}, -{{{9 \ {z \sp 4}}+{6 \ {z \sp 3}}+{{55} \ {z \sp 2}}+{{15} \ z} -{90}}=0} -\right]} -\right] -$$ -\returnType{Type: List List Equation Fraction Polynomial Integer} - -Use {\bf radicalSolve} if you want your solutions expressed -in terms of radicals. - -\spadcommand{radicalSolve([3*x**3 + y + 1,y**2 -4],[x,y])} -$$ -\begin{array}{@{}l} -\left[ -{\left[ {x={{{\sqrt {-3}}+1} \over 2}}, {y=2} \right]}, -{\left[ {x={{-{\sqrt {-3}}+1} \over 2}}, {y=2} \right]}, -\right. -\\ -\\ -\displaystyle -{\left[ {x={{-{{\sqrt {-1}} \ {\sqrt {3}}} -1} \over {2 \ {\root {3} \of -{3}}}}}, {y=-2} \right]}, -{\left[ {x={{{{\sqrt {-1}} \ {\sqrt {3}}} -1} \over {2 \ {\root {3} \of -{3}}}}}, {y=-2} \right]}, -\\ -\\ -\displaystyle -\left. -{\left[ {x={1 \over {\root {3} \of {3}}}}, {y=-2} \right]}, -{\left[ {x=-1}, {y=2} \right]} -\right] -\end{array} -$$ -\returnType{Type: List List Equation Expression Integer} - -To get numeric solutions you only need to give the list of -equations and the precision desired. -The list of variables would be redundant information since there -can be no parameters for the numerical solver. - -If the precision is expressed as a floating-point number you get -results expressed as floats. - -\spadcommand{solve([x**2*y - 1,x*y**2 - 2],.01)} -$$ -\left[ -{\left[ {y={1.5859375}}, {x={0.79296875}} \right]} -\right] -$$ -\returnType{Type: List List Equation Polynomial Float} - -To get complex numeric solutions, use the operation {\bf complexSolve}, -which takes the same arguments as in the real case. - -\spadcommand{complexSolve([x**2*y - 1,x*y**2 - 2],1/1000)} -$$ -\begin{array}{@{}l} -\left[ -{\left[ {y={{1625} \over {1024}}}, {x={{1625} \over {2048}}} \right]}, -\right. -\\ -\\ -\displaystyle -{\left[ {y={-{{435445573689} \over {549755813888}} -{{{1407} \over {1024}} -\ i}}}, -{x={-{{435445573689} \over {1099511627776}} -{{{1407} \over {2048}} \ i}}} -\right]}, -\\ -\\ -\displaystyle -\left. -{\left[ {y={-{{435445573689} \over {549755813888}}+{{{1407} \over {1024}} -\ i}}}, -{x={-{{435445573689} \over {1099511627776}}+{{{1407} \over {2048}} \ i}}} -\right]} -\right] -\end{array} -$$ -\returnType{Type: List List Equation Polynomial Complex Fraction Integer} - - -It is also possible to solve systems of equations in rational functions -over the rational numbers. -Note that $[x = 0.0, a = 0.0]$ is not returned as a solution since -the denominator vanishes there. - -\spadcommand{solve([x**2/a = a,a = a*x],.001)} -$$ -\left[ -{\left[ {x={1.0}}, {a=-{1.0}} \right]}, -{\left[ {x={1.0}}, {a={1.0}} \right]} -\right] -$$ -\returnType{Type: List List Equation Polynomial Float} - - -When solving equations with -denominators, all solutions where the denominator vanishes are -discarded. - -\spadcommand{radicalSolve([x**2/a + a + y**3 - 1,a*y + a + 1],[x,y])} -$$ -\begin{array}{@{}l} -\left[ -{\left[ {x=-{\sqrt {{{-{a \sp 4}+{2 \ {a \sp 3}}+{3 \ {a \sp 2}}+{3 \ -a}+1} \over {a \sp 2}}}}}, {y={{-a -1} \over a}} -\right]}, -\right. -\\ -\\ -\displaystyle -\left. -{\left[ {x={\sqrt {{{-{a \sp 4}+{2 \ {a \sp 3}}+{3 \ {a \sp 2}}+{3 \ -a}+1} \over {a \sp 2}}}}}, {y={{-a -1} \over a}} -\right]} -\right] -\end{array} -$$ -\returnType{Type: List List Equation Expression Integer} - -\section{Limits} -\label{ugProblemLimits} - -% -To compute a limit, you must specify a functional expression, -\index{limit} -a variable, and a limiting value for that variable. -If you do not specify a direction, Axiom attempts to -compute a two-sided limit. - -Issue this to compute the limit -$$\lim_{x \rightarrow 1}{{\displaystyle x^2 - 3x + -2}\over{\displaystyle x^2 - 1}}.$$ - -\spadcommand{limit((x**2 - 3*x + 2)/(x**2 - 1),x = 1)} -$$ --{1 \over 2} -$$ -\returnType{Type: Union(OrderedCompletion Fraction Polynomial Integer,...)} - -Sometimes the limit when approached from the left is different from -the limit from the right and, in this case, you may wish to ask for a -one-sided limit. Also, if you have a function that is only defined on -one side of a particular value, \index{limit!one-sided vs. two-sided} -you can compute a one-sided limit. - -The function $log(x)$ is only defined to the right of zero, that is, -for $x > 0$. Thus, when computing limits of functions involving -$log(x)$, you probably want a ``right-hand'' limit. - -\spadcommand{limit(x * log(x),x = 0,"right")} -$$ -0 -$$ -\returnType{Type: Union(OrderedCompletion Expression Integer,...)} - -When you do not specify ``$right$'' or ``$left$'' as the optional fourth -argument, {\bf limit} tries to compute a two-sided limit. Here the -limit from the left does not exist, as Axiom indicates when you try to -take a two-sided limit. - -\spadcommand{limit(x * log(x),x = 0)} -$$ -\left[ -{leftHandLimit= \mbox{\tt "failed"} }, {rightHandLimit=0} -\right] -$$ -\returnType{Type: Union(Record(leftHandLimit: -Union(OrderedCompletion Expression Integer,"failed"), -rightHandLimit: Union(OrderedCompletion Expression Integer,"failed")),...)} - -A function can be defined on both sides of a particular value, but -tend to different limits as its variable approaches that value from -the left and from the right. We can construct an example of this as -follows: Since $\sqrt{y^2}$ is simply the absolute value of $y$, the -function $\sqrt{y^2} / y$ is simply the sign ($+1$ or $-1$) of the -nonzero real number $y$. Therefore, $\sqrt{y^2} / y = -1$ for $y < 0$ -and $\sqrt{y^2} / y = +1$ for $y > 0$. - -This is what happens when we take the limit at $y = 0$. -The answer returned by Axiom gives both a -``left-hand'' and a ``right-hand'' limit. - -\spadcommand{limit(sqrt(y**2)/y,y = 0)} -$$ -\left[ -{leftHandLimit=-1}, {rightHandLimit=1} -\right] -$$ -\returnType{Type: Union(Record(leftHandLimit: -Union(OrderedCompletion Expression Integer,"failed"), -rightHandLimit: Union(OrderedCompletion Expression Integer,"failed")),...)} - -Here is another example, this time using a more complicated function. - -\spadcommand{limit(sqrt(1 - cos(t))/t,t = 0)} -$$ -\left[ -{leftHandLimit=-{1 \over {\sqrt {2}}}}, -{rightHandLimit={1 \over {\sqrt {2}}}} -\right] -$$ -\returnType{Type: Union(Record(leftHandLimit: -Union(OrderedCompletion Expression Integer,"failed"), -rightHandLimit: Union(OrderedCompletion Expression Integer,"failed")),...)} - -You can compute limits at infinity by passing either -\index{limit!at infinity} $+\infty$ or $-\infty$ as the third -argument of {\bf limit}. - -To do this, use the constants $\%plusInfinity$ and $\%minusInfinity$. - -\spadcommand{limit(sqrt(3*x**2 + 1)/(5*x),x = \%plusInfinity)} -$$ -{\sqrt {3}} \over 5 -$$ -\returnType{Type: Union(OrderedCompletion Expression Integer,...)} - -\spadcommand{limit(sqrt(3*x**2 + 1)/(5*x),x = \%minusInfinity)} -$$ --{{\sqrt {3}} \over 5} -$$ -\returnType{Type: Union(OrderedCompletion Expression Integer,...)} - -You can take limits of functions with parameters. -\index{limit!of function with parameters} -As you can see, the limit is expressed in terms of the parameters. - -\spadcommand{limit(sinh(a*x)/tan(b*x),x = 0)} -$$ -a \over b -$$ -\returnType{Type: Union(OrderedCompletion Expression Integer,...)} - -When you use {\bf limit}, you are taking the limit of a real -function of a real variable. - -When you compute this, Axiom returns $0$ because, as a function of a -real variable, $sin(1/z)$ is always between $-1$ and $1$, so -$z * sin(1/z)$ tends to $0$ as $z$ tends to $0$. - -\spadcommand{limit(z * sin(1/z),z = 0)} -$$ -0 -$$ -\returnType{Type: Union(OrderedCompletion Expression Integer,...)} - -However, as a function of a {\it complex} variable, $sin(1/z)$ is badly -\index{limit!real vs. complex} -behaved near $0$ (one says that $sin(1/z)$ has an -\index{essential singularity} -{\it essential singularity} at $z = 0$). -\index{singularity!essential} - -When viewed as a function of a complex variable, $z * sin(1/z)$ -does not approach any limit as $z$ tends to $0$ in the complex plane. -Axiom indicates this when we call {\bf complexLimit}. - -\spadcommand{complexLimit(z * sin(1/z),z = 0)} -$$ -\mbox{\tt "failed"} -$$ -\returnType{Type: Union("failed",...)} - -Here is another example. -As $x$ approaches $0$ along the real axis, $exp(-1/x**2)$ -tends to $0$. -\spadcommand{limit(exp(-1/x**2),x = 0)} -$$ -0 -$$ -\returnType{Type: Union(OrderedCompletion Expression Integer,...)} - -However, if $x$ is allowed to approach $0$ along any path in the -complex plane, the limiting value of $exp(-1/x**2)$ depends on the -path taken because the function has an essential singularity at $x=0$. -This is reflected in the error message returned by the function. -\spadcommand{complexLimit(exp(-1/x**2),x = 0)} -$$ -\mbox{\tt "failed"} -$$ -\returnType{Type: Union("failed",...)} - -You can also take complex limits at infinity, that is, limits of a -function of $z$ as $z$ approaches infinity on the Riemann sphere. Use -the symbol $\%infinity$ to denote ``complex infinity.'' - -As above, to compute complex limits rather than real limits, use -{\bf complexLimit}. - -\spadcommand{complexLimit((2 + z)/(1 - z),z = \%infinity)} -$$ --1 -$$ -\returnType{Type: OnePointCompletion Fraction Polynomial Integer} - -In many cases, a limit of a real function of a real variable exists -when the corresponding complex limit does not. This limit exists. - -\spadcommand{limit(sin(x)/x,x = \%plusInfinity)} -$$ -0 -$$ -\returnType{Type: Union(OrderedCompletion Expression Integer,...)} - -But this limit does not. - -\spadcommand{complexLimit(sin(x)/x,x = \%infinity)} -$$ -\mbox{\tt "failed"} -$$ -\returnType{Type: Union("failed",...)} - -\section{Laplace Transforms} -\label{ugProblemLaplace} - -Axiom can compute some forward Laplace transforms, mostly -\index{Laplace transform} of elementary \index{function!elementary} -functions \index{transform!Laplace} not involving logarithms, although -some cases of special functions are handled. - -To compute the forward Laplace transform of $F(t)$ with respect to -$t$ and express the result as $f(s)$, issue the command -$laplace(F(t), t, s)$. - -\spadcommand{laplace(sin(a*t)*cosh(a*t)-cos(a*t)*sinh(a*t), t, s)} -$$ -{4 \ {a \sp 3}} \over {{s \sp 4}+{4 \ {a \sp 4}}} -$$ -\returnType{Type: Expression Integer} - -Here are some other non-trivial examples. - -\spadcommand{laplace((exp(a*t) - exp(b*t))/t, t, s)} -$$ --{\log \left({{s -a}} \right)}+{\log\left({{s -b}} \right)} -$$ -\returnType{Type: Expression Integer} - -\spadcommand{laplace(2/t * (1 - cos(a*t)), t, s)} -$$ -{\log \left({{{s \sp 2}+{a \sp 2}}} \right)}-{2 \ {\log \left({s} \right)}} -$$ -\returnType{Type: Expression Integer} - -\spadcommand{laplace(exp(-a*t) * sin(b*t) / b**2, t, s)} -$$ -1 \over {{b \ {s \sp 2}}+{2 \ a \ b \ s}+{b \sp 3}+{{a \sp 2} \ b}} -$$ -\returnType{Type: Expression Integer} - -\spadcommand{laplace((cos(a*t) - cos(b*t))/t, t, s)} -$$ -{{\log \left({{{s \sp 2}+{b \sp 2}}} \right)}- -{\log \left({{{s \sp 2}+{a \sp 2}}} \right)}} -\over 2 -$$ -\returnType{Type: Expression Integer} - -Axiom also knows about a few special functions. - -\spadcommand{laplace(exp(a*t+b)*Ei(c*t), t, s)} -$$ -{{e \sp b} \ {\log \left({{{s+c -a} \over c}} \right)}}\over {s -a} -$$ -\returnType{Type: Expression Integer} - -\spadcommand{laplace(a*Ci(b*t) + c*Si(d*t), t, s)} -$$ -{{a \ {\log \left({{{{s \sp 2}+{b \sp 2}} \over {b \sp 2}}} \right)}}+ -{2\ c \ {\arctan \left({{d \over s}} \right)}}}\over {2 \ s} -$$ -\returnType{Type: Expression Integer} - -When Axiom does not know about a particular transform, -it keeps it as a formal transform in the answer. - -\spadcommand{laplace(sin(a*t) - a*t*cos(a*t) + exp(t**2), t, s)} -$$ -{{{\left( {s \sp 4}+{2 \ {a \sp 2} \ {s \sp 2}}+{a \sp 4} \right)} -\ {laplace \left({{e \sp {t \sp 2}}, t, s} \right)}}+ -{2\ {a \sp 3}}} \over {{s \sp 4}+{2 \ {a \sp 2} \ {s \sp 2}}+{a \sp 4}} -$$ -\returnType{Type: Expression Integer} - -\section{Integration} -\label{ugProblemIntegration} - -% -Integration is the reverse process of differentiation, that is, -\index{integration} an {\it integral} of a function $f$ with respect -to a variable $x$ is any function $g$ such that $D(g,x)$ is equal to -$f$. - -The package {\tt FunctionSpaceIntegration} provides the top-level -integration operation, \spadfunFrom{integrate}{FunctionSpaceIntegration}, -for integrating real-valued elementary functions. -\index{FunctionSpaceIntegration} - -\spadcommand{integrate(cosh(a*x)*sinh(a*x), x)} -$$ -{{{\sinh \left({{a \ x}} \right)}\sp 2}+ -{{\cosh \left({{a \ x}} \right)}\sp 2}} -\over {4 \ a} -$$ -\returnType{Type: Union(Expression Integer,...)} - -Unfortunately, antiderivatives of most functions cannot be expressed in -terms of elementary functions. - -\spadcommand{integrate(log(1 + sqrt(a * x + b)) / x, x)} -$$ -\int \sp{\displaystyle x} {{{\log -\left( -{{{\sqrt {{b+{ \%M \ a}}}}+1}} -\right)} -\over \%M} \ {d \%M}} -$$ -\returnType{Type: Union(Expression Integer,...)} - -Given an elementary function to integrate, Axiom returns a formal -integral as above only when it can prove that the integral is not -elementary and not when it cannot determine the integral. -In this rare case it prints a message that it cannot -determine if an elementary integral exists. - -Similar functions may have antiderivatives \index{antiderivative} -that look quite different because the form of the antiderivative -depends on the sign of a constant that appears in the function. - -\spadcommand{integrate(1/(x**2 - 2),x)} -$$ -{\log \left({{{{{\left( {x \sp 2}+2 \right)}\ {\sqrt {2}}} -{4 \ x}} -\over {{x \sp 2} -2}}} \right)} -\over {2 \ {\sqrt {2}}} -$$ -\returnType{Type: Union(Expression Integer,...)} - -\spadcommand{integrate(1/(x**2 + 2),x)} -$$ -{\arctan \left({{{x \ {\sqrt {2}}} \over 2}} \right)}\over {\sqrt {2}} -$$ -\returnType{Type: Union(Expression Integer,...)} - -If the integrand contains parameters, then there may be several possible -antiderivatives, depending on the signs of expressions of the parameters. - -In this case Axiom returns a list of answers that cover all the -possible cases. Here you use the answer involving the square root of -$a$ when $a > 0$ and \index{integration!result as list of real -functions} the answer involving the square root of $-a$ when $a < 0$. - -\spadcommand{integrate(x**2 / (x**4 - a**2), x)} -$$ -\begin{array}{@{}l} -\left[ -{{{\log -\left( -{{{{{\left( {x \sp 2}+a \right)} -\ {\sqrt {a}}} -{2 \ a \ x}} \over {{x \sp 2} -a}}} -\right)}+ -{2\ {\arctan \left({{{x \ {\sqrt {a}}} \over a}} \right)}}} -\over {4 \ {\sqrt {a}}}}, -\right. -\\ -\\ -\displaystyle -\left. -{{{\log \left({{{{{\left( {x \sp 2} -a \right)} -\ {\sqrt {-a}}}+{2 \ a \ x}} \over {{x \sp 2}+a}}} -\right)} --{2 \ {\arctan \left({{{x \ {\sqrt {-a}}} \over a}} \right)}}} -\over {4 \ {\sqrt {-a}}}} -\right] -\end{array} -$$ -\returnType{Type: Union(List Expression Integer,...)} - -If the parameters and the variables of integration can be complex -numbers rather than real, then the notion of sign is not defined. In -this case all the possible answers can be expressed as one complex -function. To get that function, rather than a list of real functions, -use \spadfunFrom{complexIntegrate}{FunctionSpaceComplexIntegration}, -which is provided by the package \index{integration!result as a -complex functions} {\tt FunctionSpaceComplexIntegration}. -\index{FunctionSpaceComplexIntegration} - -This operation is used for integrating complex-valued elementary -functions. - -\spadcommand{complexIntegrate(x**2 / (x**4 - a**2), x)} -$$ -\left( -\begin{array}{@{}l} -{{\sqrt {{4 \ a}}} \ {\log -\left( -{{{{x \ {\sqrt {-{4 \ a}}}}+{2 \ a}} \over {\sqrt {-{4 \ a}}}}} -\right)}} - -{{\sqrt {-{4 \ a}}} \ {\log -\left( -{{{{x \ {\sqrt {{4 \ a}}}}+{2 \ a}} \over {\sqrt {{4 \ a}}}}} -\right)}}+ -\\ -\\ -\displaystyle -{{\sqrt{-{4 \ a}}} \ {\log -\left( -{{{{x \ {\sqrt {{4 \ a}}}} -{2 \ a}} \over {\sqrt {{4 \ a}}}}} -\right)}} --{{\sqrt {{4 \ a}}} \ {\log -\left( -{{{{x \ {\sqrt {-{4 \ a}}}} -{2 \ a}} \over {\sqrt {-{4 \ a}}}}} -\right)}} -\end{array} -\right) -\over {2 \ {\sqrt {-{4 \ a}}} \ {\sqrt {{4 \ a}}}} -$$ -\returnType{Type: Expression Integer} - -As with the real case, antiderivatives for most complex-valued -functions cannot be expressed in terms of elementary functions. - -\spadcommand{complexIntegrate(log(1 + sqrt(a * x + b)) / x, x)} -$$ -\int \sp{\displaystyle x} -{{{\log \left({{{\sqrt {{b+{ \%M \ a}}}}+1}} \right)} -\over \%M} \ {d \%M}} -$$ -\returnType{Type: Expression Integer} - -Sometimes {\bf integrate} can involve symbolic algebraic numbers -such as those returned by \spadfunFrom{rootOf}{Expression}. -To see how to work with these strange generated symbols (such as -$\%\%a0$), see -\ref{ugxProblemSymRootAll} on page~\pageref{ugxProblemSymRootAll}. - -Definite integration is the process of computing the area between -\index{integration!definite} -the $x$-axis and the curve of a function $f(x)$. -The fundamental theorem of calculus states that if $f$ is -continuous on an interval $a..b$ and if there exists a function $g$ -that is differentiable on $a..b$ and such that $D(g, x)$ -is equal to $f$, then the definite integral of $f$ -for $x$ in the interval $a..b$ is equal to $g(b) - g(a)$. - -The package {\tt RationalFunctionDefiniteIntegration} provides -the top-level definite integration operation, -\spadfunFrom{integrate}{RationalFunctionDefiniteIntegration}, -for integrating real-valued rational functions. - -\spadcommand{integrate((x**4 - 3*x**2 + 6)/(x**6-5*x**4+5*x**2+4), x = 1..2)} -$$ -{{2 \ {\arctan \left({8} \right)}}+ -{2\ {\arctan \left({5} \right)}}+ -{2\ {\arctan \left({2} \right)}}+ -{2\ {\arctan \left({{1 \over 2}} \right)}} --\pi} \over 2 -$$ -\returnType{Type: Union(f1: OrderedCompletion Expression Integer,...)} - -Axiom checks beforehand that the function you are integrating is -defined on the interval $a..b$, and prints an error message if it -finds that this is not case, as in the following example: -\begin{verbatim} -integrate(1/(x**2-2), x = 1..2) - - >> Error detected within library code: - Pole in path of integration - You are being returned to the top level - of the interpreter. -\end{verbatim} -When parameters are present in the function, the function may or may not be -defined on the interval of integration. - -If this is the case, Axiom issues a warning that a pole might -lie in the path of integration, and does not compute the integral. - -\spadcommand{integrate(1/(x**2-a), x = 1..2)} -$$ -potentialPole -$$ -\returnType{Type: Union(pole: potentialPole,...)} - -If you know that you are using values of the parameter for which -the function has no pole in the interval of integration, use the -string {\tt ``noPole''} as a third argument to -\spadfunFrom{integrate}{RationalFunctionDefiniteIntegration}: - -The value here is, of course, incorrect if $sqrt(a)$ is between -$1$ and $2.$ - -\spadcommand{integrate(1/(x**2-a), x = 1..2, "noPole")} -$$ -\begin{array}{@{}l} -\left[ -\left( -\begin{array}{@{}l} --{\log \left({{{{{\left( -{4 \ {a \sp 2}} -{4 \ a} \right)} -\ {\sqrt {a}}}+{a \sp 3}+{6 \ {a \sp 2}}+a} \over {{a \sp 2} -{2 \ a}+1}}} -\right)}+ -\\ -\\ -\displaystyle -{\log\left({{{{{\left( -{8 \ {a \sp 2}} -{{32} \ a} \right)} -\ {\sqrt {a}}}+{a \sp 3}+{{24} \ {a \sp 2}}+{{16} \ a}} \over {{a \sp 2} --{8 \ a}+{16}}}} -\right)} -\end{array} -\right) -\over {4 \ {\sqrt {a}}}, -\right. -\\ -\\ -\displaystyle -\left. -{{-{\arctan \left({{{2 \ {\sqrt {-a}}} \over a}} \right)}+ -{\arctan\left({{{\sqrt {-a}} \over a}} \right)}} -\over {\sqrt {-a}}} -\right] -\end{array} -$$ -\returnType{Type: Union(f2: List OrderedCompletion Expression Integer,...)} - -\section{Working with Power Series} -\label{ugProblemSeries} -% -Axiom has very sophisticated facilities for working with power -\index{series} -series. -\index{power series} - -Infinite series are represented by a list of the coefficients that -have already been determined, together with a function for computing -the additional coefficients if needed. - -The system command that determines how many terms of a series is -displayed is {\tt )set streams calculate}. For the purposes of this -book, we have used this system command to display fewer than ten -terms. \index{set streams calculate} Series can be created from -expressions, from functions for the series coefficients, and from -applications of operations on existing series. The most general -function for creating a series is called {\bf series}, although you -can also use {\bf taylor}, {\bf laurent} and {\bf puiseux} in -situations where you know what kind of exponents are involved. - -For information about solving differential equations in terms of -power series, see -\ref{ugxProblemDEQSeries} on page~\pageref{ugxProblemDEQSeries}. - -\subsection{Creation of Power Series} -\label{ugxProblemSeriesCreate} - -This is the easiest way to create a power series. This tells Axiom -that $x$ is to be treated as a power series, \index{series!creating} -so functions of $x$ are again power series. - -\spadcommand{x := series 'x } -$$ -x -$$ -\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} - -We didn't say anything about the coefficients of the power series, so -the coefficients are general expressions over the integers. This -allows us to introduce denominators, symbolic constants, and other -variables as needed. - -Here the coefficients are integers (note that the coefficients are the -Fibonacci \index{Fibonacci numbers} numbers). - -\spadcommand{1/(1 - x - x**2) } -$$ -1+x+ -{2 \ {x \sp 2}}+ -{3 \ {x \sp 3}}+ -{5 \ {x \sp 4}}+ -{8 \ {x \sp 5}}+ -{{13} \ {x \sp 6}}+ -{{21} \ {x \sp 7}}+ -{{34} \ {x \sp 8}}+ -{{55} \ {x \sp 9}}+ -{{89} \ {x \sp {10}}}+ -{O \left({{x \sp {11}}} \right)} -$$ -\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} - -This series has coefficients that are rational numbers. - -\spadcommand{sin(x) } -$$ -x - -{{1 \over 6} \ {x \sp 3}}+ -{{1 \over {120}} \ {x \sp 5}} - -{{1 \over {5040}} \ {x \sp 7}}+ -{{1 \over {362880}} \ {x \sp 9}} - -{{1 \over {39916800}} \ {x \sp {11}}}+ -{O \left({{x \sp {12}}} \right)} -$$ -\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} - -When you enter this expression you introduce the symbolic constants -$sin(1)$ and $cos(1).$ - -\spadcommand{sin(1 + x) } -$$ -\begin{array}{@{}l} -{\sin \left({1} \right)}+ -{{\cos\left({1} \right)}\ x} - -{{{\sin \left({1} \right)}\over 2} \ {x \sp 2}} - -{{{\cos \left({1} \right)}\over 6} \ {x \sp 3}}+ -{{{\sin \left({1} \right)}\over {24}} \ {x \sp 4}}+ -{{{\cos \left({1} \right)}\over {120}} \ {x \sp 5}} - -{{{\sin \left({1} \right)}\over {720}} \ {x \sp 6}} - -\\ -\\ -\displaystyle -{{{\cos \left({1} \right)}\over {5040}} \ {x \sp 7}}+ -{{{\sin \left({1} \right)}\over {40320}} \ {x \sp 8}}+ -{{{\cos \left({1} \right)}\over {362880}} \ {x \sp 9}} - -{{{\sin \left({1} \right)}\over {3628800}} \ {x \sp {10}}}+ -{O \left({{x \sp {11}}} \right)} -\end{array} -$$ -\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} - -When you enter the expression -the variable $a$ appears in the resulting series expansion. - -\spadcommand{sin(a * x) } -$$ -{a \ x} - -{{{a \sp 3} \over 6} \ {x \sp 3}}+ -{{{a \sp 5} \over {120}} \ {x \sp 5}} - -{{{a \sp 7} \over {5040}} \ {x \sp 7}}+ -{{{a \sp 9} \over {362880}} \ {x \sp 9}} - -{{{a \sp {11}} \over {39916800}} \ {x \sp {11}}}+ -{O \left({{x \sp {12}}} \right)} -$$ -\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} - -You can also convert an expression into a series expansion. This -expression creates the series expansion of $1/log(y)$ about $y = 1$. -For details and more examples, see \ref{ugxProblemSeriesConversions} -on page~\pageref{ugxProblemSeriesConversions}. - -\spadcommand{series(1/log(y),y = 1)} -$$ -\begin{array}{@{}l} -{{\left( y -1 \right)}\sp {\left( -1 \right)}}+ -{1\over 2} -{{1 \over {12}} \ {\left( y -1 \right)}}+ -{{1\over {24}} \ {{\left( y -1 \right)}\sp 2}} - -{{{19} \over {720}} \ {{\left( y -1 \right)}\sp 3}}+ -{{3 \over {160}} \ {{\left( y -1 \right)}\sp 4}} - -\\ -\\ -\displaystyle -{{{863} \over {60480}} \ {{\left( y -1 \right)}\sp 5}}+ -{{{275} \over {24192}} \ {{\left( y -1 \right)}\sp 6}} - -{{{33953} \over {3628800}} \ {{\left( y -1 \right)}\sp 7}}+ -\\ -\\ -\displaystyle -{{{8183} \over {1036800}} \ {{\left( y -1 \right)}\sp 8}} - -{{{3250433} \over {479001600}} \ {{\left( y -1 \right)}\sp 9}}+ -{O \left({{{\left( y -1 \right)}\sp {10}}} \right)} -\end{array} -$$ -\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,y,1)} - -You can create power series with more general coefficients. You -normally accomplish this via a type declaration (see -\ref{ugTypesDeclare} on page~\pageref{ugTypesDeclare}). -See \ref{ugxProblemSeriesFunctions} on -page~\pageref{ugxProblemSeriesFunctions} for some warnings about working with -declared series. - -We declare that $y$ is a one-variable Taylor series -\index{series!Taylor} ({\tt UTS} is the abbreviation for -{\tt UnivariateTaylorSeries}) in the variable $z$ with {\tt FLOAT} -(that is, floating-point) coefficients, centered about $0.$ Then, by -assignment, we obtain the Taylor expansion of $exp(z)$ with -floating-point coefficients. \index{UnivariateTaylorSeries} - -\spadcommand{y : UTS(FLOAT,'z,0) := exp(z) } -$$ -\begin{array}{@{}l} -{1.0}+z+ -{{0.5} \ {z \sp 2}}+ -{{0.1666666666\ 6666666667} \ {z \sp 3}}+ -\\ -\\ -\displaystyle -{{0.0416666666\ 6666666666 7} \ {z \sp 4}}+ -{{0.0083333333\ 3333333333 34} \ {z \sp 5}}+ -\\ -\\ -\displaystyle -{{0.0013888888\ 8888888888 89} \ {z \sp 6}}+ -{{0.0001984126\ 9841269841 27} \ {z \sp 7}}+ -\\ -\\ -\displaystyle -{{0.0000248015\ 8730158730 1587} \ {z \sp 8}}+ -{{0.0000027557\ 3192239858 90653} \ {z \sp 9}}+ -\\ -\\ -\displaystyle -{{0.2755731922\ 3985890653 E -6} \ {z \sp {10}}}+ -{O \left({{z \sp {11}}} \right)} -\end{array} -$$ -\returnType{Type: UnivariateTaylorSeries(Float,z,0.0)} - -You can also create a power series by giving an explicit formula for -its $n$-th coefficient. For details and more examples, see -\ref{ugxProblemSeriesFormula} on page~\pageref{ugxProblemSeriesFormula}. - -To create a series about $w = 0$ whose $n$-th Taylor coefficient is -$1/n!$, you can evaluate this expression. This is the Taylor -expansion of $exp(w)$ at $w = 0$. - -\spadcommand{series(1/factorial(n),n,w = 0)} -$$ -\begin{array}{@{}l} -1+w+ -{{1 \over 2} \ {w \sp 2}}+ -{{1 \over 6} \ {w \sp 3}}+ -{{1 \over {24}} \ {w \sp 4}}+ -{{1 \over {120}} \ {w \sp 5}}+ -{{1 \over {720}} \ {w \sp 6}}+ -{{1 \over {5040}} \ {w \sp 7}}+ -\\ -\\ -\displaystyle -{{1 \over {40320}} \ {w \sp 8}}+ -{{1 \over {362880}} \ {w \sp 9}}+ -{{1 \over {3628800}} \ {w \sp {10}}}+ -{O \left({{w \sp {11}}} \right)} -\end{array} -$$ -\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,w,0)} - -\subsection{Coefficients of Power Series} -\label{ugxProblemSeriesCoefficients} - -You can extract any coefficient from a power series---even one that -hasn't been computed yet. This is possible because in Axiom, infinite -series are represented by a list of the coefficients that have already -been determined, together with a function for computing the additional -coefficients. (This is known as {\it lazy evaluation}.) When you ask -for a \index{series!lazy evaluation} coefficient that hasn't yet been -computed, Axiom computes \index{lazy evaluation} whatever additional -coefficients it needs and then stores them in the representation of -the power series. - -Here's an example of how to extract the coefficients of a power series. -\index{series!extracting coefficients} - -\spadcommand{x := series(x) } -$$ -x -$$ -\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} - -\spadcommand{y := exp(x) * sin(x) } -$$ -\begin{array}{@{}l} -x+ -{x \sp 2}+ -{{1 \over 3} \ {x \sp 3}} - -{{1 \over {30}} \ {x \sp 5}} - -{{1 \over {90}} \ {x \sp 6}} - -{{1 \over {630}} \ {x \sp 7}}+ -{{1 \over {22680}} \ {x \sp 9}}+ -\\ -\\ -\displaystyle -{{1 \over {113400}} \ {x \sp {10}}}+ -{{1 \over {1247400}} \ {x \sp {11}}}+ -{O \left({{x \sp {12}}} \right)} -\end{array} -$$ -\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} - -This coefficient is readily available. - -\spadcommand{coefficient(y,6) } -$$ --{1 \over {90}} -$$ -\returnType{Type: Expression Integer} - -But let's get the fifteenth coefficient of $y$. - -\spadcommand{coefficient(y,15) } -$$ --{1 \over {10216206000}} -$$ -\returnType{Type: Expression Integer} - -If you look at $y$ then you see that the coefficients up to order $15$ -have all been computed. - -\spadcommand{y } -$$ -\begin{array}{@{}l} -x+ -{x \sp 2}+ -{{1 \over 3} \ {x \sp 3}} - -{{1 \over {30}} \ {x \sp 5}} - -{{1 \over {90}} \ {x \sp 6}} - -{{1 \over {630}} \ {x \sp 7}}+ -{{1 \over {22680}} \ {x \sp 9}}+ -{{1 \over {113400}} \ {x \sp {10}}}+ -\\ -\\ -\displaystyle -{{1 \over {1247400}} \ {x \sp {11}}} - -{{1 \over {97297200}} \ {x \sp {13}}} - -{{1 \over {681080400}} \ {x \sp {14}}} - -{{1 \over {10216206000}} \ {x \sp {15}}}+ -{O \left({{x \sp {16}}} \right)} -\end{array} -$$ -\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} - -\subsection{Power Series Arithmetic} -\label{ugxProblemSeriesArithmetic} - -You can manipulate power series using the usual arithmetic operations -\index{series!arithmetic} -$+$, $-$, $*$, and $/$ (from UnivariatePuiseuxSeries) - -The results of these operations are also power series. - -\spadcommand{x := series x } -$$ -x -$$ -\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} - -\spadcommand{(3 + x) / (1 + 7*x)} -$$ -\begin{array}{@{}l} -3 - -{{20} \ x}+ -{{140} \ {x \sp 2}} - -{{980} \ {x \sp 3}}+ -{{6860} \ {x \sp 4}} - -{{48020} \ {x \sp 5}}+ -{{336140} \ {x \sp 6}} - -{{2352980} \ {x \sp 7}}+ -\\ -\\ -\displaystyle -{{16470860} \ {x \sp 8}} - -{{115296020} \ {x \sp 9}}+ -{{807072140} \ {x \sp {10}}}+ -{O \left({{x \sp {11}}} \right)} -\end{array} -$$ -\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} - -You can also compute $f(x) ** g(x)$, where $f(x)$ and $g(x)$ -are two power series. - -\spadcommand{base := 1 / (1 - x) } -$$ -1+x+ -{x \sp 2}+ -{x \sp 3}+ -{x \sp 4}+ -{x \sp 5}+ -{x \sp 6}+ -{x \sp 7}+ -{x \sp 8}+ -{x \sp 9}+ -{x \sp {10}}+ -{O \left({{x \sp {11}}} \right)} -$$ -\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} - -\spadcommand{expon := x * base } -$$ -x+ -{x \sp 2}+ -{x \sp 3}+ -{x \sp 4}+ -{x \sp 5}+ -{x \sp 6}+ -{x \sp 7}+ -{x \sp 8}+ -{x \sp 9}+ -{x \sp {10}}+ -{x \sp {11}}+ -{O \left({{x \sp {12}}} \right)} -$$ -\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} - -\spadcommand{base ** expon } -$$ -\begin{array}{@{}l} -1+ -{x \sp 2}+ -{{3 \over 2} \ {x \sp 3}}+ -{{7 \over 3} \ {x \sp 4}}+ -{{{43} \over {12}} \ {x \sp 5}}+ -{{{649} \over {120}} \ {x \sp 6}}+ -{{{241} \over {30}} \ {x \sp 7}}+ -{{{3706} \over {315}} \ {x \sp 8}}+ -\\ -\\ -\displaystyle -{{{85763} \over {5040}} \ {x \sp 9}}+ -{{{245339} \over {10080}} \ {x \sp {10}}}+ -{O \left({{x \sp {11}}} \right)} -\end{array} -$$ -\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} - -\subsection{Functions on Power Series} -\label{ugxProblemSeriesFunctions} - -Once you have created a power series, you can apply transcendental -functions -(for example, {\bf exp}, {\bf log}, {\bf sin}, {\bf tan}, -{\bf cosh}, etc.) to it. - -To demonstrate this, we first create the power series -expansion of the rational function - -$${\displaystyle x^2} \over {\displaystyle 1 - 6x + x^2}$$ - -about $x = 0$. - -\spadcommand{x := series 'x } -$$ -x -$$ -\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} - -\spadcommand{rat := x**2 / (1 - 6*x + x**2) } -$$ -\begin{array}{@{}l} -{x \sp 2}+ -{6 \ {x \sp 3}}+ -{{35} \ {x \sp 4}}+ -{{204} \ {x \sp 5}}+ -{{1189} \ {x \sp 6}}+ -{{6930} \ {x \sp 7}}+ -{{40391} \ {x \sp 8}}+ -{{235416} \ {x \sp 9}}+ -\\ -\\ -\displaystyle -{{1372105} \ {x \sp {10}}}+ -{{7997214} \ {x \sp {11}}}+ -{{46611179} \ {x \sp {12}}}+ -{O \left({{x \sp {13}}} \right)} -\end{array} -$$ -\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} - -If you want to compute the series expansion of - -$$\sin\left({\displaystyle x^2} \over {\displaystyle 1 - 6x + x^2}\right)$$ - -you simply compute the sine of $rat$. - -\spadcommand{sin(rat) } -$$ -\begin{array}{@{}l} -{x \sp 2}+ -{6 \ {x \sp 3}}+ -{{35} \ {x \sp 4}}+ -{{204} \ {x \sp 5}}+ -{{{7133} \over 6} \ {x \sp 6}}+ -{{6927} \ {x \sp 7}}+ -{{{80711} \over 2} \ {x \sp 8}}+ -{{235068} \ {x \sp 9}}+ -\\ -\\ -\displaystyle -{{{164285281} \over {120}} \ {x \sp {10}}}+ -{{{31888513} \over 4} \ {x \sp {11}}}+ -{{{371324777} \over 8} \ {x \sp {12}}}+ -{O \left({{x \sp {13}}} \right)} -\end{array} -$$ -\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} - -\boxed{4.6in}{ -\vskip 0.1cm -\noindent {\bf Warning:} -the type of the coefficients of a power series may -affect the kind of computations that you can do with that series. -This can only happen when you have made a declaration to -specify a series domain with a certain type of coefficient.\\ -} - -If you evaluate then you have declared that $y$ is a one variable -Taylor series \index{series!Taylor} ({\tt UTS} is the abbreviation for -{\tt UnivariateTaylorSeries}) in the variable $y$ with {\tt FRAC INT} -(that is, fractions of integer) coefficients, centered about $0$. - -\spadcommand{y : UTS(FRAC INT,y,0) := y } -$$ -y -$$ -\returnType{Type: UnivariateTaylorSeries(Fraction Integer,y,0)} - -You can now compute certain power series in $y$, {\it provided} that -these series have rational coefficients. - -\spadcommand{exp(y) } -$$ -\begin{array}{@{}l} -1+y+ -{{1 \over 2} \ {y \sp 2}}+ -{{1 \over 6} \ {y \sp 3}}+ -{{1 \over {24}} \ {y \sp 4}}+ -{{1 \over {120}} \ {y \sp 5}}+ -{{1 \over {720}} \ {y \sp 6}}+ -{{1 \over {5040}} \ {y \sp 7}}+ -{{1 \over {40320}} \ {y \sp 8}}+ -\\ -\\ -\displaystyle -{{1 \over {362880}} \ {y \sp 9}}+ -{{1 \over {3628800}} \ {y \sp {10}}}+ -{O \left({{y \sp {11}}} \right)} -\end{array} -$$ -\returnType{Type: UnivariateTaylorSeries(Fraction Integer,y,0)} - -You can get examples of such series by applying transcendental -functions to series in $y$ that have no constant terms. - -\spadcommand{tan(y**2) } -$$ -{y \sp 2}+ -{{1 \over 3} \ {y \sp 6}}+ -{{2 \over {15}} \ {y \sp {10}}}+ -{O \left({{y \sp {11}}} \right)} -$$ -\returnType{Type: UnivariateTaylorSeries(Fraction Integer,y,0)} - -\spadcommand{cos(y + y**5) } -$$ -1 - -{{1 \over 2} \ {y \sp 2}}+ -{{1 \over {24}} \ {y \sp 4}} - -{{{721} \over {720}} \ {y \sp 6}}+ -{{{6721} \over {40320}} \ {y \sp 8}} - -{{{1844641} \over {3628800}} \ {y \sp {10}}}+ -{O \left({{y \sp {11}}} \right)} -$$ -\returnType{Type: UnivariateTaylorSeries(Fraction Integer,y,0)} - -Similarly, you can compute the logarithm of a power series with rational -coefficients if the constant coefficient is $1.$ - -\spadcommand{log(1 + sin(y)) } -$$ -\begin{array}{@{}l} -y - -{{1 \over 2} \ {y \sp 2}}+ -{{1 \over 6} \ {y \sp 3}} - -{{1 \over {12}} \ {y \sp 4}}+ -{{1 \over {24}} \ {y \sp 5}} - -{{1 \over {45}} \ {y \sp 6}}+ -{{{61} \over {5040}} \ {y \sp 7}} - -{{{17} \over {2520}} \ {y \sp 8}}+ -{{{277} \over {72576}} \ {y \sp 9}} - -\\ -\\ -\displaystyle -{{{31} \over {14175}} \ {y \sp {10}}}+ -{O \left({{y \sp {11}}} \right)} -\end{array} -$$ -\returnType{Type: UnivariateTaylorSeries(Fraction Integer,y,0)} - -If you wanted to apply, say, the operation {\bf exp} to a power series -with a nonzero constant coefficient $a_0$, then the constant -coefficient of the result would be $e^{a_0}$, which is {\it not} a -rational number. Therefore, evaluating $exp(2 + tan(y))$ would -generate an error message. - -If you want to compute the Taylor expansion of $exp(2 + tan(y))$, you -must ensure that the coefficient domain has an operation {\bf exp} -defined for it. An example of such a domain is {\tt Expression -Integer}, the type of formal functional expressions over the integers. - -When working with coefficients of this type, - -\spadcommand{z : UTS(EXPR INT,z,0) := z } -$$ -z -$$ -\returnType{Type: UnivariateTaylorSeries(Expression Integer,z,0)} - -this presents no problems. - -\spadcommand{exp(2 + tan(z)) } -$$ -\begin{array}{@{}l} -{e \sp 2}+ -{{e \sp 2} \ z}+ -{{{e \sp 2} \over 2} \ {z \sp 2}}+ -{{{e \sp 2} \over 2} \ {z \sp 3}}+ -{{{3 \ {e \sp 2}} \over 8} \ {z \sp 4}}+ -{{{{37} \ {e \sp 2}} \over {120}} \ {z \sp 5}}+ -{{{{59} \ {e \sp 2}} \over {240}} \ {z \sp 6}}+ -{{{{137} \ {e \sp 2}} \over {720}} \ {z \sp 7}}+ -\\ -\\ -\displaystyle -{{{{871} \ {e \sp 2}} \over {5760}} \ {z \sp 8}}+ -{{{{41641} \ {e \sp 2}} \over {362880}} \ {z \sp 9}}+ -{{{{325249} \ {e \sp 2}} \over {3628800}} \ {z \sp {10}}}+ -{O \left({{z \sp {11}}} \right)} -\end{array} -$$ -\returnType{Type: UnivariateTaylorSeries(Expression Integer,z,0)} - -Another way to create Taylor series whose coefficients are expressions -over the integers is to use {\bf taylor} which works similarly to -\index{series!Taylor} {\bf series}. - -This is equivalent to the previous computation, except that now we -are using the variable $w$ instead of $z$. - -\spadcommand{w := taylor 'w } -$$ -w -$$ -\returnType{Type: UnivariateTaylorSeries(Expression Integer,w,0)} - -\spadcommand{exp(2 + tan(w)) } -$$ -\begin{array}{@{}l} -{e \sp 2}+ -{{e \sp 2} \ w}+ -{{{e \sp 2} \over 2} \ {w \sp 2}}+ -{{{e \sp 2} \over 2} \ {w \sp 3}}+ -{{{3 \ {e \sp 2}} \over 8} \ {w \sp 4}}+ -{{{{37} \ {e \sp 2}} \over {120}} \ {w \sp 5}}+ -{{{{59} \ {e \sp 2}} \over {240}} \ {w \sp 6}}+ -{{{{137} \ {e \sp 2}} \over {720}} \ {w \sp 7}}+ -\\ -\\ -\displaystyle -{{{{871} \ {e \sp 2}} \over {5760}} \ {w \sp 8}}+ -{{{{41641} \ {e \sp 2}} \over {362880}} \ {w \sp 9}}+ -{{{{325249} \ {e \sp 2}} \over {3628800}} \ {w \sp {10}}}+ -{O \left({{w \sp {11}}} \right)} -\end{array} -$$ -\returnType{Type: UnivariateTaylorSeries(Expression Integer,w,0)} - -\subsection{Converting to Power Series} -\label{ugxProblemSeriesConversions} - -The {\tt ExpressionToUnivariatePowerSeries} package provides -operations for computing series expansions of functions. -\index{ExpressionToUnivariatePowerSeries} - -Evaluate this to compute the Taylor expansion of $sin x$ about -\index{series!Taylor} $x = 0$. The first argument, $sin(x)$, -specifies the function whose series expansion is to be computed and -the second argument, $x = 0$, specifies that the series is to be -expanded in power of $(x - 0)$, that is, in power of $x$. - -\spadcommand{taylor(sin(x),x = 0)} -$$ -x - -{{1 \over 6} \ {x \sp 3}}+ -{{1 \over {120}} \ {x \sp 5}} - -{{1 \over {5040}} \ {x \sp 7}}+ -{{1 \over {362880}} \ {x \sp 9}}+ -{O \left({{x \sp {11}}} \right)} -$$ -\returnType{Type: UnivariateTaylorSeries(Expression Integer,x,0)} - -Here is the Taylor expansion of $sin x$ about $x = \frac{\pi}{6}$: - -\spadcommand{taylor(sin(x),x = \%pi/6)} -$$ -\begin{array}{@{}l} -{1 \over 2}+ -{{{\sqrt {3}} \over 2} \ {\left( x -{\pi \over 6} \right)}} --{{1 \over 4} \ {{\left( x -{\pi \over 6} \right)}\sp 2}} - -{{{\sqrt {3}} \over {12}} \ {{\left( x -{\pi \over 6} \right)}\sp 3}}+ -{{1 \over {48}} \ {{\left( x -{\pi \over 6} \right)}\sp 4}}+ -\\ -\\ -\displaystyle -{{{\sqrt {3}} \over {240}} \ {{\left( x -{\pi \over 6} \right)}\sp 5}} - -{{1 \over {1440}} \ {{\left( x -{\pi \over 6} \right)}\sp 6}} - -{{{\sqrt {3}} \over {10080}} \ {{\left( x -{\pi \over 6} \right)}\sp 7}}+ -{{1 \over {80640}} \ {{\left( x -{\pi \over 6} \right)}\sp 8}}+ -\\ -\\ -\displaystyle -{{{\sqrt {3}} \over {725760}} \ {{\left( x -{\pi \over 6} \right)}\sp 9}} - -{{1 \over {7257600}} \ {{\left( x -{\pi \over 6} \right)}\sp {10}}}+ -{O \left({{{\left( x -{\pi \over 6} \right)}\sp {11}}} \right)} -\end{array} -$$ -\returnType{Type: UnivariateTaylorSeries(Expression Integer,x,pi/6)} - - -The function to be expanded into a series may have variables other -than \index{series!multiple variables} the series variable. - -For example, we may expand $tan(x*y)$ as a Taylor series in $x$ - -\spadcommand{taylor(tan(x*y),x = 0)} -$$ -{y \ x}+ -{{{y \sp 3} \over 3} \ {x \sp 3}}+ -{{{2 \ {y \sp 5}} \over {15}} \ {x \sp 5}}+ -{{{{17} \ {y \sp 7}} \over {315}} \ {x \sp 7}}+ -{{{{62} \ {y \sp 9}} \over {2835}} \ {x \sp 9}}+ -{O \left({{x \sp {11}}} \right)} -$$ -\returnType{Type: UnivariateTaylorSeries(Expression Integer,x,0)} - -or as a Taylor series in $y$. - -\spadcommand{taylor(tan(x*y),y = 0)} -$$ -{x \ y}+ -{{{x \sp 3} \over 3} \ {y \sp 3}}+ -{{{2 \ {x \sp 5}} \over {15}} \ {y \sp 5}}+ -{{{{17} \ {x \sp 7}} \over {315}} \ {y \sp 7}}+ -{{{{62} \ {x \sp 9}} \over {2835}} \ {y \sp 9}}+ -{O \left({{y \sp {11}}} \right)} -$$ -\returnType{Type: UnivariateTaylorSeries(Expression Integer,y,0)} - -A more interesting function is -$${\displaystyle t e^{x t}} \over{\displaystyle e^t - 1}$$ -When we expand this function as a Taylor -series in $t$ the $n$-th order coefficient is the $n$-th Bernoulli -\index{Bernoulli!polynomial} polynomial \index{polynomial!Bernoulli} -divided by $n!$. - -\spadcommand{bern := taylor(t*exp(x*t)/(exp(t) - 1),t = 0) } -$$ -\begin{array}{@{}l} -1+ -{{{{2 \ x} -1} \over 2} \ t}+ -{{{{6 \ {x \sp 2}} -{6 \ x}+1} \over {12}} \ {t \sp 2}}+ -{{{{2 \ {x \sp 3}} -{3 \ {x \sp 2}}+x} \over {12}} \ {t \sp 3}}+ -\\ -\\ -\displaystyle -{{{{{30} \ {x \sp 4}} -{{60} \ {x \sp 3}}+{{30} \ {x \sp 2}} -1} \over -{720}} \ {t \sp 4}}+ -{{{{6 \ {x \sp 5}} -{{15} \ {x \sp 4}}+{{10} \ {x \sp -3}} -x} \over {720}} \ {t \sp 5}}+ -\\ -\\ -\displaystyle -{{{{{42} \ {x \sp 6}} -{{126} \ {x \sp 5}}+{{105} \ {x \sp 4}} -{{21} -\ {x \sp 2}}+1} \over {30240}} \ {t \sp 6}}+ -{{{{6 \ {x \sp 7}} -{{21} \ {x \sp 6}}+{{21} \ {x \sp 5}} -{7 \ {x -\sp 3}}+x} \over {30240}} \ {t \sp 7}}+ -\\ -\\ -\displaystyle -{{{{{30} \ {x \sp 8}} -{{120} \ {x \sp 7}}+{{140} \ {x \sp 6}} - -{{70} \ {x \sp 4}}+{{20} \ {x \sp 2}} -1} \over {1209600}} \ {t \sp 8}}+ -\\ -\\ -\displaystyle -{{{{{10} \ {x \sp 9}} -{{45} \ {x \sp 8}}+{{60} \ {x \sp 7}} - -{{42} \ {x \sp 5}}+{{20} \ {x \sp 3}} -{3 \ x}} -\over {3628800}} \ {t \sp 9}}+ -\\ -\\ -\displaystyle -{{{{{66} \ {x \sp {10}}} -{{330} \ {x \sp 9}}+{{495} \ {x \sp 8}} - -{{462} \ {x \sp 6}}+{{330} \ {x \sp 4}} -{{99} \ {x \sp 2}}+5} -\over {239500800}} \ {t \sp {10}}}+ -{O \left({{t \sp {11}}} \right)} -\end{array} -$$ -\returnType{Type: UnivariateTaylorSeries(Expression Integer,t,0)} - -Therefore, this and the next expression produce the same result. - -\spadcommand{factorial(6) * coefficient(bern,6) } -$$ -{{{42} \ {x \sp 6}} - -{{126} \ {x \sp 5}}+ -{{105} \ {x \sp 4}} - -{{21} \ {x \sp 2}}+1} -\over {42} -$$ -\returnType{Type: Expression Integer} - -\spadcommand{bernoulliB(6,x)} -$$ -{x \sp 6} - -{3 \ {x \sp 5}}+ -{{5 \over 2} \ {x \sp 4}} - -{{1 \over 2} \ {x \sp 2}}+ -{1 \over {42}} -$$ -\returnType{Type: Polynomial Fraction Integer} - -Technically, a series with terms of negative degree is not considered -to be a Taylor series, but, rather, a \index{series!Laurent} -{\it Laurent series}. \index{Laurent series} If you try to compute a -Taylor series expansion of $\frac{x}{\log x}$ at $x = 1$ via -$taylor(x/log(x),x = 1)$ you get an error message. The reason is that -the function has a {\it pole} at $x = 1$, meaning that its series -expansion about this point has terms of negative degree. A series -with finitely many terms of negative degree is called a Laurent -series. - -You get the desired series expansion by issuing this. - -\spadcommand{laurent(x/log(x),x = 1)} -$$ -\begin{array}{@{}l} -{{\left( x -1 \right)}\sp {\left( -1\right)}}+ -{3\over 2}+ -{{5 \over {12}} \ {\left( x -1 \right)}} --{{1 \over {24}} \ {{\left( x -1 \right)}\sp 2}}+ -{{{11} \over {720}} \ {{\left( x -1 \right)}\sp 3}} - -{{{11} \over {1440}} \ {{\left( x -1 \right)}\sp 4}}+ -\\ -\\ -\displaystyle -{{{271} \over {60480}} \ {{\left( x -1 \right)}\sp 5}} - -{{{13} \over {4480}} \ {{\left( x -1 \right)}\sp 6}}+ -{{{7297} \over {3628800}} \ {{\left( x -1 \right)}\sp 7}} - -{{{425} \over {290304}} \ {{\left( x -1 \right)}\sp 8}}+ -\\ -\\ -\displaystyle -{{{530113} \over {479001600}} \ {{\left( x -1 \right)}\sp 9}}+ -{O \left({{{\left( x -1 \right)}\sp {10}}} \right)} -\end{array} -$$ -\returnType{Type: UnivariateLaurentSeries(Expression Integer,x,1)} - -Similarly, a series with terms of fractional degree is neither a -Taylor series nor a Laurent series. Such a series is called a -\index{series!Puiseux} {\it Puiseux series}. \index{Puiseux series} -The expression $laurent(sqrt(sec(x)),x = 3 * \%pi/2)$ results in an -error message because the series expansion about this point has terms -of fractional degree. - -However, this command produces what you want. - -\spadcommand{puiseux(sqrt(sec(x)),x = 3 * \%pi/2)} -$$ -{{\left( x -{{3 \ \pi} \over 2} \right)}\sp {\left( -{1 \over 2} \right)}}+ -{{1\over {12}} \ {{\left( x -{{3 \ \pi} \over 2} \right)}\sp {3 \over 2}}}+ -{{1 \over {160}} \ {{\left( x -{{3 \ \pi} \over 2} \right)}\sp {7 \over 2}}}+ -{O \left({{{\left( x -{{3 \ \pi} \over 2} \right)}\sp 5}} \right)} -$$ -\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,(3*pi)/2)} - -Finally, consider the case of functions that do not have Puiseux -expansions about certain points. An example of this is $x^x$ about $x -= 0$. $puiseux(x**x,x=0)$ produces an error message because of the -type of singularity of the function at $x = 0$. - -The general function {\bf series} can be used in this case. -Notice that the series returned is not, strictly speaking, a power series -because of the $log(x)$ in the expansion. - -\spadcommand{series(x**x,x=0)} -$$ -\begin{array}{@{}l} -1+ -{{\log \left({x} \right)}\ x}+ -{{{{\log \left({x} \right)}\sp 2} \over 2} \ {x \sp 2}}+ -{{{{\log \left({x} \right)}\sp 3} \over 6} \ {x \sp 3}}+ -{{{{\log \left({x} \right)}\sp 4} \over {24}} \ {x \sp 4}}+ -{{{{\log \left({x} \right)}\sp 5} \over {120}} \ {x \sp 5}}+ -{{{{\log \left({x} \right)}\sp 6} \over {720}} \ {x \sp 6}}+ -\\ -\\ -\displaystyle -{{{{\log \left({x} \right)}\sp 7} \over {5040}} \ {x \sp 7}}+ -{{{{\log \left({x} \right)}\sp 8} \over {40320}} \ {x \sp 8}}+ -{{{{\log \left({x} \right)}\sp 9} \over {362880}} \ {x \sp 9}}+ -{{{{\log \left({x} \right)}\sp {10}} \over {3628800}} \ {x \sp {10}}}+ -{O \left({{x \sp {11}}} \right)} -\end{array} -$$ -\returnType{Type: GeneralUnivariatePowerSeries(Expression Integer,x,0)} - -\boxed{4.6in}{ -\vskip 0.1cm -The operation {\bf series} returns the most general type of -infinite series. -The user who is not interested in distinguishing -between various types of infinite series may wish to use this operation -exclusively.\\ -} - -\subsection{Power Series from Formulas} -\label{ugxProblemSeriesFormula} - -The {\tt GenerateUnivariatePowerSeries} package enables you to -\index{series!giving formula for coefficients} create power series -from explicit formulas for their $n$-th coefficients. In what -follows, we construct series expansions for certain transcendental -functions by giving formulas for their coefficients. You can also -compute such series expansions directly simply by specifying the -function and the point about which the series is to be expanded. -\index{GenerateUnivariatePowerSeries} See -\ref{ugxProblemSeriesConversions} on -page~\pageref{ugxProblemSeriesConversions} for more information. - -Consider the Taylor expansion of $e^x$ \index{series!Taylor} -about $x = 0$: - -$$ -\begin{array}{ccl} -e^x &=& \displaystyle 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \cdots \\ \\ - &=& \displaystyle\sum_{n=0}^\infty \frac{x^n}{n!} -\end{array} -$$ - -The $n$-th Taylor coefficient is $1/n!$. - -This is how you create this series in Axiom. - -\spadcommand{series(n +-> 1/factorial(n),x = 0)} -$$ -\begin{array}{@{}l} -1+x+ -{{1 \over 2} \ {x \sp 2}}+ -{{1 \over 6} \ {x \sp 3}}+ -{{1 \over {24}} \ {x \sp 4}}+ -{{1 \over {120}} \ {x \sp 5}}+ -{{1 \over {720}} \ {x \sp 6}}+ -{{1 \over {5040}} \ {x \sp 7}}+ -{{1 \over {40320}} \ {x \sp 8}}+ -\\ -\\ -\displaystyle -{{1 \over {362880}} \ {x \sp 9}}+ -{{1 \over {3628800}} \ {x \sp {10}}}+ -{O \left({{x \sp {11}}} \right)} -\end{array} -$$ -\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} - -The first argument specifies a formula for the $n$-th coefficient by -giving a function that maps $n$ to $1/n!$. The second argument -specifies that the series is to be expanded in powers of $(x - 0)$, -that is, in powers of $x$. Since we did not specify an initial -degree, the first term in the series was the term of degree 0 (the -constant term). Note that the formula was given as an anonymous -function. These are discussed in \ref{ugUserAnon} on -page~\pageref{ugUserAnon}. - -Consider the Taylor expansion of $log x$ about $x = 1$: - -$$ -\begin{array}{ccl} -\log(x) &=& \displaystyle (x - 1) - \frac{(x - 1)^2}{2} + \frac{(x - 1)^3}{3} - \cdots \\ \\ - &=& \displaystyle\sum_{n = 1}^\infty (-1)^{n-1} \frac{(x - 1)^n}{n} -\end{array}$$ - -If you were to evaluate the expression -$series(n +-> (-1)**(n-1) / n, x = 1)$ -you would get an error message because Axiom would try to -calculate a term of degree $0$ and therefore divide by $0.$ - -Instead, evaluate this. -The third argument, $1..$, indicates that only terms of degree -$n = 1, ...$ are to be computed. - -\spadcommand{series(n +-> (-1)**(n-1)/n,x = 1,1..)} -$$ -\begin{array}{@{}l} -{\left( x -1 \right)} --{{1 \over 2} \ {{\left( x -1 \right)}\sp 2}}+ -{{1 \over 3} \ {{\left( x -1 \right)}\sp 3}} - -{{1 \over 4} \ {{\left( x -1 \right)}\sp 4}}+ -{{1 \over 5} \ {{\left( x -1 \right)}\sp 5}} - -{{1 \over 6} \ {{\left( x -1 \right)}\sp 6}}+ -\\ -\\ -\displaystyle -{{1 \over 7} \ {{\left( x -1 \right)}\sp 7}} - -{{1 \over 8} \ {{\left( x -1 \right)}\sp 8}}+ -{{1 \over 9} \ {{\left( x -1 \right)}\sp 9}} - -{{1 \over {10}} \ {{\left( x -1 \right)}\sp {10}}}+ -{{1 \over {11}} \ {{\left( x -1 \right)}\sp {11}}}+ -\\ -\\ -\displaystyle -{O \left({{{\left( x -1 \right)}\sp {12}}} \right)} -\end{array} -$$ -\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,1)} - -Next consider the Taylor expansion of an odd function, say, $sin(x)$: - -$$\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots$$ - -Here every other coefficient is zero and we would like to give an -explicit formula only for the odd Taylor coefficients. - -This is one way to do it. The third argument, $1..$, specifies that -the first term to be computed is the term of degree 1. The fourth -argument, $2$, specifies that we increment by $2$ to find the degrees -of subsequent terms, that is, the next term is of degree $1 + 2$, the -next of degree $1 + 2 + 2$, etc. - -\spadcommand{series(n +-> (-1)**((n-1)/2)/factorial(n),x = 0,1..,2)} -$$ -x - -{{1 \over 6} \ {x \sp 3}}+ -{{1 \over {120}} \ {x \sp 5}} - -{{1 \over {5040}} \ {x \sp 7}}+ -{{1 \over {362880}} \ {x \sp 9}} - -{{1 \over {39916800}} \ {x \sp {11}}}+ -{O \left({{x \sp {12}}} \right)} -$$ -\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} - -The initial degree and the increment do not have to be integers. -For example, this expression produces a series expansion of -$\sin(x^{\frac{1}{3}})$. - -\spadcommand{series(n +-> (-1)**((3*n-1)/2)/factorial(3*n),x = 0,1/3..,2/3)} -$$ -{x \sp {1 \over 3}} - -{{1 \over 6} \ x}+ -{{1 \over {120}} \ {x \sp {5 \over 3}}} - -{{1 \over {5040}} \ {x \sp {7 \over 3}}}+ -{{1 \over {362880}} \ {x \sp 3}} - -{{1 \over {39916800}} \ {x \sp {{11} \over 3}}}+ -{O \left({{x \sp 4}} \right)} -$$ -\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} - -While the increment must be positive, the initial degree may be -negative. This yields the Laurent expansion of $csc(x)$ at $x = 0$. -(bernoulli(numer(n+1)) is necessary because bernoulli takes integer -arguments.) - -\spadcommand{cscx := series(n +-> (-1)**((n-1)/2) * 2 * (2**n-1) * bernoulli(numer(n+1)) / factorial(n+1), x=0, -1..,2) } -$$ -{x \sp {\left( -1 \right)}}+ -{{1\over 6} \ x}+ -{{7 \over {360}} \ {x \sp 3}}+ -{{{31} \over {15120}} \ {x \sp 5}}+ -{{{127} \over {604800}} \ {x \sp 7}}+ -{{{73} \over {3421440}} \ {x \sp 9}}+ -{O \left({{x \sp {10}}} \right)} -$$ -\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} - -Of course, the reciprocal of this power series is the Taylor expansion -of $sin(x)$. - -\spadcommand{1/cscx } -$$ -x - -{{1 \over 6} \ {x \sp 3}}+ -{{1 \over {120}} \ {x \sp 5}} - -{{1 \over {5040}} \ {x \sp 7}}+ -{{1 \over {362880}} \ {x \sp 9}} - -{{1 \over {39916800}} \ {x \sp {11}}}+ -{O \left({{x \sp {12}}} \right)} -$$ -\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} - -As a final example,here is the Taylor expansion of $asin(x)$ about $x = 0$. - -\spadcommand{asinx := series(n +-> binomial(n-1,(n-1)/2)/(n*2**(n-1)),x=0,1..,2) } -$$ -x+ -{{1 \over 6} \ {x \sp 3}}+ -{{3 \over {40}} \ {x \sp 5}}+ -{{5 \over {112}} \ {x \sp 7}}+ -{{{35} \over {1152}} \ {x \sp 9}}+ -{{{63} \over {2816}} \ {x \sp {11}}}+ -{O \left({{x \sp {12}}} \right)} -$$ -\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} - -When we compute the $sin$ of this series, we get $x$ -(in the sense that all higher terms computed so far are zero). - -\spadcommand{sin(asinx) } -$$ -x+{O \left({{x \sp {12}}} \right)} -$$ -\returnType{Type: UnivariatePuiseuxSeries(Expression Integer,x,0)} - -\boxed{4.6in}{ -\vskip 0.1cm -Axiom isn't sufficiently ``symbolic'' in the sense we might wish. It -is an open problem to decide that ``x'' is the only surviving -term. Two attacks on the problem might be: - -(1) Notice that all of the higher terms are identically zero but -Axiom can't decide that from the information it knows. Presumably -we could attack this problem by looking at the sin function as -a taylor series around x=0 and seeing the term cancellation occur. -This uses a term-difference mechanism. - -(2) Notice that there is no way to decide that the stream for asinx -is actually the definition of asin(x). But we could recognize that -the stream for asin(x) has a generator term and so will a taylor -series expansion of sin(x). From these two generators it may be -possible in certain cases to decide that the application of one -generator to the other will yield only ``x''. This trick involves -finding the correct inverse for the stream functions. If we can -find an inverse for the ``remaining tail'' of the stream we could -conclude cancellation and thus turn an infinite stream into a -finite object. - -In general this is the zero-equivalence problem and is undecidable.\\ -} - -As we discussed in \ref{ugxProblemSeriesConversions} on -page~\pageref{ugxProblemSeriesConversions}, you can also use -the operations {\bf taylor}, {\bf laurent} and {\bf puiseux} instead -of {\bf series} if you know ahead of time what kind of exponents a -series has. You can't go wrong using {\bf series}, though. - -\subsection{Substituting Numerical Values in Power Series} -\label{ugxProblemSeriesSubstitute} - -Use \spadfunFrom{eval}{UnivariatePowerSeriesCategory} -\index{approximation} to substitute a numerical value for a variable -in \index{series!numerical approximation} a power series. For -example, here's a way to obtain numerical approximations of $\%e$ from -the Taylor series expansion of $exp(x)$. - -First you create the desired Taylor expansion. - -\spadcommand{f := taylor(exp(x)) } -$$ -\begin{array}{@{}l} -1+x+ -{{1 \over 2} \ {x \sp 2}}+ -{{1 \over 6} \ {x \sp 3}}+ -{{1 \over {24}} \ {x \sp 4}}+ -{{1 \over {120}} \ {x \sp 5}}+ -{{1 \over {720}} \ {x \sp 6}}+ -{{1 \over {5040}} \ {x \sp 7}}+ -\\ -\\ -\displaystyle -{{1 \over {40320}} \ {x \sp 8}}+ -{{1 \over {362880}} \ {x \sp 9}}+ -{{1 \over {3628800}} \ {x \sp {10}}}+ -{O \left({{x \sp {11}}} \right)} -\end{array} -$$ -\returnType{Type: UnivariateTaylorSeries(Expression Integer,x,0)} - - -Then you evaluate the series at the value $1.0$. -The result is a sequence of the partial sums. - -\spadcommand{eval(f,1.0)} -$$ -\begin{array}{@{}l} -\left[ -{1.0}, -{2.0}, -{2.5}, -{2.6666666666\ 666666667}, -{2.7083333333\ 333333333}, -\right. -\\ -\\ -\displaystyle -{2.7166666666\ 666666667}, -{2.7180555555\ 555555556}, -{2.7182539682\ 53968254}, -\\ -\\ -\displaystyle -\left. -{2.7182787698\ 412698413}, -{2.7182815255\ 731922399}, -\ldots -\right] -\end{array} -$$ -\returnType{Type: Stream Expression Float} - -\subsection{Example: Bernoulli Polynomials and Sums of Powers} -\label{ugxProblemSeriesBernoulli} - -Axiom provides operations for computing definite and -\index{summation!definite} indefinite sums. -\index{summation!indefinite} - -You can compute the sum of the first ten fourth powers by evaluating -this. This creates a list whose entries are $m^4$ as $m$ ranges from -1 to 10, and then computes the sum of the entries of that list. - -\spadcommand{reduce(+,[m**4 for m in 1..10])} -$$ -25333 -$$ -\returnType{Type: PositiveInteger} - -You can also compute a formula for the sum of the first $k$ fourth -powers, where $k$ is an unspecified positive integer. - -\spadcommand{sum4 := sum(m**4, m = 1..k) } -$$ -{{6 \ {k \sp 5}}+{{15} \ {k \sp 4}}+{{10} \ {k \sp 3}} -k} \over {30} -$$ -\returnType{Type: Fraction Polynomial Integer} - -This formula is valid for any positive integer $k$. For instance, if -we replace $k$ by 10, \index{summation!definite} we obtain the number -we computed earlier. - -\spadcommand{eval(sum4, k = 10) } -$$ -25333 -$$ -\returnType{Type: Fraction Polynomial Integer} - -You can compute a formula for the sum of the first $k$ $n$-th powers -in a similar fashion. Just replace the $4$ in the definition of -{\bf sum4} by any expression not involving $k$. Axiom computes these -formulas using Bernoulli polynomials; \index{Bernoulli!polynomial} we -\index{polynomial!Bernoulli} use the rest of this section to describe -this method. - -First consider this function of $t$ and $x$. - -\spadcommand{f := t*exp(x*t) / (exp(t) - 1) } -$$ -{t \ {e \sp {\left( t \ x \right)}}}\over {{e \sp t} -1} -$$ -\returnType{Type: Expression Integer} - -Since the expressions involved get quite large, we tell -Axiom to show us only terms of degree up to $5.$ - -\spadcommand{)set streams calculate 5 } - -\index{set streams calculate} - -If we look at the Taylor expansion of $f(x, t)$ about $t = 0,$ -we see that the coefficients of the powers of $t$ are polynomials -in $x$. - -\spadcommand{ff := taylor(f,t = 0) } -$$ -\begin{array}{@{}l} -1+ -{{{{2 \ x} -1} \over 2} \ t}+ -{{{{6 \ {x \sp 2}} -{6 \ x}+1} \over {12}} \ {t \sp 2}}+ -{{{{2 \ {x \sp 3}} -{3 \ {x \sp 2}}+x} \over {12}} \ {t \sp 3}}+ -\\ -\\ -\displaystyle -{{{{{30} \ {x \sp 4}} -{{60} \ {x \sp 3}}+{{30} \ {x \sp 2}} -1} \over -{720}} \ {t \sp 4}}+ -{{{{6 \ {x \sp 5}} -{{15} \ {x \sp 4}}+{{10} \ {x \sp -3}} -x} \over {720}} \ {t \sp 5}}+ -{O \left({{t \sp 6}} \right)} -\end{array} -$$ - - Type: UnivariateTaylorSeries(Expression Integer,t,0) - - -In fact, the $n$-th coefficient in this series is essentially the -$n$-th Bernoulli polynomial: the $n$-th coefficient of the series is -${1 \over {n!}} B_n(x)$, where $B_n(x)$ is the $n$-th Bernoulli -polynomial. Thus, to obtain the $n$-th Bernoulli polynomial, we -multiply the $n$-th coefficient of the series $ff$ by $n!$. - -For example, the sixth Bernoulli polynomial is this. - -\spadcommand{factorial(6) * coefficient(ff,6) } -$$ -{{{42} \ {x \sp 6}} -{{126} \ {x \sp 5}}+{{105} \ {x \sp 4}} - -{{21} \ {x \sp 2}}+1} \over {42} -$$ -\returnType{Type: Expression Integer} - -We derive some properties of the function $f(x,t)$. -First we compute $f(x + 1,t) - f(x,t)$. - -\spadcommand{g := eval(f, x = x + 1) - f } -$$ -{{t \ {e \sp {\left( {t \ x}+t \right)}}} --{t \ {e \sp {\left( t \ x \right)}}}} -\over {{e \sp t} -1} -$$ -\returnType{Type: Expression Integer} - -If we normalize $g$, we see that it has a particularly simple form. - -\spadcommand{normalize(g) } -$$ -t \ {e \sp {\left( t \ x \right)}} -$$ -\returnType{Type: Expression Integer} - -From this it follows that the $n$-th coefficient in the Taylor -expansion of $g(x,t)$ at $t = 0$ is $${1\over{(n-1)!}}x^{n-1}$$. - -If you want to check this, evaluate the next expression. - -\spadcommand{taylor(g,t = 0) } -$$ -t+ -{x \ {t \sp 2}}+ -{{{x \sp 2} \over 2} \ {t \sp 3}}+ -{{{x \sp 3} \over 6} \ {t \sp 4}}+ -{{{x \sp 4} \over {24}} \ {t \sp 5}}+ -{O \left({{t \sp 6}} \right)} -$$ -\returnType{Type: UnivariateTaylorSeries(Expression Integer,t,0)} - -However, since -$$g(x,t) = f(x+1,t)-f(x,t)$$ -it follows that the $n$-th coefficient is -$${1 \over {n!}}(B_n(x+1)-B_n(x))$$ Equating -coefficients, we see that -$${1\over{(n-1)!}}x^{n-1} = {1\over{n!}}(B_n(x + 1) - B_n(x))$$ -and, therefore, -$$x^{n-1} = {1\over{n}}(B_n(x + 1) - B_n(x))$$ - -Let's apply this formula repeatedly, letting $x$ vary between two -integers $a$ and $b$, with $a < b$: - -$$ -\begin{array}{lcl} - a^{n-1} & = & {1 \over n} (B_n(a + 1) - B_n(a)) \\ - (a + 1)^{n-1} & = & {1 \over n} (B_n(a + 2) - B_n(a + 1)) \\ - (a + 2)^{n-1} & = & {1 \over n} (B_n(a + 3) - B_n(a + 2)) \\ - & \vdots & \\ - (b - 1)^{n-1} & = & {1 \over n} (B_n(b) - B_n(b - 1)) \\ - b^{n-1} & = & {1 \over n} (B_n(b + 1) - B_n(b)) -\end{array} -$$ - -When we add these equations we find that the sum of the left-hand -sides is -$$\sum_{m=a}^{b} m^{n-1},$$ -the sum of the -$$(n-1)^{\hbox{\small\rm st}}$$ -powers from $a$ to $b$. The sum of the right-hand sides is a -``telescoping series.'' After cancellation, the sum is simply -$${1\over{n}}(B_n(b + 1) - B_n(a))$$ - -Replacing $n$ by $n + 1$, we have shown that -$$ -\sum_{m = a}^{b} m^n = {1 \over {\displaystyle n + 1}} -(B_{n+1}(b + 1) - B_{n+1}(a)) -$$ - -Let's use this to obtain the formula for the sum of fourth powers. - -First we obtain the Bernoulli polynomial $B_5$. - -\spadcommand{B5 := factorial(5) * coefficient(ff,5) } -$$ -{{6 \ {x \sp 5}} -{{15} \ {x \sp 4}}+{{10} \ {x \sp 3}} -x} \over 6 -$$ -\returnType{Type: Expression Integer} - -To find the sum of the first $k$ 4th powers, -we multiply $1/5$ by $B_5(k+1) - B_5(1)$. - -\spadcommand{1/5 * (eval(B5, x = k + 1) - eval(B5, x = 1)) } -$$ -{{6 \ {k \sp 5}}+{{15} \ {k \sp 4}}+{{10} \ {k \sp 3}} -k} \over {30} -$$ -\returnType{Type: Expression Integer} - -This is the same formula that we obtained via $sum(m**4, m = 1..k)$. - -\spadcommand{sum4 } -$$ -{{6 \ {k \sp 5}}+{{15} \ {k \sp 4}}+{{10} \ {k \sp 3}} -k} \over {30} -$$ -\returnType{Type: Fraction Polynomial Integer} - -At this point you may want to do the same computation, but with an -exponent other than $4.$ For example, you might try to find a formula -for the sum of the first $k$ 20th powers. - -\section{Solution of Differential Equations} -\label{ugProblemDEQ} - -In this section we discuss Axiom's facilities for -\index{equation!differential!solving} solving \index{differential -equation} differential equations in closed-form and in series. - -Axiom provides facilities for closed-form solution of -\index{equation!differential!solving in closed-form} single -differential equations of the following kinds: -\begin{itemize} -\item linear ordinary differential equations, and -\item non-linear first order ordinary differential equations -when integrating factors can be found just by integration. -\end{itemize} - -For a discussion of the solution of systems of linear and polynomial -equations, see \ref{ugProblemLinPolEqn} on page~\pageref{ugProblemLinPolEqn}. - -\subsection{Closed-Form Solutions of Linear Differential Equations} -\label{ugxProblemLDEQClosed} - -A {\it differential equation} is an equation involving an unknown -{\it function} and one or more of its derivatives. -\index{differential equation} The equation is called {\it ordinary} -if derivatives with respect to \index{equation!differential} only -one dependent variable appear in the equation (it is called -{\it partial} otherwise). The package {\tt ElementaryFunctionODESolver} -provides the top-level operation {\bf solve} for finding closed-form -solutions of ordinary differential equations. -\index{ElementaryFunctionODESolver} - -To solve a differential equation, you must first create an operator -for \index{operator} the unknown function. - -We let $y$ be the unknown function in terms of $x$. - -\spadcommand{y := operator 'y } -$$ -y -$$ -\returnType{Type: BasicOperator} - -You then type the equation using {\tt D} to create the -derivatives of the unknown function $y(x)$ where $x$ is any -symbol you choose (the so-called {\it dependent variable}). - -This is how you enter -the equation $y'' + y' + y = 0$. - -\spadcommand{deq := D(y x, x, 2) + D(y x, x) + y x = 0} -$$ -{{{y \sb {{\ }} \sp {,,}} \left({x} \right)}+ -{{y \sb {{\ }} \sp {,}} \left({x} \right)}+ -{y \left({x} \right)}}=0 -$$ -\returnType{Type: Equation Expression Integer} - -The simplest way to invoke the {\bf solve} command is with three -arguments. -\begin{itemize} -\item the differential equation, -\item the operator representing the unknown function, -\item the dependent variable. -\end{itemize} - -So, to solve the above equation, we enter this. - -\spadcommand{solve(deq, y, x) } -$$ -\left[ -{particular=0}, {basis={\left[ -{{\cos \left({{{x \ {\sqrt {3}}} \over 2}} \right)} -\ {e \sp {\left( -{x \over 2} \right)}}}, -{{e \sp {\left( -{x \over 2} \right)}} -\ {\sin \left({{{x \ {\sqrt {3}}} \over 2}} \right)}}\right]}} -\right] -$$ -\returnType{Type: Union(Record(particular: Expression Integer,basis: -List Expression Integer),...)} - -Since linear ordinary differential equations have infinitely many -solutions, {\bf solve} returns a {\it particular solution} $f_p$ and a -basis $f_1,\dots,f_n$ for the solutions of the corresponding -homogenuous equation. Any expression of the form -$$f_p + c_1 f_1 + \dots c_n f_n$$ -where the $c_i$ do not involve the dependent variable -is also a solution. This is similar to what you get when you solve -systems of linear algebraic equations. - -A way to select a unique solution is to specify {\it initial -conditions}: choose a value $a$ for the dependent variable and specify -the values of the unknown function and its derivatives at $a$. If the -number of initial conditions is equal to the order of the equation, -then the solution is unique (if it exists in closed form!) and {\bf -solve} tries to find it. To specify initial conditions to {\bf -solve}, use an {\tt Equation} of the form $x = a$ for the third -parameter instead of the dependent variable, and add a fourth -parameter consisting of the list of values $y(a), y'(a), ...$. - -To find the solution of $y'' + y = 0$ satisfying $y(0) = y'(0) = 1$, -do this. - -\spadcommand{deq := D(y x, x, 2) + y x } -$$ -{{y \sb {{\ }} \sp {,,}} \left({x} \right)}+{y\left({x} \right)} -$$ -\returnType{Type: Expression Integer} - -You can omit the $= 0$ when you enter the equation to be solved. - -\spadcommand{solve(deq, y, x = 0, [1, 1]) } -$$ -{\sin \left({x} \right)}+{\cos\left({x} \right)} -$$ -\returnType{Type: Union(Expression Integer,...)} - -Axiom is not limited to linear differential equations with constant -coefficients. It can also find solutions when the coefficients are -rational or algebraic functions of the dependent variable. -Furthermore, Axiom is not limited by the order of the equation. - -Axiom can solve the following third order equations with -polynomial coefficients. - -\spadcommand{deq := x**3 * D(y x, x, 3) + x**2 * D(y x, x, 2) - 2 * x * D(y x, x) + 2 * y x = 2 * x**4 } -$$ -{{{x \sp 3} \ {{y \sb {{\ }} \sp {,,,}} \left({x} \right)}}+ -{{x\sp 2} \ {{y \sb {{\ }} \sp {,,}} \left({x} \right)}}- -{2 \ x \ {{y \sb {{\ }} \sp {,}} \left({x} \right)}}+ -{2\ {y \left({x} \right)}}}= -{2\ {x \sp 4}} -$$ -\returnType{Type: Equation Expression Integer} - -\spadcommand{solve(deq, y, x) } -$$ -\begin{array}{@{}l} -\left[ -{particular={{{x \sp 5} -{{10} \ {x \sp 3}}+{{20} \ {x \sp 2}}+4} \over -{{15} \ x}}}, -\right. -\\ -\\ -\displaystyle -\left. -{basis={\left[ -{{{2 \ {x \sp 3}} -{3 \ {x \sp 2}}+1} \over x}, -{{{x \sp 3} -1} \over x}, -{{{x \sp 3} -{3 \ {x \sp 2}} -1} \over x} -\right]}} -\right] -\end{array} -$$ -\returnType{Type: Union(Record(particular: Expression Integer,basis: -List Expression Integer),...)} - -Here we are solving a homogeneous equation. - -\spadcommand{deq := (x**9+x**3) * D(y x, x, 3) + 18 * x**8 * D(y x, x, 2) - 90 * x * D(y x, x) - 30 * (11 * x**6 - 3) * y x } -$$ -{{\left( {x \sp 9}+{x \sp 3} \right)}\ {{y \sb {{\ }} \sp {,,,}} -\left({x} \right)}}+ -{{18}\ {x \sp 8} \ {{y \sb {{\ }} \sp {,,}} \left({x} \right)}}- -{{90} \ x \ {{y \sb {{\ }} \sp {,}} \left({x} \right)}}+ -{{\left(-{{330} \ {x \sp 6}}+{90} \right)}\ {y \left({x} \right)}} -$$ -\returnType{Type: Expression Integer} - -\spadcommand{solve(deq, y, x) } -$$ -\left[ -{particular=0}, -{basis={\left[ -{x \over {{x \sp 6}+1}}, -{{x \ {e \sp {\left( -{{\sqrt {{91}}} \ {\log \left({x} \right)}}\right)}}} -\over {{x \sp 6}+1}}, -{{x \ {e \sp {\left( {\sqrt {{91}}} \ {\log \left({x} \right)}\right)}}} -\over {{x \sp 6}+1}} -\right]}} -\right] -$$ -\returnType{Type: Union(Record(particular: Expression Integer,basis: -List Expression Integer),...)} - -On the other hand, and in contrast with the operation {\bf integrate}, -it can happen that Axiom finds no solution and that some closed-form -solution still exists. While it is mathematically complicated to -describe exactly when the solutions are guaranteed to be found, the -following statements are correct and form good guidelines for linear -ordinary differential equations: -\begin{itemize} -\item If the coefficients are constants, Axiom finds a complete basis -of solutions (i,e, all solutions). -\item If the coefficients are rational functions in the dependent variable, -Axiom at least finds all solutions that do not involve algebraic -functions. -\end{itemize} - -Note that this last statement does not mean that Axiom does not find -the solutions that are algebraic functions. It means that it is not -guaranteed that the algebraic function solutions will be found. - -This is an example where all the algebraic solutions are found. - -\spadcommand{deq := (x**2 + 1) * D(y x, x, 2) + 3 * x * D(y x, x) + y x = 0 } -$$ -{{{\left( {x \sp 2}+1 \right)}\ {{y \sb {{\ }} \sp {,,}} \left({x} \right)}}+ -{3\ x \ {{y \sb {{\ }} \sp {,}} \left({x} \right)}}+ -{y\left({x} \right)}}=0 -$$ -\returnType{Type: Equation Expression Integer} - -\spadcommand{solve(deq, y, x) } -$$ -\left[ -{particular=0}, -{basis={\left[ {1 \over {\sqrt {{{x \sp 2}+1}}}}, -{{\log \left({{{\sqrt {{{x \sp 2}+1}}} -x}} \right)} -\over {\sqrt {{{x \sp 2}+1}}}} \right]}} -\right] -$$ -\returnType{Type: Union(Record(particular: Expression Integer,basis: -List Expression Integer),...)} - -\subsection{Closed-Form Solutions of Non-Linear Differential Equations} -\label{ugxProblemNLDEQClosed} - -This is an example that shows how to solve a non-linear first order -ordinary differential equation manually when an integrating factor can -be found just by integration. At the end, we show you how to solve it -directly. - -Let's solve the differential equation $y' = y / (x + y log y)$. - -Using the notation $m(x, y) + n(x, y) y' = 0$, we have $m = -y$ and -$n = x + y log y$. - -\spadcommand{m := -y } -$$ --y -$$ -\returnType{Type: Polynomial Integer} - -\spadcommand{n := x + y * log y } -$$ -{y \ {\log \left({y} \right)}}+x -$$ -\returnType{Type: Expression Integer} - -We first check for exactness, that is, does $dm/dy = dn/dx$? - -\spadcommand{D(m, y) - D(n, x) } -$$ --2 -$$ -\returnType{Type: Expression Integer} - -This is not zero, so the equation is not exact. Therefore we must -look for an integrating factor: a function $mu(x,y)$ such that -$d(mu m)/dy = d(mu n)/dx$. Normally, we first search for $mu(x,y)$ -depending only on $x$ or only on $y$. - -Let's search for such a $mu(x)$ first. - -\spadcommand{mu := operator 'mu } -$$ -mu -$$ -\returnType{Type: BasicOperator} - -\spadcommand{a := D(mu(x) * m, y) - D(mu(x) * n, x) } -$$ -{{\left( -{y \ {\log \left({y} \right)}}-x \right)} -\ {{mu \sb {{\ }} \sp {,}} \left({x} \right)}}- -{2 \ {mu \left({x} \right)}} -$$ -\returnType{Type: Expression Integer} - -If the above is zero for a function $mu$ that does {\it not} depend on -$y$, then $mu(x)$ is an integrating factor. - -\spadcommand{solve(a = 0, mu, x) } -$$ -\left[ -{particular=0}, -{basis={\left[ -{1 \over -{{{y \sp 2} \ {{\log \left({y} \right)}\sp 2}}+ -{2 \ x \ y \ {\log \left({y} \right)}}+ -{x\sp 2}}} \right]}} -\right] -$$ -\returnType{Type: Union(Record(particular: Expression Integer,basis: -List Expression Integer),...)} - -The solution depends on $y$, so there is no integrating factor that -depends on $x$ only. - -Let's look for one that depends on $y$ only. - -\spadcommand{b := D(mu(y) * m, y) - D(mu(y) * n, x) } -$$ --{y \ {{mu \sb {{\ }} \sp {,}} \left({y} \right)}}- -{2 \ {mu \left({y} \right)}} -$$ -\returnType{Type: Expression Integer} - -\spadcommand{sb := solve(b = 0, mu, y) } -$$ -\left[ -{particular=0}, -{basis={\left[ {1 \over {y \sp 2}} \right]}} -\right] -$$ -\returnType{Type: Union(Record(particular: Expression Integer,basis: List Expression Integer),...)} - -\noindent -We've found one! - -The above $mu(y)$ is an integrating factor. We must multiply our -initial equation (that is, $m$ and $n$) by the integrating factor. - -\spadcommand{intFactor := sb.basis.1 } -$$ -1 \over {y \sp 2} -$$ -\returnType{Type: Expression Integer} - -\spadcommand{m := intFactor * m } -$$ --{1 \over y} -$$ -\returnType{Type: Expression Integer} - -\spadcommand{n := intFactor * n } -$$ -{{y \ {\log \left({y} \right)}}+x}\over {y \sp 2} -$$ -\returnType{Type: Expression Integer} - -Let's check for exactness. - -\spadcommand{D(m, y) - D(n, x) } -$$ -0 -$$ -\returnType{Type: Expression Integer} - -We must solve the exact equation, that is, find a function $s(x,y)$ -such that $ds/dx = m$ and $ds/dy = n$. - -We start by writing $s(x, y) = h(y) + integrate(m, x)$ where $h(y)$ is -an unknown function of $y$. This guarantees that $ds/dx = m$. - -\spadcommand{h := operator 'h } -$$ -h -$$ -\returnType{Type: BasicOperator} - -\spadcommand{sol := h y + integrate(m, x) } -$$ -{{y \ {h \left({y} \right)}}-x} \over y -$$ -\returnType{Type: Expression Integer} - -All we want is to find $h(y)$ such that $ds/dy = n$. - -\spadcommand{dsol := D(sol, y) } -$$ -{{{y \sp 2} \ {{h \sb {{\ }} \sp {,}} -\left({y} \right)}}+x}\over {y \sp 2} -$$ -\returnType{Type: Expression Integer} - -\spadcommand{nsol := solve(dsol = n, h, y) } -$$ -\left[ -{particular={{{\log \left({y} \right)}\sp 2} \over 2}}, -{basis={\left[ 1 \right]}} -\right] -$$ -\returnType{Type: Union(Record(particular: Expression Integer,basis: -List Expression Integer),...)} - -The above particular solution is the $h(y)$ we want, so we just replace -$h(y)$ by it in the implicit solution. - -\spadcommand{eval(sol, h y = nsol.particular) } -$$ -{{y \ {{\log \left({y} \right)}\sp 2}} -{2 \ x}} \over {2 \ y} -$$ -\returnType{Type: Expression Integer} - -A first integral of the initial equation is obtained by setting this -result equal to an arbitrary constant. - -Now that we've seen how to solve the equation ``by hand,'' we show you -how to do it with the {\bf solve} operation. - -First define $y$ to be an operator. - -\spadcommand{y := operator 'y } -$$ -y -$$ -\returnType{Type: BasicOperator} - -Next we create the differential equation. - -\spadcommand{deq := D(y x, x) = y(x) / (x + y(x) * log y x) } -$$ -{{y \sb {{\ }} \sp {,}} \left({x} \right)}= -{{y\left({x} \right)}\over -{{{y \left({x} \right)}\ {\log \left({{y \left({x} \right)}}\right)}}+x}} -$$ -\returnType{Type: Equation Expression Integer} - -Finally, we solve it. - -\spadcommand{solve(deq, y, x) } -$$ -{{{y \left({x} \right)}\ {{\log \left({{y \left({x} \right)}}\right)}\sp 2}}- -{2 \ x}} \over {2 \ {y \left({x} \right)}} -$$ -\returnType{Type: Union(Expression Integer,...)} - -\subsection{Power Series Solutions of Differential Equations} -\label{ugxProblemDEQSeries} - -The command to solve differential equations in power -\index{equation!differential!solving in power series} series -\index{power series} around \index{series!power} a particular initial -point with specific initial conditions is called {\bf seriesSolve}. -It can take a variety of parameters, so we illustrate its use with -some examples. - -Since the coefficients of some solutions are quite large, we reset the -default to compute only seven terms. - -\spadcommand{)set streams calculate 7 } - -You can solve a single nonlinear equation of any order. For example, -we solve -$$y''' = sin(y'') * exp(y) + cos(x)$$ subject to -$$y(0) = 1, y'(0) = 0, y''(0) = 0$$ - -We first tell Axiom that the symbol $'y$ denotes a new operator. - -\spadcommand{y := operator 'y } -$$ -y -$$ -\returnType{Type: BasicOperator} - -Enter the differential equation using $y$ like any system function. - -\spadcommand{eq := D(y(x), x, 3) - sin(D(y(x), x, 2))*exp(y(x)) = cos(x)} -$$ -{{{y \sb {{\ }} \sp {,,,}} \left({x} \right)}- -{{e \sp {y \left({x} \right)}}\ -{\sin \left({{{y \sb {{\ }} \sp {,,}} \left({x} \right)}}\right)}}}= -{\cos\left({x} \right)} -$$ -\returnType{Type: Equation Expression Integer} - -Solve it around $x = 0$ with the initial conditions -$y(0) = 1, y'(0) = y''(0) = 0$. - -\spadcommand{seriesSolve(eq, y, x = 0, [1, 0, 0])} -$$ -1+ -{{1 \over 6} \ {x \sp 3}}+ -{{e \over {24}} \ {x \sp 4}}+ -{{{{e \sp 2} -1} \over {120}} \ {x \sp 5}}+ -{{{{e \sp 3} -{2 \ e}} \over {720}} \ {x \sp 6}}+ -{{{{e \sp 4} -{8 \ {e \sp 2}}+{4 \ e}+1} \over {5040}} \ {x \sp 7}}+ -{O \left({{x \sp 8}} \right)} -$$ -\returnType{Type: UnivariateTaylorSeries(Expression Integer,x,0)} - -You can also solve a system of nonlinear first order equations. For -example, we solve a system that has $tan(t)$ and $sec(t)$ as -solutions. - -We tell Axiom that $x$ is also an operator. - -\spadcommand{x := operator 'x} -$$ -x -$$ -\returnType{Type: BasicOperator} - -Enter the two equations forming our system. - -\spadcommand{eq1 := D(x(t), t) = 1 + x(t)**2} -$$ -{{x \sb {{\ }} \sp {,}} \left({t} \right)}= -{{{x\left({t} \right)}\sp 2}+1} -$$ -\returnType{Type: Equation Expression Integer} - -\spadcommand{eq2 := D(y(t), t) = x(t) * y(t)} -$$ -{{y \sb {{\ }} \sp {,}} \left({t} \right)}= -{{x\left({t} \right)}\ {y \left({t} \right)}} -$$ -\returnType{Type: Equation Expression Integer} - -Solve the system around $t = 0$ with the initial conditions $x(0) = 0$ -and $y(0) = 1$. Notice that since we give the unknowns in the order -$[x, y]$, the answer is a list of two series in the order -$$[{\rm series\ for\ } x(t), {\rm \ series\ for\ }y(t)]$$ - -\spadcommand{seriesSolve([eq2, eq1], [x, y], t = 0, [y(0) = 1, x(0) = 0])} -\begin{verbatim} - Compiling function %BZ with type List UnivariateTaylorSeries( - Expression Integer,t,0) -> UnivariateTaylorSeries(Expression - Integer,t,0) - Compiling function %CA with type List UnivariateTaylorSeries( - Expression Integer,t,0) -> UnivariateTaylorSeries(Expression - Integer,t,0) -\end{verbatim} -$$ -\left[ -{t+ -{{1 \over 3} \ {t \sp 3}}+ -{{2 \over {15}} \ {t \sp 5}}+ -{{{17} \over {315}} \ {t \sp 7}}+ -{O \left({{t \sp 8}} \right)}}, -{1+{{1 \over 2} \ {t \sp 2}}+ -{{5 \over {24}} \ {t \sp 4}}+ -{{{61} \over {720}} \ {t \sp 6}}+ -{O \left({{t \sp 8}} \right)}} -\right] -$$ -\returnType{Type: List UnivariateTaylorSeries(Expression Integer,t,0)} - -\noindent -The order in which we give the equations and the initial conditions -has no effect on the order of the solution. - -\section{Finite Fields} -\label{ugProblemFinite} - -A {\it finite field} (also called a {\it Galois field}) is a finite -algebraic structure where one can add, multiply and divide under the -same laws (for example, commutativity, associativity or -distributivity) as apply to the rational, real or complex numbers. -Unlike those three fields, for any finite field there exists a -positive prime integer $p$, called the {\bf characteristic}, such that -$p x = 0$ for any element $x$ in the finite field. In fact, the -number of elements in a finite field is a power of the characteristic -and for each prime $p$ and positive integer $n$ there exists exactly -one finite field with $p^n$ elements, up to isomorphism.\footnote{For -more information about the algebraic structure and properties of -finite fields, see, for example, S. Lang, {\it Algebra}, Second -Edition, New York: Addison-Wesley Publishing Company, Inc., 1984, ISBN -0 201 05487 6; or R. Lidl, H. Niederreiter, {\it Finite Fields}, -Encyclopedia of Mathematics and Its Applications, Vol. 20, Cambridge: -Cambridge Univ. Press, 1983, ISBN 0 521 30240 4.} - -When $n = 1,$ the field has $p$ elements and is called a {\it prime -field}, discussed in the next section. There are several ways of -implementing extensions of finite fields, and Axiom provides quite a -bit of freedom to allow you to choose the one that is best for your -application. Moreover, we provide operations for converting among the -different representations of extensions and different extensions of a -single field. Finally, note that you usually need to package-call -operations from finite fields if the operations do not take as an -argument an object of the field. See -\ref{ugTypesPkgCall} on page~\pageref{ugTypesPkgCall} for more -information on package-calling. - -\subsection{Modular Arithmetic and Prime Fields} -\label{ugxProblemFinitePrime} -\index{finite field} -\index{Galois!field} -\index{field!finite!prime} -\index{field!prime} -\index{field!Galois} -\index{prime field} -\index{modular arithmetic} -\index{arithmetic!modular} - -Let $n$ be a positive integer. It is well known that you can get the -same result if you perform addition, subtraction or multiplication of -integers and then take the remainder on dividing by $n$ as if you had -first done such remaindering on the operands, performed the arithmetic -and then (if necessary) done remaindering again. This allows us to -speak of arithmetic {\it modulo} $n$ or, more simply {\it mod} $n$. - -In Axiom, you use {\tt IntegerMod} to do such arithmetic. - -\spadcommand{(a,b) : IntegerMod 12 } -\returnType{Type: Void} - -\spadcommand{(a, b) := (16, 7) } -$$ -7 -$$ -\returnType{Type: IntegerMod 12} - -\spadcommand{[a - b, a * b] } -$$ -\left[ -9, 4 -\right] -$$ -\returnType{Type: List IntegerMod 12} - -If $n$ is not prime, there is only a limited notion of reciprocals and -division. - -\spadcommand{a / b } -\begin{verbatim} - There are 12 exposed and 13 unexposed library operations named / - having 2 argument(s) but none was determined to be applicable. - Use HyperDoc Browse, or issue - )display op / - to learn more about the available operations. Perhaps - package-calling the operation or using coercions on the arguments - will allow you to apply the operation. - - Cannot find a definition or applicable library operation named / - with argument type(s) - IntegerMod 12 - IntegerMod 12 - - Perhaps you should use "@" to indicate the required return type, - or "$" to specify which version of the function you need. -\end{verbatim} - -\spadcommand{recip a } -$$ -\mbox{\tt "failed"} -$$ -\returnType{Type: Union("failed",...)} - -Here $7$ and $12$ are relatively prime, so $7$ has a multiplicative -inverse modulo $12$. - -\spadcommand{recip b } -$$ -7 -$$ -\returnType{Type: Union(IntegerMod 12,...)} - -If we take $n$ to be a prime number $p$, then taking inverses and, -therefore, division are generally defined. - -Use {\tt PrimeField} instead of {\tt IntegerMod} for $n$ prime. - -\spadcommand{c : PrimeField 11 := 8 } -$$ -8 -$$ -\returnType{Type: PrimeField 11} - -\spadcommand{inv c } -$$ -7 -$$ -\returnType{Type: PrimeField 11} - -You can also use $1/c$ and $c**(-1)$ for the inverse of $c$. - -\spadcommand{9/c } -$$ -8 -$$ -\returnType{Type: PrimeField 11} - -{\tt PrimeField} (abbreviation {\tt PF}) checks if its argument is -prime when you try to use an operation from it. If you know the -argument is prime (particularly if it is large), {\tt InnerPrimeField} -(abbreviation {\tt IPF}) assumes the argument has already been -verified to be prime. If you do use a number that is not prime, you -will eventually get an error message, most likely a division by zero -message. For computer science applications, the most important finite -fields are {\tt PrimeField 2} and its extensions. - -In the following examples, we work with the finite field with -$p = 101$ elements. - -\spadcommand{GF101 := PF 101 } -$$ -\mbox{\rm PrimeField 101} -$$ -\returnType{Type: Domain} - -Like many domains in Axiom, finite fields provide an operation for -returning a random element of the domain. - -\spadcommand{x := random()\$GF101 } -$$ -8 -$$ -\returnType{Type: PrimeField 101} - -\spadcommand{y : GF101 := 37 } -$$ -37 -$$ -\returnType{Type: PrimeField 101} - -\spadcommand{z := x/y } -$$ -63 -$$ -\returnType{Type: PrimeField 101} - -\spadcommand{z * y - x } -$$ -0 -$$ -\returnType{Type: PrimeField 101} - -The element $2$ is a {\it primitive element} of this field, -\index{primitive element} -\index{element!primitive} - -\spadcommand{pe := primitiveElement()\$GF101 } -$$ -2 -$$ -\returnType{Type: PrimeField 101} - -in the sense that its powers enumerate all nonzero elements. - -\spadcommand{[pe**i for i in 0..99] } -$$ -\begin{array}{@{}l} -\left[ -1, 2, 4, 8, {16}, {32}, {64}, {27}, {54}, 7, {14}, {28}, {56}, -{11}, {22}, {44}, {88}, {75}, {49}, {98}, -\right. -\\ -\displaystyle -{95}, {89}, {77}, {53}, 5, {10}, {20}, {40}, {80}, {59}, {17}, -{34}, {68}, {35}, {70}, {39}, {78}, {55}, 9, -\\ -\displaystyle -{18}, {36}, {72}, {43}, {86}, {71}, {41}, {82}, {63}, {25}, -{50}, {100}, {99}, {97}, {93}, {85}, {69}, {37}, -\\ -\displaystyle -{74}, {47}, {94}, {87}, {73}, {45}, {90}, {79}, {57}, {13}, -{26}, {52}, 3, 6, {12}, {24}, {48}, {96}, {91}, -\\ -\displaystyle -{81}, {61}, {21}, {42}, {84}, {67}, {33}, {66}, {31}, {62}, -{23}, {46}, {92}, {83}, {65}, {29}, {58}, {15}, {30}, -\\ -\displaystyle -\left. -{60}, {19}, {38}, {76}, {51} -\right] -\end{array} -$$ -\returnType{Type: List PrimeField 101} - -If every nonzero element is a power of a primitive element, how do you -determine what the exponent is? Use \index{discrete logarithm} -{\bf discreteLog}. \index{logarithm!discrete} - -\spadcommand{ex := discreteLog(y) } -$$ -56 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{pe ** ex } -$$ -37 -$$ -\returnType{Type: PrimeField 101} - -The {\bf order} of a nonzero element $x$ is the smallest positive -integer $t$ such $x^t = 1$. - -\spadcommand{order y } -$$ -25 -$$ -\returnType{Type: PositiveInteger} - -The order of a primitive element is the defining $p-1$. - -\spadcommand{order pe } -$$ -100 -$$ -\returnType{Type: PositiveInteger} - -\subsection{Extensions of Finite Fields} -\label{ugxProblemFiniteExtensionFinite} -\index{finite field} -\index{field!finite!extension of} - -When you want to work with an extension of a finite field in Axiom, -you have three choices to make: -\begin{enumerate} -\item Do you want to generate an extension of the prime field -(for example, {\tt PrimeField 2}) or an extension of a given field? -\item Do you want to use a representation that is particularly -efficient for multiplication, exponentiation and addition but uses a -lot of computer memory (a representation that models the cyclic group -structure of the multiplicative group of the field extension and uses -a Zech logarithm table), one that \index{Zech logarithm} uses a normal -basis for the vector space structure of the field extension, or one -that performs arithmetic modulo an irreducible polynomial? The cyclic -group representation is only usable up to ``medium'' (relative to your -machine's performance) sized fields. If the field is large and the -normal basis is relatively simple, the normal basis representation is -more efficient for exponentiation than the irreducible polynomial -representation. -\item Do you want to provide a polynomial explicitly, a root of which -``generates'' the extension in one of the three senses in (2), or do -you wish to have the polynomial generated for you? -\end{enumerate} - -This illustrates one of the most important features of Axiom: you can -choose exactly the right data-type and representation to suit your -application best. - -We first tell you what domain constructors to use for each case above, -and then give some examples. - -\hangafter=1\hangindent=2pc -Constructors that automatically generate extensions of the prime field: -\newline -{\tt FiniteField} \newline -{\tt FiniteFieldCyclicGroup} \newline -{\tt FiniteFieldNormalBasis} - -\hangafter=1\hangindent=2pc -Constructors that generate extensions of an arbitrary field: -\newline -{\tt FiniteFieldExtension} \newline -{\tt FiniteFieldExtensionByPolynomial} \newline -{\tt FiniteFieldCyclicGroupExtension} \newline -{\tt FiniteFieldCyclicGroupExtensionByPolynomial} \newline -{\tt FiniteFieldNormalBasisExtension} \newline -{\tt FiniteFieldNormalBasisExtensionByPolynomial} - -\hangafter=1\hangindent=2pc -Constructors that use a cyclic group representation: -\newline -{\tt FiniteFieldCyclicGroup} \newline -{\tt FiniteFieldCyclicGroupExtension} \newline -{\tt FiniteFieldCyclicGroupExtensionByPolynomial} - -\hangafter=1\hangindent=2pc -Constructors that use a normal basis representation: -\newline -{\tt FiniteFieldNormalBasis} \newline -{\tt FiniteFieldNormalBasisExtension} \newline -{\tt FiniteFieldNormalBasisExtensionByPolynomial} - -\hangafter=1\hangindent=2pc -Constructors that use an irreducible modulus polynomial representation: -\newline -{\tt FiniteField} \newline -{\tt FiniteFieldExtension} \newline -{\tt FiniteFieldExtensionByPolynomial} - -\hangafter=1\hangindent=2pc -Constructors that generate a polynomial for you: -\newline -{\tt FiniteField} \newline -{\tt FiniteFieldExtension} \newline -{\tt FiniteFieldCyclicGroup} \newline -{\tt FiniteFieldCyclicGroupExtension} \newline -{\tt FiniteFieldNormalBasis} \newline -{\tt FiniteFieldNormalBasisExtension} - -\hangafter=1\hangindent=2pc -Constructors for which you provide a polynomial: -\newline -{\tt FiniteFieldExtensionByPolynomial} \newline -{\tt FiniteFieldCyclicGroupExtensionByPolynomial} \newline -{\tt FiniteFieldNormalBasisExtensionByPolynomial} - -These constructors are discussed in the following sections where we -collect together descriptions of extension fields that have the same -underlying representation.\footnote{For more information on the -implementation aspects of finite fields, see J. Grabmeier, -A. Scheerhorn, {\it Finite Fields in AXIOM,} Technical Report, IBM -Heidelberg Scientific Center, 1992.} - -If you don't really care about all this detail, just use {\tt -FiniteField}. As your knowledge of your application and its Axiom -implementation grows, you can come back and choose an alternative -constructor that may improve the efficiency of your code. Note that -the exported operations are almost the same for all constructors of -finite field extensions and include the operations exported by {\tt -PrimeField}. - -\subsection{Irreducible Modulus Polynomial Representations} -\label{ugxProblemFiniteModulus} - -All finite field extension constructors discussed in this -\index{finite field} section \index{field!finite!extension of} use a -representation that performs arithmetic with univariate (one-variable) -polynomials modulo an irreducible polynomial. This polynomial may be -given explicitly by you or automatically generated. The ground field -may be the prime field or one you specify. See -\ref{ugxProblemFiniteExtensionFinite} on -page~\pageref{ugxProblemFiniteExtensionFinite} for general information about -finite field extensions. - -For {\tt FiniteField} (abbreviation {\tt FF}) you provide a prime -number $p$ and an extension degree $n$. This degree can be 1. - -Axiom uses the prime field {\tt PrimeField(p)}, here {\tt PrimeField 2}, -and it chooses an irreducible polynomial of degree $n$, here 12, -over the ground field. - -\spadcommand{GF4096 := FF(2,12); } -\returnType{Type: Domain} - -The objects in the generated field extension are polynomials of degree -at most $n-1$ with coefficients in the prime field. The polynomial -indeterminate is automatically chosen by Axiom and is typically -something like $\%A$ or $\%D$. These (strange) variables are -{\it only} for output display; there are several ways to construct -elements of this field. - -The operation {\bf index} enumerates the elements of the field -extension and accepts as argument the integers from 1 to $p ^ n$. - -The expression $index(p)$ always gives the indeterminate. - -\spadcommand{a := index(2)\$GF4096 } -$$ -\%A -$$ -\returnType{Type: FiniteField(2,12)} - -You can build polynomials in $a$ and calculate in $GF4096$. - -\spadcommand{b := a**12 - a**5 + a } -$$ -{ \%A \sp 5}+{ \%A \sp 3}+ \%A+1 -$$ -\returnType{Type: FiniteField(2,12)} - -\spadcommand{b ** 1000 } -$$ -{ \%A \sp {10}}+ -{ \%A \sp 9}+ -{ \%A \sp 7}+ -{ \%A \sp 5}+ -{ \%A \sp 4}+ -{ \%A \sp 3}+ -\%A -$$ -\returnType{Type: FiniteField(2,12)} - -\spadcommand{c := a/b } -$$ -{ \%A \sp {11}}+ -{ \%A \sp 8}+ -{ \%A \sp 7}+ -{ \%A \sp 5}+ -{ \%A \sp 4}+ -{ \%A \sp 3}+ -{ \%A \sp 2} -$$ -\returnType{Type: FiniteField(2,12)} - -Among the available operations are {\bf norm} and {\bf trace}. - -\spadcommand{norm c } -$$ -1 -$$ -\returnType{Type: PrimeField 2} - -\spadcommand{trace c } -$$ -0 -$$ -\returnType{Type: PrimeField 2} - -Since any nonzero element is a power of a primitive element, how do we -discover what the exponent is? - -The operation {\bf discreteLog} calculates \index{discrete logarithm} -the exponent and, \index{logarithm!discrete} if it is called with only -one argument, always refers to the primitive element returned by {\bf -primitiveElement}. - -\spadcommand{dL := discreteLog a } -$$ -1729 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{g ** dL } -$$ -g \sp {1729} -$$ -\returnType{Type: Polynomial Integer} - -{\tt FiniteFieldExtension} (abbreviation {\tt FFX}) is similar to {\tt -FiniteField} except that the ground-field for {\tt FiniteFieldExtension} -is arbitrary and chosen by you. - -In case you select the prime field as ground field, there is -essentially no difference between the constructed two finite field -extensions. - -\spadcommand{GF16 := FF(2,4); } -\returnType{Type: Domain} - -\spadcommand{GF4096 := FFX(GF16,3); } -\returnType{Type: Domain} - -\spadcommand{r := (random()\$GF4096) ** 20 } -$$ -{{\left( { \%B \sp 2}+1 \right)}\ { \%C \sp 2}}+ -{{\left( { \%B \sp 3}+{ \%B \sp 2}+1 \right)}\ \%C}+ -{ \%B \sp 3}+ -{ \%B \sp 2}+ -\%B+1 -$$ -\returnType{Type: FiniteFieldExtension(FiniteField(2,4),3)} - -\spadcommand{norm(r) } -$$ -{ \%B \sp 2}+ \%B -$$ -\returnType{Type: FiniteField(2,4)} - -{\tt FiniteFieldExtensionByPolynomial} (abbreviation {\tt FFP}) -is similar to {\tt FiniteField} and {\tt FiniteFieldExtension} -but is more general. - -\spadcommand{GF4 := FF(2,2); } -\returnType{Type: Domain} - -\spadcommand{f := nextIrreduciblePoly(random(6)\$FFPOLY(GF4))\$FFPOLY(GF4) } -$$ -{? \sp 6}+ -{{\left( \%D+1 \right)}\ {? \sp 5}}+ -{{\left( \%D+1 \right)}\ {? \sp 4}}+ -{{\left( \%D+1 \right)}\ ?}+1 -$$ -\returnType{Type: Union(SparseUnivariatePolynomial FiniteField(2,2),...)} - -For {\tt FFP} you choose both the ground field and the irreducible -polynomial used in the representation. The degree of the extension is -the degree of the polynomial. - -\spadcommand{GF4096 := FFP(GF4,f); } -\returnType{Type: Domain} - -\spadcommand{discreteLog random()\$GF4096 } -$$ -582 -$$ -\returnType{Type: PositiveInteger} - -\subsection{Cyclic Group Representations} -\label{ugxProblemFiniteCyclic} -\index{finite field} -\index{field!finite!extension of} - -In every finite field there exist elements whose powers are all the -nonzero elements of the field. Such an element is called a -{\it primitive element}. - -In {\tt FiniteFieldCyclicGroup} (abbreviation {\tt FFCG}) -\index{group!cyclic} the nonzero elements are represented by the -powers of a fixed primitive \index{element!primitive} element -\index{primitive element} of the field (that is, a generator of its -cyclic multiplicative group). Multiplication (and hence -exponentiation) using this representation is easy. To do addition, we -consider our primitive element as the root of a primitive polynomial -(an irreducible polynomial whose roots are all primitive). See -\ref{ugxProblemFiniteUtility} on page~\pageref{ugxProblemFiniteUtility} -for examples of how to compute such a polynomial. - -To use {\tt FiniteFieldCyclicGroup} you provide a prime number and an -extension degree. -\spadcommand{GF81 := FFCG(3,4); } -\returnType{Type: Domain} - -Axiom uses the prime field, here {\tt PrimeField 3}, as the ground -field and it chooses a primitive polynomial of degree $n$, here 4, -over the prime field. - -\spadcommand{a := primitiveElement()\$GF81 } -$$ - \%F \sp 1 -$$ -\returnType{Type: FiniteFieldCyclicGroup(3,4)} - -You can calculate in $GF81$. - -\spadcommand{b := a**12 - a**5 + a } -$$ - \%F \sp {72} -$$ -\returnType{Type: FiniteFieldCyclicGroup(3,4)} - -In this representation of finite fields the discrete logarithm of an -element can be seen directly in its output form. - -\spadcommand{b } -$$ - \%F \sp {72} -$$ -\returnType{Type: FiniteFieldCyclicGroup(3,4)} - -\spadcommand{discreteLog b } -$$ -72 -$$ -\returnType{Type: PositiveInteger} - -{\tt FiniteFieldCyclicGroupExtension} (abbreviation {\tt FFCGX}) is -similar to {\tt FiniteFieldCyclicGroup} except that the ground field -for {\tt FiniteFieldCyclicGroupExtension} is arbitrary and chosen by -you. In case you select the prime field as ground field, there is -essentially no difference between the constructed two finite field -extensions. - -\spadcommand{GF9 := FF(3,2); } -\returnType{Type: Domain} - -\spadcommand{GF729 := FFCGX(GF9,3); } -\returnType{Type: Domain} - -\spadcommand{r := (random()\$GF729) ** 20 } -$$ - \%H \sp {420} -$$ -\returnType{Type: FiniteFieldCyclicGroupExtension(FiniteField(3,2),3)} - -\spadcommand{trace(r) } -$$ -0 -$$ -\returnType{Type: FiniteField(3,2)} - -{\tt FiniteFieldCyclicGroupExtensionByPolynomial} (abbreviation -{\tt FFCGP}) is similar to {\tt FiniteFieldCyclicGroup} and -{\tt FiniteFieldCyclicGroupExtension} but is more general. For -{\tt FiniteFieldCyclicGroupExtensionByPolynomial} you choose both the -ground field and the irreducible polynomial used in the -representation. The degree of the extension is the degree of the -polynomial. - -\spadcommand{GF3 := PrimeField 3; } -\returnType{Type: Domain} - -We use a utility operation to generate an irreducible primitive -polynomial (see -\ref{ugxProblemFiniteUtility} on page~\pageref{ugxProblemFiniteUtility}). -The polynomial has one variable that is ``anonymous'': -it displays as a question mark. - -\spadcommand{f := createPrimitivePoly(4)\$FFPOLY(GF3) } -$$ -{? \sp 4}+?+2 -$$ -\returnType{Type: SparseUnivariatePolynomial PrimeField 3} - -\spadcommand{GF81 := FFCGP(GF3,f); } -\returnType{Type: Domain} - -Let's look at a random element from this field. - -\spadcommand{random()\$GF81 } -$$ - \%K \sp {13} -$$ -\returnType{Type: -FiniteFieldCyclicGroupExtensionByPolynomial(PrimeField 3,?**4+?+2)} - -\subsection{Normal Basis Representations} -\label{ugxProblemFiniteNormal} -\index{finite field} -\index{field!finite!extension of} -\index{basis!normal} -\index{normal basis} - -Let $K$ be a finite extension of degree $n$ of the finite field $F$ -and let $F$ have $q$ elements. An element $x$ of $K$ is said to be -{\it normal} over $F$ if the elements - -$$1, x^q, x^{q^2}, \ldots, x^{q^{n-1}}$$ - -form a basis of $K$ as a vector space over $F$. Such a basis is -called a {\it normal basis}.\footnote{This agrees with the general -definition of a normal basis because the $n$ distinct powers of the -automorphism $x \mapsto x^q$ constitute the Galois group of $K/F$.} - -If $x$ is normal over $F$, its minimal \index{polynomial!minimal} -polynomial is also said to be {\it normal} over $F$. -\index{minimal polynomial} -There exist normal bases for all finite extensions of arbitrary -finite fields. - -In {\tt FiniteFieldNormalBasis} (abbreviation {\tt FFNB}), the -elements of the finite field are represented by coordinate vectors -with respect to a normal basis. - -You provide a prime $p$ and an extension degree $n$. - -\spadcommand{K := FFNB(3,8) } -$$ -FiniteFieldNormalBasis(3,8) -$$ -\returnType{Type: Domain} - -Axiom uses the prime field {\tt PrimeField(p)}, here {\tt PrimeField -3}, and it chooses a normal polynomial of degree $n$, here 8, over the -ground field. The remainder class of the indeterminate is used as the -normal element. The polynomial indeterminate is automatically chosen -by Axiom and is typically something like $\%A$ or $\%D$. These -(strange) variables are only for output display; there are several -ways to construct elements of this field. The output of the basis -elements is something like $\%A^{q^i}.$ - -\spadcommand{a := normalElement()\$K } -$$ - \%I -$$ -\returnType{Type: FiniteFieldNormalBasis(3,8)} - -You can calculate in $K$ using $a$. - -\spadcommand{b := a**12 - a**5 + a } -$$ -{2 \ { \%I \sp {q \sp 7}}}+{ \%I \sp {q \sp 5}}+{ \%I \sp q} -$$ -\returnType{Type: FiniteFieldNormalBasis(3,8)} - -{\tt FiniteFieldNormalBasisExtension} (abbreviation {\tt FFNBX}) is -similar to {\tt FiniteFieldNormalBasis} except that the groundfield -for {\tt FiniteFieldNormalBasisExtension} is arbitrary and chosen by -you. In case you select the prime field as ground field, there is -essentially no difference between the constructed two finite field -extensions. - -\spadcommand{GF9 := FFNB(3,2); } -\returnType{Type: Domain} - -\spadcommand{GF729 := FFNBX(GF9,3); } -\returnType{Type: Domain} - -\spadcommand{r := random()\$GF729 } -$$ -2 \ \%K \ { \%L \sp q} -$$ -\returnType{Type: -FiniteFieldNormalBasisExtension(FiniteFieldNormalBasis(3,2),3)} - -\spadcommand{r + r**3 + r**9 + r**27 } -$$ -{2 \ \%K \ { \%L \sp {q \sp 2}}}+ -{{\left( {2 \ { \%K \sp q}}+{2 \ \%K} \right)}\ { \%L \sp q}}+ -{2 \ { \%K \sp q} \ \%L} -$$ -\returnType{Type: -FiniteFieldNormalBasisExtension(FiniteFieldNormalBasis(3,2),3)} - -{\tt FiniteFieldNormalBasisExtensionByPolynomial} (abbreviation -{\tt FFNBP}) is similar to {\tt FiniteFieldNormalBasis} and -{\tt FiniteFieldNormalBasisExtension} but is more general. For -{\tt FiniteFieldNormalBasisExtensionByPolynomial} you choose both the -ground field and the irreducible polynomial used in the representation. -The degree of the extension is the degree of the polynomial. - -\spadcommand{GF3 := PrimeField 3; } -\returnType{Type: Domain} - -We use a utility operation to generate an irreducible normal -polynomial (see -\ref{ugxProblemFiniteUtility} on page~\pageref{ugxProblemFiniteUtility}). -The polynomial has -one variable that is ``anonymous'': it displays as a question mark. - -\spadcommand{f := createNormalPoly(4)\$FFPOLY(GF3) } -$$ -{? \sp 4}+{2 \ {? \sp 3}}+2 -$$ -\returnType{Type: SparseUnivariatePolynomial PrimeField 3} - -\spadcommand{GF81 := FFNBP(GF3,f); } -\returnType{Type: Domain} - -Let's look at a random element from this field. - -\spadcommand{r := random()\$GF81 } -$$ -{ \%M \sp {q \sp 2}}+{2 \ { \%M \sp q}}+{2 \ \%M} -$$ -\returnType{Type: -FiniteFieldNormalBasisExtensionByPolynomial(PrimeField 3,?**4+2*?**3+2)} - -\spadcommand{r * r**3 * r**9 * r**27 } -$$ -{2 \ { \%M \sp {q \sp 3}}}+ -{2 \ { \%M \sp {q \sp 2}}}+ -{2 \ { \%M \sp q}}+ -{2 \ \%M} -$$ -\returnType{Type: -FiniteFieldNormalBasisExtensionByPolynomial(PrimeField 3,?**4+2*?**3+2)} - -\spadcommand{norm r } -$$ -2 -$$ -\returnType{Type: PrimeField 3} - -\subsection{Conversion Operations for Finite Fields} -\label{ugxProblemFiniteConversion} -\index{field!finite!conversions} - -Let $K$ be a finite field. - -\spadcommand{K := PrimeField 3 } -$$ -\mbox{\rm PrimeField 3} -$$ -\returnType{Type: Domain} - -An extension field $K_m$ of degree $m$ over $K$ is a subfield of an -extension field $K_n$ of degree $n$ over $K$ if and only if $m$ -divides $n$. - -\begin{center} -\begin{tabular}{ccc} -$K_n$ \\ -$|$ \\ -$K_m$ & $\Longleftrightarrow$ & $m | n$ \\ -$|$ \\ -K -\end{tabular} -\end{center} - -{\tt FiniteFieldHomomorphisms} provides conversion operations between -different extensions of one fixed finite ground field and between -different representations of these finite fields. - -Let's choose $m$ and $n$, - -\spadcommand{(m,n) := (4,8) } -$$ -8 -$$ -\returnType{Type: PositiveInteger} - -build the field extensions, - -\spadcommand{Km := FiniteFieldExtension(K,m) } -$$ -\mbox{\rm FiniteFieldExtension(PrimeField 3,4)} -$$ -\returnType{Type: Domain} - -and pick two random elements from the smaller field. - -\spadcommand{Kn := FiniteFieldExtension(K,n) } -$$ -\mbox{\rm FiniteFieldExtension(PrimeField 3,8)} -$$ -\returnType{Type: Domain} - -\spadcommand{a1 := random()\$Km } -$$ -{2 \ { \%A \sp 3}}+{ \%A \sp 2} -$$ -\returnType{Type: FiniteFieldExtension(PrimeField 3,4)} - -\spadcommand{b1 := random()\$Km } -$$ -{ \%A \sp 3}+{ \%A \sp 2}+{2 \ \%A}+1 -$$ -\returnType{Type: FiniteFieldExtension(PrimeField 3,4)} - -Since $m$ divides $n$, -$K_m$ is a subfield of $K_n$. - -\spadcommand{a2 := a1 :: Kn } -$$ - \%B \sp 4 -$$ -\returnType{Type: FiniteFieldExtension(PrimeField 3,8)} - -Therefore we can convert the elements of $K_m$ -into elements of $K_n$. - -\spadcommand{b2 := b1 :: Kn } -$$ -{2 \ { \%B \sp 6}}+{2 \ { \%B \sp 4}}+{ \%B \sp 2}+1 -$$ -\returnType{Type: FiniteFieldExtension(PrimeField 3,8)} - -To check this, let's do some arithmetic. - -\spadcommand{a1+b1 - ((a2+b2) :: Km) } -$$ -0 -$$ -\returnType{Type: FiniteFieldExtension(PrimeField 3,4)} - -\spadcommand{a1*b1 - ((a2*b2) :: Km) } -$$ -0 -$$ -\returnType{Type: FiniteFieldExtension(PrimeField 3,4)} - -There are also conversions available for the situation, when $K_m$ and -$K_n$ are represented in different ways (see -\ref{ugxProblemFiniteExtensionFinite} on -page~\pageref{ugxProblemFiniteExtensionFinite}). For example let's choose -$K_m$ where the representation is 0 plus the cyclic multiplicative -group and $K_n$ with a normal basis representation. - -\spadcommand{Km := FFCGX(K,m) } -$$ -\mbox{\rm FiniteFieldCyclicGroupExtension(PrimeField 3,4)} -$$ -\returnType{Type: Domain} - -\spadcommand{Kn := FFNBX(K,n) } -$$ -\mbox{\rm FiniteFieldNormalBasisExtension(PrimeField 3,8)} -$$ -\returnType{Type: Domain} - -\spadcommand{(a1,b1) := (random()\$Km,random()\$Km) } -$$ - \%C \sp {13} -$$ -\returnType{Type: FiniteFieldCyclicGroupExtension(PrimeField 3,4)} - -\spadcommand{a2 := a1 :: Kn } -$$ -{2 \ { \%D \sp {q \sp 6}}}+ -{2 \ { \%D \sp {q \sp 5}}}+ -{2 \ { \%D \sp {q \sp 4}}}+ -{2 \ { \%D \sp {q \sp 2}}}+ -{2 \ { \%D \sp q}}+ -{2 \ \%D} -$$ -\returnType{Type: FiniteFieldNormalBasisExtension(PrimeField 3,8)} - -\spadcommand{b2 := b1 :: Kn } -$$ -{2 \ { \%D \sp {q \sp 7}}}+ -{ \%D \sp {q \sp 6}}+ -{ \%D \sp {q \sp 5}}+ -{ \%D \sp {q \sp 4}}+ -{2 \ { \%D \sp {q \sp 3}}}+ -{ \%D \sp {q \sp 2}}+ -{ \%D \sp q}+ -\%D -$$ -\returnType{Type: FiniteFieldNormalBasisExtension(PrimeField 3,8)} - -Check the arithmetic again. - -\spadcommand{a1+b1 - ((a2+b2) :: Km) } -$$ -0 -$$ -\returnType{Type: FiniteFieldCyclicGroupExtension(PrimeField 3,4)} - -\spadcommand{a1*b1 - ((a2*b2) :: Km) } -$$ -0 -$$ -\returnType{Type: FiniteFieldCyclicGroupExtension(PrimeField 3,4)} - -\subsection{Utility Operations for Finite Fields} -\label{ugxProblemFiniteUtility} - -{\tt FiniteFieldPolynomialPackage} (abbreviation {\tt FFPOLY}) -provides operations for generating, counting and testing polynomials -over finite fields. Let's start with a couple of definitions: -\begin{itemize} -\item A polynomial is {\it primitive} if its roots are primitive -\index{polynomial!primitive} -elements in an extension of the coefficient field of degree equal -to the degree of the polynomial. -\item A polynomial is {\it normal} over its coefficient field -\index{polynomial!normal} -if its roots are linearly independent -elements in an extension of the coefficient field of degree equal -to the degree of the polynomial. -\end{itemize} - -In what follows, many of the generated polynomials have one -``anonymous'' variable. This indeterminate is displayed as a question -mark ({\tt ``?''}). - -To fix ideas, let's use the field with five elements for the first -few examples. - -\spadcommand{GF5 := PF 5; } -\returnType{Type: Domain} - -You can generate irreducible polynomials of any (positive) degree -\index{polynomial!irreducible} (within the storage capabilities of the -computer and your ability to wait) by using -\spadfunFrom{createIrreduciblePoly}{FiniteFieldPolynomialPackage}. - -\spadcommand{f := createIrreduciblePoly(8)\$FFPOLY(GF5) } -$$ -{? \sp 8}+{? \sp 4}+2 -$$ -\returnType{Type: SparseUnivariatePolynomial PrimeField 5} - -Does this polynomial have other important properties? Use -{\bf primitive?} to test whether it is a primitive polynomial. - -\spadcommand{primitive?(f)\$FFPOLY(GF5) } -$$ -{\tt false} -$$ -\returnType{Type: Boolean} - -Use {\bf normal?} to test whether it is a normal polynomial. - -\spadcommand{normal?(f)\$FFPOLY(GF5) } -$$ -{\tt false} -$$ -\returnType{Type: Boolean} - -\noindent -Note that this is actually a trivial case, because a normal polynomial -of degree $n$ must have a nonzero term of degree $n-1$. We will refer -back to this later. - -To get a primitive polynomial of degree 8 just issue this. - -\spadcommand{p := createPrimitivePoly(8)\$FFPOLY(GF5) } -$$ -{? \sp 8}+{? \sp 3}+{? \sp 2}+?+2 -$$ -\returnType{Type: SparseUnivariatePolynomial PrimeField 5} - -\spadcommand{primitive?(p)\$FFPOLY(GF5) } -$$ -{\tt true} -$$ -\returnType{Type: Boolean} - -This polynomial is not normal, - -\spadcommand{normal?(p)\$FFPOLY(GF5) } -$$ -{\tt false} -$$ -\returnType{Type: Boolean} - -but if you want a normal one simply write this. - -\spadcommand{n := createNormalPoly(8)\$FFPOLY(GF5) } -$$ -{? \sp 8}+{4 \ {? \sp 7}}+{? \sp 3}+1 -$$ -\returnType{Type: SparseUnivariatePolynomial PrimeField 5} - -This polynomial is not primitive! - -\spadcommand{primitive?(n)\$FFPOLY(GF5) } -$$ -{\tt false} -$$ -\returnType{Type: Boolean} - -This could have been seen directly, as the constant term is 1 here, -which is not a primitive element up to the factor ($-1$) raised to the -degree of the polynomial.\footnote{Cf. Lidl, R. \& Niederreiter, H., -{\it Finite Fields,} Encycl. of Math. 20, (Addison-Wesley, 1983), -p.90, Th. 3.18.} - -What about polynomials that are both primitive and normal? The -existence of such a polynomial is by no means obvious. -\footnote{The existence of such polynomials is proved in -Lenstra, H. W. \& Schoof, R. J., {\it Primitive -Normal Bases for Finite Fields,} Math. Comp. 48, 1987, pp. 217-231.} -% - -If you really need one use either -\spadfunFrom{createPrimitiveNormalPoly}{FiniteFieldPolynomialPackage} or -\spadfunFrom{createNormalPrimitivePoly}{FiniteFieldPolynomialPackage}. - -\spadcommand{createPrimitiveNormalPoly(8)\$FFPOLY(GF5) } -$$ -{? \sp 8}+{4 \ {? \sp 7}}+{2 \ {? \sp 5}}+2 -$$ -\returnType{Type: SparseUnivariatePolynomial PrimeField 5} - -If you want to obtain additional polynomials of the various types -above as given by the {\bf create...} operations above, you can use -the {\bf next...} operations. For instance, -\spadfunFrom{nextIrreduciblePoly}{FiniteFieldPolynomialPackage} yields -the next monic irreducible polynomial with the same degree as the -input polynomial. By ``next'' we mean ``next in a natural order using -the terms and coefficients.'' This will become more clear in the -following examples. - -This is the field with five elements. - -\spadcommand{GF5 := PF 5; } -\returnType{Type: Domain} - -Our first example irreducible polynomial, say of degree 3, must be -``greater'' than this. - -\spadcommand{h := monomial(1,8)\$SUP(GF5) } -$$ -? \sp 8 -$$ -\returnType{Type: SparseUnivariatePolynomial PrimeField 5} - -You can generate it by doing this. - -\spadcommand{nh := nextIrreduciblePoly(h)\$FFPOLY(GF5) } -$$ -{? \sp 8}+2 -$$ -\returnType{Type: Union(SparseUnivariatePolynomial PrimeField 5,...)} - -Notice that this polynomial is not the same as the one -\spadfunFrom{createIrreduciblePoly}{FiniteFieldPolynomialPackage}. - -\spadcommand{createIrreduciblePoly(3)\$FFPOLY(GF5) } -$$ -{? \sp 3}+?+1 -$$ -\returnType{Type: SparseUnivariatePolynomial PrimeField 5} - -You can step through all irreducible polynomials of degree 8 over -the field with 5 elements by repeatedly issuing this. - -\spadcommand{nh := nextIrreduciblePoly(nh)\$FFPOLY(GF5) } -$$ -{? \sp 8}+3 -$$ -\returnType{Type: Union(SparseUnivariatePolynomial PrimeField 5,...)} - -You could also ask for the total number of these. - -\spadcommand{numberOfIrreduciblePoly(5)\$FFPOLY(GF5) } -$$ -624 -$$ -\returnType{Type: PositiveInteger} - -We hope that ``natural order'' on polynomials is now clear: first we -compare the number of monomials of two polynomials (``more'' is -``greater''); then, if necessary, the degrees of these monomials -(lexicographically), and lastly their coefficients (also -lexicographically, and using the operation {\bf lookup} if our field -is not a prime field). Also note that we make both polynomials monic -before looking at the coefficients: multiplying either polynomial by a -nonzero constant produces the same result. - -The package {\tt FiniteFieldPolynomialPackage} also provides similar -operations for primitive and normal polynomials. With the exception of -the number of primitive normal polynomials; we're not aware of any -known formula for this. - -\spadcommand{numberOfPrimitivePoly(3)\$FFPOLY(GF5) } -$$ -20 -$$ -\returnType{Type: PositiveInteger} - -Take these, - -\spadcommand{m := monomial(1,1)\$SUP(GF5) } -$$ -? -$$ -\returnType{Type: SparseUnivariatePolynomial PrimeField 5} - -\spadcommand{f := m**3 + 4*m**2 + m + 2 } -$$ -{? \sp 3}+{4 \ {? \sp 2}}+?+2 -$$ -\returnType{Type: SparseUnivariatePolynomial PrimeField 5} - -and then we have: - -\spadcommand{f1 := nextPrimitivePoly(f)\$FFPOLY(GF5) } -$$ -{? \sp 3}+{4 \ {? \sp 2}}+{4 \ ?}+2 -$$ -\returnType{Type: Union(SparseUnivariatePolynomial PrimeField 5,...)} - -What happened? - -\spadcommand{nextPrimitivePoly(f1)\$FFPOLY(GF5) } -$$ -{? \sp 3}+{2 \ {? \sp 2}}+3 -$$ -\returnType{Type: Union(SparseUnivariatePolynomial PrimeField 5,...)} - -Well, for the ordering used in -\spadfunFrom{nextPrimitivePoly}{FiniteFieldPolynomialPackage} we use -as first criterion a comparison of the constant terms of the -polynomials. Analogously, in -\spadfunFrom{nextNormalPoly}{FiniteFieldPolynomialPackage} we first -compare the monomials of degree 1 less than the degree of the -polynomials (which is nonzero, by an earlier remark). - -\spadcommand{f := m**3 + m**2 + 4*m + 1 } -$$ -{? \sp 3}+{? \sp 2}+{4 \ ?}+1 -$$ -\returnType{Type: SparseUnivariatePolynomial PrimeField 5} - -\spadcommand{f1 := nextNormalPoly(f)\$FFPOLY(GF5) } -$$ -{? \sp 3}+{? \sp 2}+{4 \ ?}+3 -$$ -\returnType{Type: Union(SparseUnivariatePolynomial PrimeField 5,...)} - -\spadcommand{nextNormalPoly(f1)\$FFPOLY(GF5) } -$$ -{? \sp 3}+{2 \ {? \sp 2}}+1 -$$ -\returnType{Type: Union(SparseUnivariatePolynomial PrimeField 5,...)} - -\noindent -We don't have to restrict ourselves to prime fields. - -Let's consider, say, a field with 16 elements. - -\spadcommand{GF16 := FFX(FFX(PF 2,2),2); } -\returnType{Type: Domain} - -We can apply any of the operations described above. - -\spadcommand{createIrreduciblePoly(5)\$FFPOLY(GF16) } -$$ -{? \sp 5}+ \%G -$$ -\returnType{Type: SparseUnivariatePolynomial -FiniteFieldExtension(FiniteFieldExtension(PrimeField 2,2),2)} - -Axiom also provides operations for producing random polynomials of a -given degree - -\spadcommand{random(5)\$FFPOLY(GF16) } -$$ -\begin{array}{@{}l} -{? \sp 5}+ -{{\left( { \%F \ \%G}+1 \right)}\ {? \sp 4}}+ -{ \%F \ \%G \ {? \sp 3}}+ -{{\left( \%G+ \%F+1 \right)}\ {? \sp 2}}+ -\\ -\\ -\displaystyle -{{\left( {{\left( \%F+1 \right)}\ \%G}+ \%F \right)}\ ?}+1 -\end{array} -$$ -\returnType{Type: SparseUnivariatePolynomial -FiniteFieldExtension(FiniteFieldExtension(PrimeField 2,2),2)} - -or with degree between two given bounds. - -\spadcommand{random(3,9)\$FFPOLY(GF16) } -$$ -{? \sp 3}+ -{{\left( { \%F \ \%G}+1 \right)}\ {? \sp 2}}+ -{{\left( \%G+ \%F+1 \right)}\ ?}+1 -$$ -\returnType{Type: SparseUnivariatePolynomial -FiniteFieldExtension(FiniteFieldExtension(PrimeField 2,2),2)} - -{\tt FiniteFieldPolynomialPackage2} (abbreviation {\tt FFPOLY2}) -exports an operation {\bf rootOfIrreduciblePoly} for finding one root -of an irreducible polynomial $f$ \index{polynomial!root of} in an -extension field of the coefficient field. The degree of the extension -has to be a multiple of the degree of $f$. It is not checked whether -$f$ actually is irreducible. - -To illustrate this operation, we fix a ground field $GF$ - -\spadcommand{GF2 := PrimeField 2; } -\returnType{Type: Domain} - -and then an extension field. - -\spadcommand{F := FFX(GF2,12) } -$$ -\mbox{\rm FiniteFieldExtension(PrimeField 2,12)} -$$ -\returnType{Type: Domain} - -We construct an irreducible polynomial over $GF2$. - -\spadcommand{f := createIrreduciblePoly(6)\$FFPOLY(GF2) } -$$ -{? \sp 6}+?+1 -$$ -\returnType{Type: SparseUnivariatePolynomial PrimeField 2} - -We compute a root of $f$. - -\spadcommand{root := rootOfIrreduciblePoly(f)\$FFPOLY2(F,GF2) } -$$ -{ \%H \sp {11}}+{ \%H \sp 8}+{ \%H \sp 7}+{ \%H \sp 5}+ \%H+1 -$$ -\returnType{Type: FiniteFieldExtension(PrimeField 2,12)} - -and check the result -\spadcommand{eval(f, monomial(1,1)\$SUP(F) = root) } -$$ -0 -$$ -\returnType{Type: SparseUnivariatePolynomial -FiniteFieldExtension(PrimeField 2,12)} - -\section{Primary Decomposition of Ideals} -\label{ugProblemIdeal} - -Axiom provides a facility for the primary decomposition -\index{ideal!primary decomposition} of \index{primary decomposition of -ideal} polynomial ideals over fields of characteristic zero. The -algorithm -%is discussed in \cite{gtz:gbpdpi} and -works in essentially two steps: -\begin{enumerate} -\item the problem is solved for 0-dimensional ideals by ``generic'' -projection on the last coordinate -\item a ``reduction process'' uses localization and ideal quotients -to reduce the general case to the 0-dimensional one. -\end{enumerate} -The Axiom constructor {\tt PolynomialIdeals} represents ideals with -coefficients in any field and supports the basic ideal operations, -including intersection, sum and quotient. {\tt IdealDecompositionPackage} -contains the specific operations for the -primary decomposition and the computation of the radical of an ideal -with polynomial coefficients in a field of characteristic 0 with an -effective algorithm for factoring polynomials. - -The following examples illustrate the capabilities of this facility. - -First consider the ideal generated by -$x^2 + y^2 - 1$ -(which defines a circle in the $(x,y)$-plane) and the ideal -generated by $x^2 - y^2$ (corresponding to the -straight lines $x = y$ and $x = -y$. - -\spadcommand{(n,m) : List DMP([x,y],FRAC INT) } -\returnType{Type: Void} - -\spadcommand{m := [x**2+y**2-1] } -$$ -\left[ -{{x \sp 2}+{y \sp 2} -1} -\right] -$$ -\returnType{Type: List -DistributedMultivariatePolynomial([x,y],Fraction Integer)} - -\spadcommand{n := [x**2-y**2] } -$$ -\left[ -{{x \sp 2} -{y \sp 2}} -\right] -$$ -\returnType{Type: List -DistributedMultivariatePolynomial([x,y],Fraction Integer)} - -We find the equations defining the intersection of the two loci. -This correspond to the sum of the associated ideals. - -\spadcommand{id := ideal m + ideal n } -$$ -\left[ -{{x \sp 2} -{1 \over 2}}, {{y \sp 2} -{1 \over 2}} -\right] -$$ -\returnType{Type: PolynomialIdeals(Fraction Integer, -DirectProduct(2,NonNegativeInteger),OrderedVariableList [x,y], -DistributedMultivariatePolynomial([x,y],Fraction Integer))} - -We can check if the locus contains only a finite number of points, -that is, if the ideal is zero-dimensional. - -\spadcommand{zeroDim? id } -$$ -{\tt true} -$$ -\returnType{Type: Boolean} - -\spadcommand{zeroDim?(ideal m) } -$$ -{\tt false} -$$ -\returnType{Type: Boolean} - -\spadcommand{dimension ideal m } -$$ -1 -$$ -\returnType{Type: PositiveInteger} - -We can find polynomial relations among the generators ($f$ and $g$ are -the parametric equations of the knot). - -\spadcommand{(f,g):DMP([x,y],FRAC INT) } -\returnType{Type: Void} - -\spadcommand{f := x**2-1 } -$$ -{x \sp 2} -1 -$$ -\returnType{Type: DistributedMultivariatePolynomial([x,y],Fraction Integer)} - -\spadcommand{g := x*(x**2-1) } -$$ -{x \sp 3} -x -$$ -\returnType{Type: DistributedMultivariatePolynomial([x,y],Fraction Integer)} - -\spadcommand{relationsIdeal [f,g] } -$$ -{\left[ {-{ \%B \sp 2}+{ \%A \sp 3}+{ \%A \sp 2}} \right]} -\mid -{\left[ { \%A={{x \sp 2} -1}}, { \%B={{x \sp 3} -x}} \right]} -$$ -\returnType{Type: SuchThat(List Polynomial Fraction Integer, -List Equation Polynomial Fraction Integer)} - -We can compute the primary decomposition of an ideal. - -\spadcommand{l: List DMP([x,y,z],FRAC INT) } -\returnType{Type: Void} - -\spadcommand{l:=[x**2+2*y**2,x*z**2-y*z,z**2-4] } -$$ -\left[ -{{x \sp 2}+{2 \ {y \sp 2}}}, {{x \ {z \sp 2}} -{y \ z}}, {{z \sp 2} -4} -\right] -$$ -\returnType{Type: List -DistributedMultivariatePolynomial([x,y,z],Fraction Integer)} - -\spadcommand{ld:=primaryDecomp ideal l } -$$ -\left[ -{\left[ {x+{{1 \over 2} \ y}}, {y \sp 2}, {z+2} \right]}, -{\left[ {x -{{1 \over 2} \ y}}, {y \sp 2}, {z -2} \right]} -\right] -$$ -\returnType{Type: List PolynomialIdeals(Fraction Integer, -DirectProduct(3,NonNegativeInteger), -OrderedVariableList [x,y,z], -DistributedMultivariatePolynomial([x,y,z],Fraction Integer))} - -We can intersect back. - -\spadcommand{reduce(intersect,ld) } -$$ -\left[ -{x -{{1 \over 4} \ y \ z}}, {y \sp 2}, {{z \sp 2} -4} -\right] -$$ -\returnType{Type: PolynomialIdeals(Fraction Integer, -DirectProduct(3,NonNegativeInteger), -OrderedVariableList [x,y,z], -DistributedMultivariatePolynomial([x,y,z],Fraction Integer))} - -We can compute the radical of every primary component. - -\spadcommand{reduce(intersect,[radical ld.i for i in 1..2]) } -$$ -\left[ -x, y, {{z \sp 2} -4} -\right] -$$ -\returnType{Type: PolynomialIdeals(Fraction Integer, -DirectProduct(3,NonNegativeInteger), -OrderedVariableList [x,y,z], -DistributedMultivariatePolynomial([x,y,z],Fraction Integer))} - -Their intersection is equal to the radical of the ideal of $l$. - -\spadcommand{radical ideal l } -$$ -\left[x, y, {{z \sp 2} -4} \right] -$$ -\returnType{Type: PolynomialIdeals(Fraction Integer, -DirectProduct(3,NonNegativeInteger), -OrderedVariableList [x,y,z], -DistributedMultivariatePolynomial([x,y,z],Fraction Integer))} - -\section{Computation of Galois Groups} -\label{ugProblemGalois} - -As a sample use of Axiom's algebraic number facilities, -\index{group!Galois} -we compute -\index{Galois!group} -the Galois group of the polynomial -$p(x) = x^5 - 5 x + 12$. - -\spadcommand{p := x**5 - 5*x + 12 } -$$ -{x \sp 5} -{5 \ x}+{12} -$$ -\returnType{Type: Polynomial Integer} - -We would like to construct a polynomial $f(x)$ such that the splitting -\index{field!splitting} field \index{splitting field} of $p(x)$ is -generated by one root of $f(x)$. First we construct a polynomial -$r = r(x)$ such that one root of $r(x)$ generates the field generated by -two roots of the polynomial $p(x)$. (As it will turn out, the field -generated by two roots of $p(x)$ is, in fact, the splitting field of -$p(x)$.) - -From the proof of the primitive element theorem we know that if $a$ -and $b$ are algebraic numbers, then the field ${\bf Q}(a,b)$ is equal -to ${\bf Q}(a+kb)$ for an appropriately chosen integer $k$. In our -case, we construct the minimal polynomial of $a_i - a_j$, where $a_i$ -and $a_j$ are two roots of $p(x)$. We construct this polynomial using -{\bf resultant}. The main result we need is the following: If $f(x)$ -is a polynomial with roots $a_i \ldots a_m$ and $g(x)$ is a polynomial -with roots $b_i \ldots b_n$, then the polynomial $h(x) = -resultant(f(y), g(x-y), y)$ is a polynomial of degree $m*n$ with roots -$a_i + b_j, i = 1 \ldots m, j = 1 \ldots n$. - -For $f(x)$ we use the polynomial $p(x)$. For $g(x)$ we use the -polynomial $-p(-x)$. Thus, the polynomial we first construct is -$resultant(p(y), -p(y-x), y)$. - -\spadcommand{q := resultant(eval(p,x,y),-eval(p,x,y-x),y) } -$$ -\begin{array}{@{}l} -{x \sp {25}} - -{{50} \ {x \sp {21}}} - -{{2375} \ {x \sp {17}}}+ -{{90000} \ {x \sp {15}}} - -{{5000} \ {x \sp {13}}}+ -{{2700000} \ {x \sp {11}}}+ -{{250000} \ {x \sp 9}}+ -\\ -\\ -\displaystyle -{{18000000} \ {x \sp 7}}+ -{{64000000} \ {x \sp 5}} -\end{array} -$$ -\returnType{Type: Polynomial Integer} - -The roots of $q(x)$ are $a_i - a_j, i \leq 1, j \leq 5$. Of course, -there are five pairs $(i,j)$ with $i = j$, so $0$ is a 5-fold root of -$q(x)$. - -Let's get rid of this factor. - -\spadcommand{q1 := exquo(q, x**5) } -$$ -\begin{array}{@{}l} -{x \sp {20}} - -{{50} \ {x \sp {16}}} - -{{2375} \ {x \sp {12}}}+ -{{90000} \ {x \sp {10}}} - -{{5000} \ {x \sp 8}}+ -{{2700000} \ {x \sp 6}}+ -\\ -\\ -\displaystyle -{{250000} \ {x \sp 4}}+ -{{18000000} \ {x \sp 2}}+ -{64000000} -\end{array} -$$ -\returnType{Type: Union(Polynomial Integer,...)} - -Factor the polynomial $q1$. - -\spadcommand{factoredQ := factor q1 } -$$ -\begin{array}{@{}l} -{\left( -{x \sp {10}} - -{{10} \ {x \sp 8}} - -{{75} \ {x \sp 6}}+ -{{1500} \ {x \sp 4}} - -{{5500} \ {x \sp 2}}+ -{16000} -\right)} * -\\ -\\ -\displaystyle -{\left( -{x \sp {10}}+ -{{10} \ {x \sp 8}}+ -{{125} \ {x \sp 6}}+ -{{500} \ {x \sp 4}}+ -{{2500} \ {x \sp 2}}+ -{4000} -\right)} -\end{array} -$$ -\returnType{Type: Factored Polynomial Integer} - -We see that $q1$ has two irreducible factors, each of degree $10$. -(The fact that the polynomial $q1$ has two factors of degree $10$ is -enough to show that the Galois group of $p(x)$ is the dihedral group -of order $10$.\footnote{See McKay, Soicher, Computing Galois Groups -over the Rationals, Journal of Number Theory 20, 273-281 (1983). We -do not assume the results of this paper, however, and we continue with -the computation.} Note that the type of $factoredQ$ is {\tt FR POLY -INT}, that is, {\tt Factored Polynomial Integer}. \index{Factored} -This is a special data type for recording factorizations of -polynomials with integer coefficients. - -We can access the individual factors using the operation -\spadfunFrom{nthFactor}{Factored}. - -\spadcommand{r := nthFactor(factoredQ,1) } -$$ -{x \sp {10}} -{{10} \ {x \sp 8}} -{{75} \ {x \sp 6}}+{{1500} \ {x \sp 4}} --{{5500} \ {x \sp 2}}+{16000} -$$ -\returnType{Type: Polynomial Integer} - -Consider the polynomial $r = r(x)$. This is the minimal polynomial of -the difference of two roots of $p(x)$. Thus, the splitting field of -$p(x)$ contains a subfield of degree $10$. We show that this subfield -is, in fact, the splitting field of $p(x)$ by showing that $p(x)$ -factors completely over this field. - -First we create a symbolic root of the polynomial $r(x)$. (We -replaced $x$ by $b$ in the polynomial $r$ so that our symbolic root -would be printed as $b$.) - -\spadcommand{beta:AN := rootOf(eval(r,x,b)) } -$$ -b -$$ -\returnType{Type: AlgebraicNumber} - -We next tell Axiom to view $p(x)$ as a univariate polynomial in $x$ -with algebraic number coefficients. This is accomplished with this -type declaration. - -\spadcommand{p := p::UP(x,INT)::UP(x,AN) } -$$ -{x \sp 5} -{5 \ x}+{12} -$$ -\returnType{Type: UnivariatePolynomial(x,AlgebraicNumber)} - -Factor $p(x)$ over the field ${\bf Q}(\beta)$. -(This computation will take some time!) - -\spadcommand{algFactors := factor(p,[beta]) } -$$ -\begin{array}{@{}l} -{\left( -x+ -{\left( -\begin{array}{@{}l} --{{85} \ {b \sp 9}} - -{{116} \ {b \sp 8}}+ -{{780} \ {b \sp 7}}+ -{{2640} \ {b \sp 6}}+ -{{14895} \ {b \sp 5}} - -\\ -\\ -\displaystyle -{{8820} \ {b \sp 4}} - -{{127050} \ {b \sp 3}} - -{{327000} \ {b \sp 2}} - -{{405200} \ b}+ -{2062400} -\end{array} -\right) -\over {1339200}} -\right)} -\\ -\\ -\displaystyle -{\left( -x+ -{{-{{17} \ {b \sp 8}}+ -{{156} \ {b \sp 6}}+ -{{2979} \ {b \sp 4}} - -{{25410} \ {b \sp 2}} - -{14080}} \over {66960}} -\right)} -\\ -\\ -\displaystyle -\ {\left( -x+ -{{{{143} \ {b \sp 8}} - -{{2100} \ {b \sp 6}} - -{{10485} \ {b \sp 4}}+ -{{290550} \ {b \sp 2}} - -{{334800} \ b} - -{960800}} -\over {669600}} -\right)} -\\ -\\ -\displaystyle -\ {\left( -x+ -{{{{143} \ {b \sp 8}} - -{{2100} \ {b \sp 6}} - -{{10485} \ {b \sp 4}}+ -{{290550} \ {b \sp 2}}+ -{{334800} \ b} - -{960800}} -\over {669600}} -\right)} -\\ -\\ -\displaystyle -{\left( -x+ -{\left( -\begin{array}{@{}l} -{{85} \ {b \sp 9}} - -{{116} \ {b \sp 8}} - -{{780} \ {b \sp 7}}+ -{{2640} \ {b \sp 6}} - -{{14895} \ {b \sp 5}} - -\\ -\\ -\displaystyle -{{8820} \ {b \sp 4}}+ -{{127050} \ {b \sp 3}} - -{{327000} \ {b \sp 2}}+ -{{405200} \ b}+ -{2062400} -\end{array} -\right) -\over {1339200}} -\right)} -\end{array} -$$ -\returnType{Type: Factored UnivariatePolynomial(x,AlgebraicNumber)} - -When factoring over number fields, it is important to specify the -field over which the polynomial is to be factored, as polynomials have -different factorizations over different fields. When you use the -operation {\bf factor}, the field over which the polynomial is -factored is the field generated by -\begin{enumerate} -\item the algebraic numbers that appear -in the coefficients of the polynomial, and -\item the algebraic numbers that -appear in a list passed as an optional second argument of the operation. -\end{enumerate} -In our case, the coefficients of $p$ -are all rational integers and only $beta$ -appears in the list, so the field is simply -${\bf Q}(\beta)$. - -It was necessary to give the list $[beta]$ as a second argument of the -operation because otherwise the polynomial would have been factored -over the field generated by its coefficients, namely the rational -numbers. - -\spadcommand{factor(p) } -$$ -{x \sp 5} -{5 \ x}+{12} -$$ -\returnType{Type: Factored UnivariatePolynomial(x,AlgebraicNumber)} - -We have shown that the splitting field of $p(x)$ has degree $10$. -Since the symmetric group of degree 5 has only one transitive subgroup -of order $10$, we know that the Galois group of $p(x)$ must be this -group, the dihedral group \index{group!dihedral} of order $10$. -Rather than stop here, we explicitly compute the action of the Galois -group on the roots of $p(x)$. - -First we assign the roots of $p(x)$ as the values of five \index{root} -variables. - -We can obtain an individual root by negating the constant coefficient of -one of the factors of $p(x)$. - -\spadcommand{factor1 := nthFactor(algFactors,1) } -$$ -x+ -{ -\left( -\begin{array}{@{}l} --{{85} \ {b \sp 9}} - -{{116} \ {b \sp 8}}+ -{{780} \ {b \sp 7}}+ -{{2640} \ {b \sp 6}}+ -{{14895} \ {b \sp 5}} - -\\ -\\ -\displaystyle -{{8820} \ {b \sp 4}} - -{{127050} \ {b \sp 3}} - -{{327000} \ {b \sp 2}} - -{{405200} \ b}+ -{2062400} -\end{array} -\right) -\over {1339200}} -$$ -\returnType{Type: UnivariatePolynomial(x,AlgebraicNumber)} - -\spadcommand{root1 := -coefficient(factor1,0) } -$$ -\left( -\begin{array}{@{}l} -{{85} \ {b \sp 9}}+ -{{116} \ {b \sp 8}} - -{{780} \ {b \sp 7}} - -{{2640} \ {b \sp 6}} - -{{14895} \ {b \sp 5}}+ -\\ -\\ -\displaystyle -{{8820} \ {b \sp 4}}+ -{{127050} \ {b \sp 3}}+ -{{327000} \ {b \sp 2}}+ -{{405200} \ b} - -{2062400} -\end{array} -\right) -\over {1339200} -$$ -\returnType{Type: AlgebraicNumber} - -We can obtain a list of all the roots in this way. - -\spadcommand{roots := [-coefficient(nthFactor(algFactors,i),0) for i in 1..5] } -$$ -\begin{array}{@{}l} -\left[ -\left( -\begin{array}{@{}l} -{{85} \ {b \sp 9}}+ -{{116} \ {b \sp 8}} - -{{780} \ {b \sp 7}} - -{{2640} \ {b \sp 6}} - -{{14895} \ {b \sp 5}}+ -{{8820} \ {b \sp 4}}+ -\\ -\\ -\displaystyle -{{127050} \ {b \sp 3}}+ -{{327000} \ {b \sp 2}}+ -{{405200} \ b} - -{2062400} -\end{array} -\right) -\over {1339200}, -\right. -\\ -\\ -\displaystyle -{{{{17} \ {b \sp 8}} - -{{156} \ {b \sp 6}} - -{{2979} \ {b \sp 4}}+ -{{25410} \ {b \sp 2}}+ -{14080}} -\over {66960}}, -\\ -\\ -\displaystyle -{{-{{143} \ {b \sp 8}}+ -{{2100} \ {b \sp 6}}+ -{{10485} \ {b \sp 4}} - -{{290550} \ {b \sp 2}}+ -{{334800} \ b}+ -{960800}} -\over {669600}}, -\\ -\\ -\displaystyle -{{-{{143} \ {b \sp 8}}+ -{{2100} \ {b \sp 6}}+ -{{10485} \ {b \sp 4}} - -{{290550} \ {b \sp 2}} - -{{334800} \ b}+{960800}} -\over {669600}}, -\\ -\\ -\displaystyle -\left. -\left( -\begin{array}{@{}l} --{{85} \ {b \sp 9}}+ -{{116} \ {b \sp 8}}+ -{{780} \ {b \sp 7}} - -{{2640} \ {b \sp 6}}+ -{{14895} \ {b \sp 5}}+ -{{8820} \ {b \sp 4}} - -\\ -\\ -\displaystyle -{{127050} \ {b \sp 3}}+ -{{327000} \ {b \sp 2}}- -{{405200} \ b} - -{2062400} -\end{array} -\right) -\over {1339200} -\right] -\end{array} -$$ -\returnType{Type: List AlgebraicNumber} - -The expression -\begin{verbatim} -- coefficient(nthFactor(algFactors, i), 0)} -\end{verbatim} -is the $i $-th root of $p(x)$ and the elements of $roots$ are the -$i$-th roots of $p(x)$ as $i$ ranges from $1$ to $5$. - -Assign the roots as the values of the variables $a1,...,a5$. - -\spadcommand{(a1,a2,a3,a4,a5) := (roots.1,roots.2,roots.3,roots.4,roots.5) } -$$ -\left( -\begin{array}{@{}l} --{{85} \ {b \sp 9}}+ -{{116} \ {b \sp 8}}+ -{{780} \ {b \sp 7}} - -{{2640} \ {b \sp 6}}+ -{{14895} \ {b \sp 5}}+ -{{8820} \ {b \sp 4}} - -\\ -\\ -\displaystyle -{{127050} \ {b \sp 3}}+ -{{327000} \ {b \sp 2}}- -{{405200} \ b} - -{2062400} -\end{array} -\right) -\over {1339200} -$$ -\returnType{Type: AlgebraicNumber} - -Next we express the roots of $r(x)$ as polynomials in $beta$. We -could obtain these roots by calling the operation {\bf factor}: -$factor(r, [beta])$ factors $r(x)$ over ${\bf Q}(\beta)$. However, -this is a lengthy computation and we can obtain the roots of $r(x)$ as -differences of the roots $a1,...,a5$ of $p(x)$. Only ten of these -differences are roots of $r(x)$ and the other ten are roots of the -other irreducible factor of $q1$. We can determine if a given value -is a root of $r(x)$ by evaluating $r(x)$ at that particular value. -(Of course, the order in which factors are returned by the operation -{\bf factor} is unimportant and may change with different -implementations of the operation. Therefore, we cannot predict in -advance which differences are roots of $r(x)$ and which are not.) - -Let's look at four examples (two are roots of $r(x)$ and -two are not). - -\spadcommand{eval(r,x,a1 - a2) } -$$ -0 -$$ -\returnType{Type: Polynomial AlgebraicNumber} - -\spadcommand{eval(r,x,a1 - a3) } -$$ -\left( -\begin{array}{@{}l} -{{47905} \ {b \sp 9}}+ -{{66920} \ {b \sp 8}} - -{{536100} \ {b \sp 7}} - -{{980400} \ {b \sp 6}} - -{{3345075} \ {b \sp 5}} - -{{5787000} \ {b \sp 4}}+ -\\ -\\ -\displaystyle -{{75572250} \ {b \sp 3}}+ -{{161688000} \ {b \sp 2}} - -{{184600000} \ b} - -{710912000} -\end{array} -\right) -\over {4464} -$$ -\returnType{Type: Polynomial AlgebraicNumber} - -\spadcommand{eval(r,x,a1 - a4) } -$$ -0 -$$ -\returnType{Type: Polynomial AlgebraicNumber} - -\spadcommand{eval(r,x,a1 - a5) } -$$ -{{{405} \ {b \sp 8}}+ -{{3450} \ {b \sp 6}} - -{{19875} \ {b \sp 4}} - -{{198000} \ {b \sp 2}} - -{588000}} -\over {31} -$$ -\returnType{Type: Polynomial AlgebraicNumber} - -Take one of the differences that was a root of $r(x)$ and assign it to -the variable $bb$. - -For example, if $eval(r,x,a1 - a4)$ returned $0$, you would enter this. - -\spadcommand{bb := a1 - a4 } -$$ -\left( -\begin{array}{@{}l} -{{85} \ {b \sp 9}}+ -{{402} \ {b \sp 8}} - -{{780} \ {b \sp 7}} - -{{6840} \ {b \sp 6}} - -{{14895} \ {b \sp 5}} - -{{12150} \ {b \sp 4}}+ -\\ -\\ -\displaystyle -{{127050} \ {b \sp 3}}+ -{{908100} \ {b \sp 2}}+ -{{1074800} \ b} - -{3984000} -\end{array} -\right) -\over {1339200} -$$ -\returnType{Type: AlgebraicNumber} - -Of course, if the difference is, in fact, equal to the root $beta$, -you should choose another root of $r(x)$. - -Automorphisms of the splitting field are given by mapping a generator -of the field, namely $beta$, to other roots of its minimal polynomial. -Let's see what happens when $beta$ is mapped to $bb$. - -We compute the images of the roots $a1,...,a5$ under this automorphism: - -\spadcommand{aa1 := subst(a1,beta = bb) } -$$ -{-{{143} \ {b \sp 8}}+ -{{2100} \ {b \sp 6}}+ -{{10485} \ {b \sp 4}}- -{{290550} \ {b \sp 2}}+ -{{334800} \ b}+ -{960800}} -\over {669600} -$$ -\returnType{Type: AlgebraicNumber} - -\spadcommand{aa2 := subst(a2,beta = bb) } -$$ -\left( -\begin{array}{@{}l} --{{85} \ {b \sp 9}}+ -{{116} \ {b \sp 8}}+ -{{780} \ {b \sp 7}} - -{{2640} \ {b \sp 6}}+ -{{14895} \ {b \sp 5}}+ -{{8820} \ {b \sp 4}} - -\\ -\\ -\displaystyle -{{127050} \ {b \sp 3}}+ -{{327000} \ {b \sp 2}} - -{{405200} \ b} - -{2062400} -\end{array} -\right) -\over {1339200} -$$ -\returnType{Type: AlgebraicNumber} - -\spadcommand{aa3 := subst(a3,beta = bb) } -$$ -\left( -\begin{array}{@{}l} -{{85} \ {b \sp 9}}+ -{{116} \ {b \sp 8}} - -{{780} \ {b \sp 7}} - -{{2640} \ {b \sp 6}} - -{{14895} \ {b \sp 5}}+ -{{8820} \ {b \sp 4}}+ -\\ -\\ -\displaystyle -{{127050} \ {b \sp 3}}+ -{{327000} \ {b \sp 2}}+ -{{405200} \ b} - -{2062400} -\end{array} -\right) -\over {1339200} -$$ -\returnType{Type: AlgebraicNumber} - -\spadcommand{aa4 := subst(a4,beta = bb) } -$$ -{-{{143} \ {b \sp 8}}+ -{{2100} \ {b \sp 6}}+ -{{10485} \ {b \sp 4}}- -{{290550} \ {b \sp 2}} - -{{334800} \ b}+ -{960800}} -\over {669600} -$$ -\returnType{Type: AlgebraicNumber} - -\spadcommand{aa5 := subst(a5,beta = bb) } -$$ -{{{17} \ {b \sp 8}} - -{{156} \ {b \sp 6}} - -{{2979} \ {b \sp 4}}+ -{{25410} \ {b \sp 2}}+ -{14080}} -\over {66960} -$$ -\returnType{Type: AlgebraicNumber} - -Of course, the values $aa1,...,aa5$ are simply a permutation of the values -$a1,...,a5$. - -Let's find the value of $aa1$ (execute as many of the following five commands -as necessary). - -\spadcommand{(aa1 = a1) :: Boolean } -$$ -{\tt false} -$$ -\returnType{Type: Boolean} - -\spadcommand{(aa1 = a2) :: Boolean } -$$ -{\tt false} -$$ -\returnType{Type: Boolean} - -\spadcommand{(aa1 = a3) :: Boolean } -$$ -{\tt true} -$$ -\returnType{Type: Boolean} - -\spadcommand{(aa1 = a4) :: Boolean } -$$ -{\tt false} -$$ -\returnType{Type: Boolean} - -\spadcommand{(aa1 = a5) :: Boolean } -$$ -{\tt false} -$$ -\returnType{Type: Boolean} - -Proceeding in this fashion, you can find the values of -$aa2,...aa5$. You have represented the automorphism $beta -> bb$ as a -permutation of the roots $a1,...,a5$. If you wish, you can repeat -this computation for all the roots of $r(x)$ and represent the Galois -group of $p(x)$ as a subgroup of the symmetric group on five letters. - -Here are two other problems that you may attack in a similar fashion: -\begin{enumerate} -\item Show that the Galois group of -$p(x) = x^4 + 2 x^3 - 2 x^2 - 3 x + 1$ -is the dihedral group of order eight. \index{group!dihedral} -(The splitting field of this polynomial is the Hilbert class field -\index{Hilbert class field} of \index{field!Hilbert class} the quadratic field -${\bf Q}(\sqrt{145})$.) -\item Show that the Galois group of -$p(x) = x^6 + 108$ -has order 6 and is isomorphic to $S_3,$ the symmetric group on three letters. -\index{group!symmetric} (The splitting field of this polynomial is the -splitting field of $x^3 - 2$.) -\end{enumerate} - -\section{Non-Associative Algebras and Modelling Genetic Laws} -\label{ugProblemGenetic} - -Many algebraic structures of mathematics and Axiom have a -multiplication operation {\tt *} that satisfies the associativity law -\index{associativity law} $a*(b*c) = (a*b)*c$ for all $a$, $b$ and -$c$. The octonions are a well known exception. There are many other -interesting non-associative structures, such as the class of -\index{Lie algebra} Lie algebras.\footnote{Two Axiom implementations -of Lie algebras are {\tt LieSquareMatrix} and {\tt FreeNilpotentLie}.} -Lie algebras can be used, for example, to analyse Lie symmetry -algebras of \index{symmetry} partial differential \index{differential -equation!partial} equations. \index{partial differential equation} In -this section we show a different application of non-associative -algebras, \index{non-associative algebra} the modelling of genetic -laws. \index{algebra!non-associative} - -The Axiom library contains several constructors for creating -non-assoc\-i\-a\-tive structures, ranging from the categories -{\tt Monad}, {\tt NonAssociativeRng}, and {\tt FramedNonAssociativeAlgebra}, -to the domains {\tt AlgebraGivenByStructuralConstants} and -{\tt GenericNonAssociativeAlgebra}. Furthermore, the package -{\tt AlgebraPackage} provides operations for analysing the structure of -such algebras.\footnote{% The interested reader can learn more about -these aspects of the Axiom library from the paper ``Computations in -Algebras of Finite Rank,'' by Johannes Grabmeier and Robert Wisbauer, -Technical Report, IBM Heidelberg Scientific Center, 1992.} - -Mendel's genetic laws are often written in a form like - -$$Aa \times Aa = {1\over 4}AA + {1\over 2}Aa + {1\over 4}aa$$ - -The implementation of general algebras in Axiom allows us to -\index{Mendel's genetic laws} use this as the definition for -multiplication in an algebra. \index{genetics} Hence, it is possible -to study questions of genetic inheritance using Axiom. To demonstrate -this more precisely, we discuss one example from a monograph of -A. W\"orz-Busekros, where you can also find a general setting of this -theory.\footnote{% W\"{o}rz-Busekros, A., {\it Algebras in Genetics}, -Springer Lectures Notes in Biomathematics 36, Berlin e.a. (1980). In -particular, see example 1.3.} - -We assume that there is an infinitely large random mating population. -Random mating of two gametes $a_i$ and $a_j$ gives zygotes -\index{zygote} $a_ia_j$, which produce new gametes. \index{gamete} In -classical Mendelian segregation we have $a_ia_j = {1 \over 2}a_i+{1 -\over 2}a_j$. In general, we have - -$$a_ia_j = \sum_{k=1}^n \gamma_{i,j}^k\ a_k.$$ - -%{$ai aj = gammaij1 a1 + gammaij2 a2 + ... + gammaijn an$} - -The segregation rates $\gamma_{i,j}$ are the structural constants of -an $n$-dimensional algebra. This is provided in Axiom by the -constructor {\tt AlgebraGivenByStructuralConstants} (abbreviation -{\tt ALGSC}). - -Consider two coupled autosomal loci with alleles $A$, $a$, $B$, and -$b$, building four different gametes $a_1 = AB, a_2 = Ab, a_3 = aB,$ -and $a_4 = ab$ {$a1 := AB, a2 := Ab, a3 := aB,$ and $a4 := ab$}. The -zygotes $a_ia_j$ produce gametes $a_i$ and $a_j$ with classical -Mendelian segregation. Zygote $a_1a_4$ undergoes transition to -$a_2a_3$ and vice versa with probability -$0 \le \theta \le {1\over2}$. - -Define a list $[(\gamma_{i,j}^k) 1 \le k \le 4]$ of four four-by-four -matrices giving the segregation rates. We use the value $1/10$ for -$\theta$. - -\spadcommand{segregationRates : List SquareMatrix(4,FRAC INT) := [matrix [ [1, 1/2, 1/2, 9/20], [1/2, 0, 1/20, 0], [1/2, 1/20, 0, 0], [9/20, 0, 0, 0] ], matrix [ [0, 1/2, 0, 1/20], [1/2, 1, 9/20, 1/2], [0, 9/20, 0, 0], [1/20, 1/2, 0, 0] ], matrix [ [0, 0, 1/2, 1/20], [0, 0, 9/20, 0], [1/2, 9/20, 1, 1/2], [1/20, 0, 1/2, 0] ], matrix [ [0, 0, 0, 9/20], [0, 0, 1/20, 1/2], [0, 1/20, 0, 1/2], [9/20, 1/2, 1/2, 1] ] ] } -$$ -\begin{array}{@{}l} -\left[ -{\left[ -\begin{array}{cccc} -1 & {1 \over 2} & {1 \over 2} & {9 \over {20}} \\ -{1 \over 2} & 0 & {1 \over {20}} & 0 \\ -{1 \over 2} & {1 \over {20}} & 0 & 0 \\ -{9 \over {20}} & 0 & 0 & 0 -\end{array} -\right]}, -{\left[ -\begin{array}{cccc} -0 & {1 \over 2} & 0 & {1 \over {20}} \\ -{1 \over 2} & 1 & {9 \over {20}} & {1 \over 2} \\ -0 & {9 \over {20}} & 0 & 0 \\ -{1 \over {20}} & {1 \over 2} & 0 & 0 -\end{array} -\right]}, -\right. -\\ -\\ -\displaystyle -\left. -{\left[ -\begin{array}{cccc} -0 & 0 & {1 \over 2} & {1 \over {20}} \\ -0 & 0 & {9 \over {20}} & 0 \\ -{1 \over 2} & {9 \over {20}} & 1 & {1 \over 2} \\ -{1 \over {20}} & 0 & {1 \over 2} & 0 -\end{array} -\right]}, -{\left[ -\begin{array}{cccc} -0 & 0 & 0 & {9 \over {20}} \\ -0 & 0 & {1 \over {20}} & {1 \over 2} \\ -0 & {1 \over {20}} & 0 & {1 \over 2} \\ -{9 \over {20}} & {1 \over 2} & {1 \over 2} & 1 -\end{array} -\right]} -\right] -\end{array} -$$ -\returnType{Type: List SquareMatrix(4,Fraction Integer)} - -Choose the appropriate symbols for the basis of gametes, - -\spadcommand{gametes := ['AB,'Ab,'aB,'ab] } -$$ -\left[ -AB, Ab, aB, ab -\right] -$$ -\returnType{Type: List OrderedVariableList [AB,Ab,aB,ab]} - -Define the algebra. - -\spadcommand{A := ALGSC(FRAC INT, 4, gametes, segregationRates)} -$$ -\begin{array}{@{}l} -{\rm AlgebraGivenByStructuralConstants(Fraction Integer, 4, } -\\ -\displaystyle -{\rm [AB,Ab,aB,ab], [MATRIX,MATRIX,MATRIX,MATRIX])} -\end{array} -$$ -\returnType{Type: Domain} - -What are the probabilities for zygote $a_1a_4$ to produce the -different gametes? - -\spadcommand{a := basis()\$A} -$$ -\left[ -AB, Ab, aB, ab -\right] -$$ -\returnType{Type: Vector -AlgebraGivenByStructuralConstants(Fraction Integer,4,[AB,Ab,aB,ab], -[MATRIX,MATRIX,MATRIX,MATRIX])} - -\spadcommand{a.1*a.4} -$$ -{{9 \over {20}} \ ab}+ -{{1 \over {20}} \ aB}+ -{{1 \over {20}} \ Ab}+ -{{9 \over {20}} \ AB} -$$ -\returnType{Type: -AlgebraGivenByStructuralConstants(Fraction Integer,4,[AB,Ab,aB,ab], -[MATRIX,MATRIX,MATRIX,MATRIX])} - -Elements in this algebra whose coefficients sum to one play a -distinguished role. They represent a population with the distribution -of gametes reflected by the coefficients with respect to the basis of -gametes. - -Random mating of different populations $x$ and $y$ is described by -their product $x*y$. - -This product is commutative only if the gametes are not sex-dependent, -as in our example. - -\spadcommand{commutative?()\$A } -$$ -{\tt true} -$$ -\returnType{Type: Boolean} - -In general, it is not associative. - -\spadcommand{associative?()\$A } -$$ -{\tt false} -$$ -\returnType{Type: Boolean} - -Random mating within a population $x$ is described by $x*x$. The next -generation is $(x*x)*(x*x)$. - -Use decimal numbers to compare the distributions more easily. - -\spadcommand{x : ALGSC(DECIMAL, 4, gametes, segregationRates) := convert [3/10, 1/5, 1/10, 2/5]} -$$ -{{0.4} \ ab}+{{0.1} \ aB}+{{0.2} \ Ab}+{{0.3} \ AB} -$$ -\returnType{Type: -AlgebraGivenByStructuralConstants(DecimalExpansion,4,[AB,Ab,aB,ab], -[MATRIX,MATRIX,MATRIX,MATRIX])} - -To compute directly the gametic distribution in the fifth generation, -we use {\bf plenaryPower}. - -\spadcommand{plenaryPower(x,5) } -$$ -{{0.{36561}} \ ab}+{{0.{13439}} \ aB}+{{0.{23439}} \ Ab}+{{0.{26561}} \ -AB} -$$ -\returnType{Type: -AlgebraGivenByStructuralConstants(DecimalExpansion,4,[AB,Ab,aB,ab], -[MATRIX,MATRIX,MATRIX,MATRIX])} - -We now ask two questions: Does this distribution converge to an -equilibrium state? What are the distributions that are stable? - -This is an invariant of the algebra and it is used to answer the first -question. The new indeterminates describe a symbolic distribution. - -\spadcommand{q := leftRankPolynomial()\$GCNAALG(FRAC INT, 4, gametes, segregationRates) :: UP(Y, POLY FRAC INT)} -$$ -\begin{array}{@{}l} -{Y \sp 3}+ -{{\left( --{{{29} \over {20}} \ \%x4} - -{{{29} \over {20}} \ \%x3} - -{{{29} \over {20}} \ \%x2} - -{{{29} \over {20}} \ \%x1} -\right)}\ {Y \sp 2}}+ -\\ -\\ -\displaystyle -{ -\left( -\begin{array}{@{}l} -\left( {{9 \over {20}} \ { \%x4 \sp 2}}+ -{{\left( -{{9 \over {10}} \ \%x3}+ -{{9 \over {10}} \ \%x2}+ -{{9 \over {10}} \ \%x1} -\right)}\ \%x4}+ -\right. -\\ -\\ -\displaystyle -{{9 \over {20}} \ { \%x3 \sp 2}}+ -{{\left( {{9 \over {10}} \ \%x2}+{{9 \over {10}} \ \%x1} \right)}\ \%x3}+ -{{9 \over {20}} \ { \%x2 \sp 2}}+ -\\ -\\ -\displaystyle -\left. -{{9 \over {10}} \ \%x1 \ \%x2}+ -{{9 \over {20}} \ { \%x1 \sp 2}} -\right) -\end{array} -\right) -\ Y} -\end{array} -$$ -\returnType{Type: UnivariatePolynomial(Y,Polynomial Fraction Integer)} - - -Because the coefficient ${9 \over 20}$ has absolute value less than 1, -all distributions do converge, by a theorem of this theory. - -\spadcommand{factor(q :: POLY FRAC INT) } -$$ -\begin{array}{@{}l} -{\left( Y - \%x4 - \%x3 - \%x2 - \%x1 \right)} * -\\ -\\ -\displaystyle -{\left( -Y - -{{9 \over {20}} \ \%x4} - -{{9 \over {20}} \ \%x3} - -{{9 \over {20}} \ \%x2} - -{{9 \over {20}} \ \%x1} -\right)} -\ Y -\end{array} -$$ -\returnType{Type: Factored Polynomial Fraction Integer} - -The second question is answered by searching for idempotents in the algebra. - -\spadcommand{cI := conditionsForIdempotents()\$GCNAALG(FRAC INT, 4, gametes, segregationRates) } -$$ -\begin{array}{@{}l} -\left[ -{{{9 \over {10}} \ \%x1 \ \%x4}+ -{{\left( {{1 \over {10}} \ \%x2}+ \%x1 \right)}\ \%x3}+ -{ \%x1 \ \%x2}+ -{ \%x1 \sp 2} - -\%x1}, -\right. -\\ -\\ -\displaystyle -{{{\left( \%x2+{{1 \over {10}} \ \%x1} \right)}\ \%x4}+ -{{9 \over {10}} \ \%x2 \ \%x3}+ -{ \%x2 \sp 2}+ -{{\left( \%x1 -1 \right)}\ \%x2}}, -\\ -\\ -\displaystyle -{{{\left( \%x3+{{1 \over {10}} \ \%x1} \right)}\ \%x4}+ -{ \%x3 \sp 2}+ -{{\left( {{9 \over {10}} \ \%x2}+ \%x1 -1 \right)}\ \%x3}}, -\\ -\\ -\displaystyle -\left. -{{ \%x4 \sp 2}+ -{{\left( \%x3+ \%x2+{{9 \over {10}} \ \%x1} -1 \right)}\ \%x4}+ -{{1 \over {10}} \ \%x2 \ \%x3}} -\right] -\end{array} -$$ -\returnType{Type: List Polynomial Fraction Integer} - -Solve these equations and look at the first solution. - -\spadcommand{gbs:= groebnerFactorize cI} -$$ -\begin{array}{@{}l} -\left[ -\begin{array}{@{}l} -\left[ { \%x4+ \%x3+ \%x2+ \%x1 -1}, -\right. -\\ -\displaystyle -\left. -\ \ {{{\left( \%x2+ \%x1 \right)}\ \%x3}+ -{ \%x1 \ \%x2}+{ \%x1 \sp 2} - \%x1} -\right], -\end{array} -\right. -\\ -\\ -\displaystyle -{\left[ 1 \right]}, -{\left[ { \%x4+ \%x3 -1}, \%x2, \%x1 \right]}, -\\ -\\ -\displaystyle -{\left[ { \%x4+ \%x2 -1}, \%x3, \%x1 \right]}, -{\left[ \%x4, \%x3, \%x2, \%x1 \right]}, -\\ -\\ -\displaystyle -\left. -{\left[ { \%x4 -1}, \%x3, \%x2, \%x1 \right]}, -{\left[ { \%x4 -{1 \over 2}}, { \%x3 -{1 \over 2}}, \%x2, \%x1 \right]} -\right] -\end{array} -$$ -\returnType{Type: List List Polynomial Fraction Integer} - -\spadcommand{gbs.1} -$$ -\begin{array}{@{}l} -\left[ -{ \%x4+ \%x3+ \%x2+ \%x1 -1}, -\right. -\\ -\displaystyle -\left. -{{{\left( \%x2+ \%x1 \right)}\ \%x3}+{ \%x1 \ \%x2}+{ \%x1 \sp 2} - \%x1} -\right] -\end{array} -$$ -\returnType{Type: List Polynomial Fraction Integer} - - -Further analysis using the package {\tt PolynomialIdeals} shows that -there is a two-dimensional variety of equilibrium states and all other -solutions are contained in it. - -Choose one equilibrium state by setting two indeterminates to concrete -values. - -\spadcommand{sol := solve concat(gbs.1,[\%x1-1/10,\%x2-1/10]) } -$$ -\left[ -{\left[ -{ \%x4={2 \over 5}}, -{ \%x3={2 \over 5}}, -{ \%x2={1 \over {10}}}, -{ \%x1={1 \over {10}}} -\right]} -\right] -$$ -\returnType{Type: List List Equation Fraction Polynomial Integer} - -\spadcommand{e : A := represents reverse (map(rhs, sol.1) :: List FRAC INT) } -$$ -{{2 \over 5} \ ab}+ -{{2 \over 5} \ aB}+ -{{1 \over {10}} \ Ab}+ -{{1 \over {10}} \ AB} -$$ -\returnType{Type: -AlgebraGivenByStructuralConstants(Fraction Integer,4,[AB,Ab,aB,ab], -[MATRIX,MATRIX,MATRIX,MATRIX])} - -Verify the result. - -\spadcommand{e*e-e } -$$ -0 -$$ -\returnType{Type: -AlgebraGivenByStructuralConstants(Fraction Integer,4,[AB,Ab,aB,ab], -[MATRIX,MATRIX,MATRIX,MATRIX])} - - -%\setcounter{chapter}{9} % Chapter 10 -\chapter{Some Examples of Domains and Packages} -In this chapter we show examples of many of the most commonly used -AXIOM domains and packages. The sections are organized by constructor -names. - -\section{AssociationList} -\label{AssociationListXmpPage} - -The {\tt AssociationList} constructor provides a general structure for -associative storage. This type provides association lists in which -data objects can be saved according to keys of any type. For a given -association list, specific types must be chosen for the keys and -entries. You can think of the representation of an association list -as a list of records with key and entry fields. - -Association lists are a form of table and so most of the operations -available for {\tt Table} are also available for {\tt AssociationList}. -They can also be viewed as lists and can be manipulated accordingly. - -This is a {\tt Record} type with age and gender fields. - -\spadcommand{Data := Record(monthsOld : Integer, gender : String)} -$$ -\mbox{\rm Record(monthsOld: Integer,gender: String)} -$$ -\returnType{Type: Domain} - -In this expression, {\tt al} is declared to be an association -list whose keys are strings and whose entries are the above records. - -\spadcommand{al : AssociationList(String,Data)} -\returnType{Type: Void} - -The \spadfunFrom{table}{AssociationList} operation is used to create -an empty association list. - -\spadcommand{al := table()} -$$ -table() -$$ -\returnType{Type: -AssociationList(String,Record(monthsOld: Integer,gender: String))} - -You can use assignment syntax to add things to the association list. - -\spadcommand{al."bob" := [407,"male"]\$Data} -$$ -\left[ -{monthsOld={407}}, {gender= \mbox{\tt "male"} } -\right] -$$ -\returnType{Type: Record(monthsOld: Integer,gender: String)} - -\spadcommand{al."judith" := [366,"female"]\$Data} -$$ -\left[ -{monthsOld={366}}, {gender= \mbox{\tt "female"} } -\right] -$$ -\returnType{Type: Record(monthsOld: Integer,gender: String)} - -\spadcommand{al."katie" := [24,"female"]\$Data} -$$ -\left[ -{monthsOld={24}}, {gender= \mbox{\tt "female"} } -\right] -$$ -\returnType{Type: Record(monthsOld: Integer,gender: String)} - -Perhaps we should have included a species field. - -\spadcommand{al."smokie" := [200,"female"]\$Data} -$$ -\left[ -{monthsOld={200}}, {gender= \mbox{\tt "female"} } -\right] -$$ -\returnType{Type: Record(monthsOld: Integer,gender: String)} - -Now look at what is in the association list. Note that the last-added -(key, entry) pair is at the beginning of the list. - -\spadcommand{al} -$$ -\begin{array}{@{}l} -table -\left( -{ \mbox{\tt "smokie"} = -{\left[ {monthsOld={200}}, {gender= \mbox{\tt "female"} } \right]}}, -\right. -\\ -\\ -\displaystyle -\ \ \ \ \ \ \ \ { \mbox{\tt "katie"} = -{\left[ {monthsOld={24}}, {gender= \mbox{\tt "female"} } \right]}}, -\\ -\\ -\displaystyle -\ \ \ \ \ \ \ \ { \mbox{\tt "judith"} = -{\left[ {monthsOld={366}}, {gender= \mbox{\tt "female"} } \right]}}, -\\ -\\ -\displaystyle -\left. -\ \ \ \ \ \ \ \ { \mbox{\tt "bob"} = -{\left[ {monthsOld={407}}, {gender= \mbox{\tt "male"} } \right]}} -\right) -\end{array} -$$ -\returnType{Type: -AssociationList(String,Record(monthsOld: Integer,gender: String))} - -You can reset the entry for an existing key. - -\spadcommand{al."katie" := [23,"female"]\$Data} -$$ -\left[ -{monthsOld={23}}, {gender= \mbox{\tt "female"} } -\right] -$$ -\returnType{Type: Record(monthsOld: Integer,gender: String)} - -Use \spadfunFrom{delete!}{AssociationList} to destructively remove an -element of the association list. Use -\spadfunFrom{delete}{AssociationList} to return a copy of the -association list with the element deleted. The second argument is the -index of the element to delete. - -\spadcommand{delete!(al,1)} -$$ -\begin{array}{@{}l} -table -\left( -{ \mbox{\tt "katie"} = -{\left[ {monthsOld={23}}, {gender= \mbox{\tt "female"} } \right]}}, -\right. -\\ -\\ -\displaystyle -\ \ \ \ \ \ \ \ { \mbox{\tt "judith"} = -{\left[ {monthsOld={366}}, {gender= \mbox{\tt "female"} } \right]}}, -\\ -\\ -\displaystyle -\left. -\ \ \ \ \ \ \ \ { \mbox{\tt "bob"} = -{\left[ {monthsOld={407}}, {gender= \mbox{\tt "male"} } \right]}} -\right) -\end{array} -$$ -\returnType{Type: -AssociationList(String,Record(monthsOld: Integer,gender: String))} - -For more information about tables, -see \ref{TableXmpPage} on page~\pageref{TableXmpPage}. -For more information about lists, -see \ref{ListXmpPage} on page~\pageref{ListXmpPage}. - -\section{BalancedBinaryTree} -\label{BalancedBinaryTreeXmpPage} - -{\tt BalancedBinaryTrees(S)} is the domain of balanced binary trees -with elements of type {\tt S} at the nodes. A binary tree is either -{\tt empty} or else consists of a {\tt node} having a {\tt value} and -two branches, each branch a binary tree. A balanced binary tree is -one that is balanced with respect its leaves. One with $2^k$ leaves -is perfectly ``balanced'': the tree has minimum depth, and the {\tt -left} and {\tt right} branch of every interior node is identical in -shape. - -Balanced binary trees are useful in algebraic computation for -so-called ``divide-and-conquer'' algorithms. Conceptually, the data -for a problem is initially placed at the root of the tree. The -original data is then split into two subproblems, one for each -subtree. And so on. Eventually, the problem is solved at the leaves -of the tree. A solution to the original problem is obtained by some -mechanism that can reassemble the pieces. In fact, an implementation -of the Chinese Remainder Algorithm using balanced binary trees was -first proposed by David Y. Y. Yun at the IBM T. J. Watson Research -Center in Yorktown Heights, New York, in 1978. It served as the -prototype for polymorphic algorithms in Axiom. - -In what follows, rather than perform a series of computations with a -single expression, the expression is reduced modulo a number of -integer primes, a computation is done with modular arithmetic for each -prime, and the Chinese Remainder Algorithm is used to obtain the -answer to the original problem. We illustrate this principle with the -computation of $12^2 = 144$. - -A list of moduli. - -\spadcommand{lm := [3,5,7,11]} -$$ -\left[ -3, 5, 7, {11} -\right] -$$ -\returnType{Type: List PositiveInteger} - -The expression {\tt modTree(n, lm)} creates a balanced binary tree -with leaf values {\tt n mod m} for each modulus {\tt m} in {\tt lm}. - -\spadcommand{modTree(12,lm)} -$$ -\left[ -0, 2, 5, 1 -\right] -$$ -\returnType{Type: List Integer} - -Operation {\tt modTree} does this using operations on balanced binary -trees. We trace its steps. Create a balanced binary tree {\tt t} of -zeros with four leaves. - -\spadcommand{t := balancedBinaryTree(\#lm, 0)} -$$ -\left[ -{\left[ 0, 0, 0 \right]}, 0, {\left[ 0, 0, 0\right]} -\right] -$$ -\returnType{Type: BalancedBinaryTree NonNegativeInteger} - -The leaves of the tree are set to the individual moduli. - -\spadcommand{setleaves!(t,lm)} -$$ -\left[ -{\left[ 3, 0, 5\right]}, 0, {\left[ 7, 0, {11} \right]} -\right] -$$ -\returnType{Type: BalancedBinaryTree NonNegativeInteger} - -Use {\tt mapUp!} to do a bottom-up traversal of {\tt t}, setting each -interior node to the product of the values at the nodes of its -children. - -\spadcommand{mapUp!(t,\_*)} -$$ -1155 -$$ -\returnType{Type: PositiveInteger} - -The value at the node of every subtree is the product of the moduli -of the leaves of the subtree. - -\spadcommand{t} -$$ -\left[ -{\left[ 3, {15}, 5\right]}, {1155}, {\left[ 7, {77}, {11} \right]} -\right] -$$ -\returnType{Type: BalancedBinaryTree NonNegativeInteger} - -Operation {\tt mapDown!}{\tt (t,a,fn)} replaces the value {\tt v} at -each node of {\tt t} by {\tt fn(a,v)}. - -\spadcommand{mapDown!(t,12,\_rem)} -$$ -\left[ -{\left[ 0, {12}, 2\right]}, {12}, {\left[ 5, {12}, 1 \right]} -\right] -$$ -\returnType{Type: BalancedBinaryTree NonNegativeInteger} - -The operation {\tt leaves} returns the leaves of the resulting tree. -In this case, it returns the list of {\tt 12 mod m} for each modulus -{\tt m}. - -\spadcommand{leaves \%} -$$ -\left[ -0, 2, 5, 1 -\right] -$$ -\returnType{Type: List NonNegativeInteger} - -Compute the square of the images of {\tt 12} modulo each {\tt m}. - -\spadcommand{squares := [x**2 rem m for x in \% for m in lm]} -$$ -\left[ -0, 4, 4, 1 -\right] -$$ -\returnType{Type: List NonNegativeInteger} - -Call the Chinese Remainder Algorithm to get the answer for $12^2$. - -\spadcommand{chineseRemainder(\%,lm)} -$$ -144 -$$ -\returnType{Type: PositiveInteger} - -\section{BasicOperator} -\label{BasicOperatorXmpPage} - -A basic operator is an object that can be symbolically applied to a -list of arguments from a set, the result being a kernel over that set -or an expression. In addition to this section, please see -\ref{ExpressionXmpPage} on page~\pageref{ExpressionXmpPage} and -\ref{KernelXmpPage} on page~\pageref{KernelXmpPage} for additional -information and examples. - -You create an object of type {\tt BasicOperator} by using the -\spadfunFrom{operator}{BasicOperator} operation. This first form of -this operation has one argument and it must be a symbol. The symbol -should be quoted in case the name has been used as an identifier to -which a value has been assigned. - -A frequent application of {\tt BasicOperator} is the creation of an -operator to represent the unknown function when solving a differential -equation. - -Let {\tt y} be the unknown function in terms of {\tt x}. - -\spadcommand{y := operator 'y} -$$ -y -$$ -\returnType{Type: BasicOperator} - -This is how you enter the equation {\tt y'' + y' + y = 0}. - -\spadcommand{deq := D(y x, x, 2) + D(y x, x) + y x = 0} -$$ -{{{y \sb {{\ }} \sp {,,}} \left({x} \right)}+ -{{y\sb {{\ }} \sp {,}} \left({x} \right)}+ -{y\left({x} \right)}}=0 -$$ -\returnType{Type: Equation Expression Integer} - -To solve the above equation, enter this. - -\spadcommand{solve(deq, y, x)} -$$ -\left[ -{particular=0}, -{basis={\left[ {{\cos \left({{{x \ {\sqrt {3}}} \over 2}} \right)} -\ {e \sp {\left( -{x \over 2} \right)}}}, -{{e \sp {\left( -{x \over 2} \right)}} -\ {\sin \left({{{x \ {\sqrt {3}}} \over 2}} \right)}} -\right]}} -\right] -$$ -\returnType{Type: Union(Record(particular: Expression Integer, -basis: List Expression Integer),...)} - -See \ref{ugProblemDEQPage} on page~\pageref{ugProblemDEQPage} -in Section \ref{ugProblemDEQNumber} on page~\pageref{ugProblemDEQNumber} -for this kind of use of {\tt BasicOperator}. - -Use the single argument form of \spadfunFrom{operator}{BasicOperator} -(as above) when you intend to use the operator to create functional -expressions with an arbitrary number of arguments - -{\it Nary} means an arbitrary number of arguments can be used -in the functional expressions. - -\spadcommand{nary? y} -$$ -{\tt true} -$$ -\returnType{Type: Boolean} - -\spadcommand{unary? y} -$$ -{\tt false} -$$ -\returnType{Type: Boolean} - -Use the two-argument form when you want to restrict the number of -arguments in the functional expressions created with the operator. - -This operator can only be used to create functional expressions -with one argument. - -\spadcommand{opOne := operator('opOne, 1)} -$$ -opOne -$$ -\returnType{Type: BasicOperator} - -\spadcommand{nary? opOne} -$$ -{\tt false} -$$ -\returnType{Type: Boolean} - -\spadcommand{unary? opOne} -$$ -{\tt true} -$$ -\returnType{Type: Boolean} - -Use \spadfunFrom{arity}{BasicOperator} to learn the number of arguments -that can be used. It returns {\tt "false"} if the operator is nary. - -\spadcommand{arity opOne} -$$ -1 -$$ -\returnType{Type: Union(NonNegativeInteger,...)} - -Use \spadfunFrom{name}{BasicOperator} to learn the name of an operator. - -\spadcommand{name opOne} -$$ -opOne -$$ -\returnType{Type: Symbol} - -Use \spadfunFrom{is?}{BasicOperator} to learn if an operator has a -particular name. - -\spadcommand{is?(opOne, 'z2)} -$$ -{\tt false} -$$ -\returnType{Type: Boolean} - -You can also use a string as the name to be tested against. - -\spadcommand{is?(opOne, "opOne")} -$$ -{\tt true} -$$ -\returnType{Type: Boolean} - -You can attached named properties to an operator. These are rarely -used at the top-level of the Axiom interactive environment but are -used with Axiom library source code. - -By default, an operator has no properties. - -\spadcommand{properties y} -$$ -table() -$$ -\returnType{Type: AssociationList(String,None)} - -The interface for setting and getting properties is somewhat awkward -because the property values are stored as values of type {\tt None}. - -Attach a property by using \spadfunFrom{setProperty}{BasicOperator}. - -\spadcommand{setProperty(y, "use", "unknown function" :: None )} -$$ -y -$$ -\returnType{Type: BasicOperator} - -\spadcommand{properties y} -$$ -table -\left( -{{ \mbox{\tt "use"} =NONE}} -\right) -$$ -\returnType{Type: AssociationList(String,None)} - -We {\it know} the property value has type {\tt String}. - -\spadcommand{property(y, "use") :: None pretend String} -$$ -\mbox{\tt "unknown function"} -$$ -\returnType{Type: String} - -Use \spadfunFrom{deleteProperty!}{BasicOperator} to destructively -remove a property. - -\spadcommand{deleteProperty!(y, "use")} -$$ -y -$$ -\returnType{Type: BasicOperator} - -\spadcommand{properties y} -$$ -table() -$$ -\returnType{Type: AssociationList(String,None)} - -\section{BinaryExpansion} -\label{BinaryExpansionXmpPage} - -All rational numbers have repeating binary expansions. Operations to -access the individual bits of a binary expansion can be obtained by -converting the value to {\tt RadixExpansion(2)}. More examples of -expansions are available in -\ref{DecimalExpansionXmpPage} on page~\pageref{DecimalExpansionXmpPage}, -\ref{HexadecimalExpansionXmpPage} on -page~\pageref{HexadecimalExpansionXmpPage}, and -\ref{RadixExpansionXmpPage} on page~\pageref{RadixExpansionXmpPage}. - -The expansion (of type {\tt BinaryExpansion}) of a rational number -is returned by the \spadfunFrom{binary}{BinaryExpansion} operation. - -\spadcommand{r := binary(22/7)} -$$ -{11}.{\overline {001}} -$$ -\returnType{Type: BinaryExpansion} - -Arithmetic is exact. - -\spadcommand{r + binary(6/7)} -$$ -100 -$$ -\returnType{Type: BinaryExpansion} - -The period of the expansion can be short or long \ldots - -\spadcommand{[binary(1/i) for i in 102..106] } -$$ -\begin{array}{@{}l} -\left[ -{0.0{\overline {00000101}}}, -{0.{\overline {000000100111110001000101100101111001110010010101001}}}, -\right. -\\ -\\ -\displaystyle -{0.{000}{\overline {000100111011}}}, -{0.{\overline {000000100111}}}, -\\ -\\ -\displaystyle -\left. -{0.0{\overline {0000010011010100100001110011111011001010110111100011}}} -\right] -\end{array} -$$ -\returnType{Type: List BinaryExpansion} - -or very long. - -\spadcommand{binary(1/1007) } -$$ -\begin{array}{@{}l} -0. -\overline -{000000000100000100010100100101111000001111110000101111110010110001111101} -\\ -\displaystyle -\ \ \overline -{000100111001001100110001100100101010111101101001100000000110000110011110} -\\ -\displaystyle -\ \ \overline -{111000110100010111101001000111101100001010111011100111010101110011001010} -\\ -\displaystyle -\ \ \overline -{010111000000011100011110010000001001001001101110010101001110100011011101} -\\ -\displaystyle -\ \ \overline -{101011100010010000011001011011000000101100101111100010100000101010101101} -\\ -\displaystyle -\ \ \overline -{011000001101101110100101011111110101110101001100100001010011011000100110} -\\ -\displaystyle -\ \ \overline -{001000100001000011000111010011110001} -\end{array} -$$ -\returnType{Type: BinaryExpansion} - -These numbers are bona fide algebraic objects. - -\spadcommand{p := binary(1/4)*x**2 + binary(2/3)*x + binary(4/9)} -$$ -{{0.{01}} \ {x \sp 2}}+{{0.{\overline {10}}} \ x}+{0.{\overline {011100}}} -$$ -\returnType{Type: Polynomial BinaryExpansion} - -\spadcommand{q := D(p, x)} -$$ -{{0.1} \ x}+{0.{\overline {10}}} -$$ -\returnType{Type: Polynomial BinaryExpansion} - -\spadcommand{g := gcd(p, q)} -$$ -x+{1.{\overline {01}}} -$$ -\returnType{Type: Polynomial BinaryExpansion} - -\section{BinarySearchTree} -\label{BinarySearchTreeXmpPage} - -{\tt BinarySearchTree(R)} is the domain of binary trees with elements -of type {\tt R}, ordered across the nodes of the tree. A non-empty -binary search tree has a value of type {\tt R}, and {\tt right} and -{\tt left} binary search subtrees. If a subtree is empty, it is -displayed as a period (``.''). - -Define a list of values to be placed across the tree. The resulting -tree has {\tt 8} at the root; all other elements are in the left -subtree. - -\spadcommand{lv := [8,3,5,4,6,2,1,5,7]} -$$ -\left[ -8, 3, 5, 4, 6, 2, 1, 5, 7 -\right] -$$ -\returnType{Type: List PositiveInteger} - -A convenient way to create a binary search tree is to apply the -operation {\tt binarySearchTree} to a list of elements. - -\spadcommand{t := binarySearchTree lv} -$$ -\left[ -{\left[ {\left[ 1, 2, . \right]}, 3, -{\left[ 4, 5, {\left[ 5, 6, 7 \right]}\right]} -\right]}, -8, . -\right] -$$ -\returnType{Type: BinarySearchTree PositiveInteger} - -Another approach is to first create an empty binary search tree of integers. - -\spadcommand{emptybst := empty()\$BSTREE(INT)} -$$ -[\ ] -$$ -\returnType{Type: BinarySearchTree Integer} - -Insert the value {\tt 8}. This establishes {\tt 8} as the root of the -binary search tree. Values inserted later that are less than {\tt 8} -get stored in the {\tt left} subtree, others in the {\tt right} subtree. - -\spadcommand{t1 := insert!(8,emptybst)} -$$ -8 -$$ -\returnType{Type: BinarySearchTree Integer} - -Insert the value {\tt 3}. This number becomes the root of the {\tt -left} subtree of {\tt t1}. For optimal retrieval, it is thus -important to insert the middle elements first. - -\spadcommand{insert!(3,t1)} -$$ -\left[3, 8, . \right] -$$ -\returnType{Type: BinarySearchTree Integer} - -We go back to the original tree {\tt t}. The leaves of the binary -search tree are those which have empty {\tt left} and {\tt right} subtrees. - -\spadcommand{leaves t} -$$ -\left[ -1, 4, 5, 7 -\right] -$$ -\returnType{Type: List PositiveInteger} - -The operation {\tt split}{\tt (k,t)} returns a \index{record} -containing the two subtrees: one with all elements ``less'' than -{\tt k}, another with elements ``greater'' than {\tt k}. - -\spadcommand{split(3,t)} -$$ -\left[ -{less={\left[ 1, 2, . \right]}}, -{greater={\left[ {\left[ ., 3, -{\left[ 4, 5, -{\left[ 5, 6, 7 \right]} -\right]} -\right]}, -8, . -\right]}} -\right] -$$ -\returnType{Type: -Record(less: BinarySearchTree PositiveInteger,greater: -BinarySearchTree PositiveInteger)} - -Define {\tt insertRoot} to insert new elements by creating a new node. - -\spadcommand{insertRoot: (INT,BSTREE INT) -> BSTREE INT} -\returnType{Type: Void} - -The new node puts the inserted value between its ``less'' tree and -``greater'' tree. - -\begin{verbatim} -insertRoot(x, t) == - a := split(x, t) - node(a.less, x, a.greater) -\end{verbatim} - -Function {\tt buildFromRoot} builds a binary search tree from a list -of elements {\tt ls} and the empty tree {\tt emptybst}. - -\spadcommand{buildFromRoot ls == reduce(insertRoot,ls,emptybst)} -\returnType{Type: Void} - -Apply this to the reverse of the list {\tt lv}. - -\spadcommand{rt := buildFromRoot reverse lv} -$$ -\left[ -{\left[ {\left[ 1, 2, . \right]}, 3, -{\left[ 4, 5, {\left[ 5, 6, 7\right]}\right]} -\right]}, -8, . -\right] -$$ -\returnType{Type: BinarySearchTree Integer} - -Have Axiom check that these are equal. - -\spadcommand{(t = rt)@Boolean} -$$ -{\tt true} -$$ -\returnType{Type: Boolean} - -\section{CardinalNumber} -\label{CardinalNumberXmpPage} - -The {\tt CardinalNumber} domain can be used for values indicating the -cardinality of sets, both finite and infinite. For example, the -\spadfunFrom{dimension}{VectorSpace} operation in the category -{\tt VectorSpace} returns a cardinal number. - -The non-negative integers have a natural construction as cardinals -\begin{verbatim} -0 = #{ }, 1 = {0}, 2 = {0, 1}, ..., n = {i | 0 <= i < n}. -\end{verbatim} - -The fact that {\tt 0} acts as a zero for the multiplication of cardinals is -equivalent to the axiom of choice. - -Cardinal numbers can be created by conversion from non-negative integers. - -\spadcommand{c0 := 0 :: CardinalNumber} -$$ -0 -$$ -\returnType{Type: CardinalNumber} - -\spadcommand{c1 := 1 :: CardinalNumber} -$$ -1 -$$ -\returnType{Type: CardinalNumber} - -\spadcommand{c2 := 2 :: CardinalNumber} -$$ -2 -$$ -\returnType{Type: CardinalNumber} - -\spadcommand{c3 := 3 :: CardinalNumber} -$$ -3 -$$ -\returnType{Type: CardinalNumber} - -They can also be obtained as the named cardinal {\tt Aleph(n)}. - -\spadcommand{A0 := Aleph 0} -$$ -Aleph -\left( -{0} -\right) -$$ -\returnType{Type: CardinalNumber} - -\spadcommand{A1 := Aleph 1} -$$ -Aleph -\left( -{1} -\right) -$$ -\returnType{Type: CardinalNumber} - -The \spadfunFrom{finite?}{CardinalNumber} operation tests whether a -value is a finite cardinal, that is, a non-negative integer. - -\spadcommand{finite? c2} -$$ -{\tt true} -$$ -\returnType{Type: Boolean} - -\spadcommand{finite? A0} -$$ -{\tt false} -$$ -\returnType{Type: Boolean} - -Similarly, the \spadfunFrom{countable?}{CardinalNumber} -operation determines whether a value is -a countable cardinal, that is, finite or {\tt Aleph(0)}. - -\spadcommand{countable? c2} -$$ -{\tt true} -$$ -\returnType{Type: Boolean} - -\spadcommand{countable? A0} -$$ -{\tt true} -$$ -\returnType{Type: Boolean} - -\spadcommand{countable? A1} -$$ -{\tt false} -$$ -\returnType{Type: Boolean} - -Arithmetic operations are defined on cardinal numbers as follows: -If {\tt x = \#X} and {\tt y = \#Y} then - -\noindent -$ -\begin{array}{lr} -{\tt x+y = \#(X+Y)} & cardinality of the disjoint union\\ -{\tt x-y = \#(X-Y)} & cardinality of the relative complement \\ -{\tt x*y = \#(X*Y)} & cardinality of the Cartesian product \\ -{\tt x**y = \#(X**Y)} & -cardinality of the set of maps from {\tt Y} to {\tt X} \\ -\end{array} -$ - -Here are some arithmetic examples. - -\spadcommand{[c2 + c2, c2 + A1]} -$$ -\left[4, {Aleph \left({1} \right)}\right] -$$ -\returnType{Type: List CardinalNumber} - -\spadcommand{[c0*c2, c1*c2, c2*c2, c0*A1, c1*A1, c2*A1, A0*A1]} -$$ -\left[ -0, 2, 4, 0, {Aleph \left({1} \right)}, -{Aleph \left({1} \right)}, -{Aleph \left({1} \right)} -\right] -$$ -\returnType{Type: List CardinalNumber} - -\spadcommand{[c2**c0, c2**c1, c2**c2, A1**c0, A1**c1, A1**c2]} -$$ -\left[ -1, 2, 4, 1, {Aleph \left({1} \right)}, -{Aleph \left({1} \right)} -\right] -$$ -\returnType{Type: List CardinalNumber} - -Subtraction is a partial operation: it is not defined -when subtracting a larger cardinal from a smaller one, nor -when subtracting two equal infinite cardinals. - -\spadcommand{[c2-c1, c2-c2, c2-c3, A1-c2, A1-A0, A1-A1]} -$$ -\left[ -1, 0, \mbox{\tt "failed"} , {Aleph \left({1} \right)}, -{Aleph \left({1} \right)}, -\mbox{\tt "failed"} -\right] -$$ -\returnType{Type: List Union(CardinalNumber,"failed")} - -The generalized continuum hypothesis asserts that -\begin{verbatim} -2**Aleph i = Aleph(i+1) -\end{verbatim} -and is independent of the axioms of set theory.\footnote{Goedel, -{\it The consistency of the continuum hypothesis,} -Ann. Math. Studies, Princeton Univ. Press, 1940.} - -The {\tt CardinalNumber} domain provides an operation to assert -whether the hypothesis is to be assumed. - -\spadcommand{generalizedContinuumHypothesisAssumed true} - -When the generalized continuum hypothesis -is assumed, exponentiation to a transfinite power is allowed. - -\spadcommand{[c0**A0, c1**A0, c2**A0, A0**A0, A0**A1, A1**A0, A1**A1]} -$$ -\left[ -0, 1, {Aleph \left({1} \right)}, -{Aleph \left({1} \right)}, -{Aleph \left({2} \right)}, -{Aleph \left({1} \right)}, -{Aleph \left({2} \right)} -\right] -$$ -\returnType{Type: List CardinalNumber} - -Three commonly encountered cardinal numbers are - -\noindent -$ -\begin{array}{lr} -{\tt a = \#}{\bf Z} & countable infinity \\ -{\tt c = \#}{\bf R} & the continuum \\ -{\tt f = \#\{g| g: [0,1] -> {\bf R}\}} \\ -\end{array} -$ - -In this domain, these values are obtained under the generalized -continuum hypothesis in this way. - -\spadcommand{a := Aleph 0} -$$ -Aleph \left({0} \right) -$$ -\returnType{Type: CardinalNumber} - -\spadcommand{c := 2**a} -$$ -Aleph \left({1} \right) -$$ -\returnType{Type: CardinalNumber} - -\spadcommand{f := 2**c} -$$ -Aleph \left({2} \right) -$$ -\returnType{Type: CardinalNumber} - -\section{CartesianTensor} -\label{CartesianTensorXmpPage} - -{\tt CartesianTensor(i0,dim,R)} provides Cartesian tensors with -components belonging to a commutative ring {\tt R}. Tensors can be -described as a generalization of vectors and matrices. This gives a -concise {\it tensor algebra} for multilinear objects supported by the -{\tt CartesianTensor} domain. You can form the inner or outer product -of any two tensors and you can add or subtract tensors with the same -number of components. Additionally, various forms of traces and -transpositions are useful. - -The {\tt CartesianTensor} constructor allows you to specify the -minimum index for subscripting. In what follows we discuss in detail -how to manipulate tensors. - -Here we construct the domain of Cartesian tensors of dimension 2 over the -integers, with indices starting at 1. - -\spadcommand{CT := CARTEN(i0 := 1, 2, Integer)} -$$ -CartesianTensor(1,2,Integer) -$$ -\returnType{Type: Domain} - -\subsubsection{Forming tensors} - -Scalars can be converted to tensors of rank zero. - -\spadcommand{t0: CT := 8} -$$ -8 -$$ -\returnType{Type: CartesianTensor(1,2,Integer)} - -\spadcommand{rank t0} -$$ -0 -$$ -\returnType{Type: NonNegativeInteger} - -Vectors (mathematical direct products, rather than one dimensional array -structures) can be converted to tensors of rank one. - -\spadcommand{v: DirectProduct(2, Integer) := directProduct [3,4]} -$$ -\left[ -3, 4 -\right] -$$ -\returnType{Type: DirectProduct(2,Integer)} - -\spadcommand{Tv: CT := v} -$$ -\left[ -3, 4 -\right] -$$ -\returnType{Type: CartesianTensor(1,2,Integer)} - -Matrices can be converted to tensors of rank two. - -\spadcommand{m: SquareMatrix(2, Integer) := matrix [ [1,2],[4,5] ]} -$$ -\left[ -\begin{array}{cc} -1 & 2 \\ -4 & 5 -\end{array} -\right] -$$ -\returnType{Type: SquareMatrix(2,Integer)} - -\spadcommand{Tm: CT := m} -$$ -\left[ -\begin{array}{cc} -1 & 2 \\ -4 & 5 -\end{array} -\right] -$$ -\returnType{Type: CartesianTensor(1,2,Integer)} - -\spadcommand{n: SquareMatrix(2, Integer) := matrix [ [2,3],[0,1] ]} -$$ -\left[ -\begin{array}{cc} -2 & 3 \\ -0 & 1 -\end{array} -\right] -$$ -\returnType{Type: SquareMatrix(2,Integer)} - -\spadcommand{Tn: CT := n} -$$ -\left[ -\begin{array}{cc} -2 & 3 \\ -0 & 1 -\end{array} -\right] -$$ -\returnType{Type: CartesianTensor(1,2,Integer)} - -In general, a tensor of rank {\tt k} can be formed by making a list of -rank {\tt k-1} tensors or, alternatively, a {\tt k}-deep nested list -of lists. - -\spadcommand{t1: CT := [2, 3]} -$$ -\left[ -2, 3 -\right] -$$ -\returnType{Type: CartesianTensor(1,2,Integer)} - -\spadcommand{rank t1} -$$ -1 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{t2: CT := [t1, t1]} -$$ -\left[ -\begin{array}{cc} -2 & 3 \\ -2 & 3 -\end{array} -\right] -$$ -\returnType{Type: CartesianTensor(1,2,Integer)} - -\spadcommand{t3: CT := [t2, t2]} -$$ -\left[ -{\left[ -\begin{array}{cc} -2 & 3 \\ -2 & 3 -\end{array} -\right]}, -{\left[ -\begin{array}{cc} -2 & 3 \\ -2 & 3 -\end{array} -\right]} -\right] -$$ -\returnType{Type: CartesianTensor(1,2,Integer)} - -\spadcommand{tt: CT := [t3, t3]; tt := [tt, tt]} -$$ -\left[ -{\left[ -\begin{array}{cc} -{\left[ -\begin{array}{cc} -2 & 3 \\ -2 & 3 -\end{array} -\right]} -& {\left[ -\begin{array}{cc} -2 & 3 \\ -2 & 3 -\end{array} -\right]} -\\ -{\left[ -\begin{array}{cc} -2 & 3 \\ -2 & 3 -\end{array} -\right]} -& {\left[ -\begin{array}{cc} -2 & 3 \\ -2 & 3 -\end{array} -\right]} -\end{array} -\right]}, -{\left[ -\begin{array}{cc} -{\left[ -\begin{array}{cc} -2 & 3 \\ -2 & 3 -\end{array} -\right]} -& {\left[ -\begin{array}{cc} -2 & 3 \\ -2 & 3 -\end{array} -\right]} -\\ -{\left[ -\begin{array}{cc} -2 & 3 \\ -2 & 3 -\end{array} -\right]} -& {\left[ -\begin{array}{cc} -2 & 3 \\ -2 & 3 -\end{array} -\right]} -\end{array} -\right]} -\right] -$$ -\returnType{Type: CartesianTensor(1,2,Integer)} - -\spadcommand{rank tt} -$$ -5 -$$ -\returnType{Type: PositiveInteger} - -\subsubsection{Multiplication} - -Given two tensors of rank {\tt k1} and {\tt k2}, the outer -\spadfunFrom{product}{CartesianTensor} forms a new tensor of rank -{\tt k1+k2}. Here - -$$T_{mn}(i,j,k,l) = T_m(i,j) \ T_n(k,l)$$ - -\spadcommand{Tmn := product(Tm, Tn)} -$$ -\left[ -\begin{array}{cc} -{\left[ -\begin{array}{cc} -2 & 3 \\ -0 & 1 -\end{array} -\right]} -& {\left[ -\begin{array}{cc} -4 & 6 \\ -0 & 2 -\end{array} -\right]} -\\ -{\left[ -\begin{array}{cc} -8 & {12} \\ -0 & 4 -\end{array} -\right]} -& {\left[ -\begin{array}{cc} -{10} & {15} \\ -0 & 5 -\end{array} -\right]} -\end{array} -\right] -$$ -\returnType{Type: CartesianTensor(1,2,Integer)} - -The inner product (\spadfunFrom{contract}{CartesianTensor}) forms a -tensor of rank {\tt k1+k2-2}. This product generalizes the vector dot -product and matrix-vector product by summing component products along -two indices. - -Here we sum along the second index of $T_m$ and the first index of -$T_v$. Here - -$$T_{mv} = \sum_{j=1}^{\hbox{\tiny\rm dim}} T_m(i,j) \ T_v(j)$$ - -\spadcommand{Tmv := contract(Tm,2,Tv,1)} -$$ -\left[ -{11}, {32} -\right] -$$ -\returnType{Type: CartesianTensor(1,2,Integer)} - -The multiplication operator \spadopFrom{*}{CartesianTensor} is scalar -multiplication or an inner product depending on the ranks of the arguments. - -If either argument is rank zero it is treated as scalar multiplication. -Otherwise, {\tt a*b} is the inner product summing the last index of -{\tt a} with the first index of {\tt b}. - -\spadcommand{Tm*Tv} -$$ -\left[ -{11}, {32} -\right] -$$ -\returnType{Type: CartesianTensor(1,2,Integer)} - -This definition is consistent with the inner product on matrices -and vectors. - -\spadcommand{Tmv = m * v} -$$ -{\left[ {11}, {32} \right]}= -{\left[{11}, {32} \right]} -$$ -\returnType{Type: Equation CartesianTensor(1,2,Integer)} - -\subsubsection{Selecting Components} - -For tensors of low rank (that is, four or less), components can be selected -by applying the tensor to its indices. - -\spadcommand{t0()} -$$ -8 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{t1(1+1)} -$$ -3 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{t2(2,1)} -$$ -2 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{t3(2,1,2)} -$$ -3 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{Tmn(2,1,2,1)} -$$ -0 -$$ -\returnType{Type: NonNegativeInteger} - -A general indexing mechanism is provided for a list of indices. - -\spadcommand{t0[]} -$$ -8 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{t1[2]} -$$ -3 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{t2[2,1]} -$$ -2 -$$ -\returnType{Type: PositiveInteger} - -The general mechanism works for tensors of arbitrary rank, but is -somewhat less efficient since the intermediate index list must be created. - -\spadcommand{t3[2,1,2]} -$$ -3 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{Tmn[2,1,2,1]} -$$ -0 -$$ -\returnType{Type: NonNegativeInteger} - -\subsubsection{Contraction} - -A ``contraction'' between two tensors is an inner product, as we have -seen above. You can also contract a pair of indices of a single -tensor. This corresponds to a ``trace'' in linear algebra. The -expression {\tt contract(t,k1,k2)} forms a new tensor by summing the -diagonal given by indices in position {\tt k1} and {\tt k2}. - -This is the tensor given by -$$xT_{mn} = \sum_{k=1}^{\hbox{\tiny\rm dim}} T_{mn}(k,k,i,j)$$ - -\spadcommand{cTmn := contract(Tmn,1,2)} -$$ -\left[ -\begin{array}{cc} -{12} & {18} \\ -0 & 6 -\end{array} -\right] -$$ -\returnType{Type: CartesianTensor(1,2,Integer)} - -Since {\tt Tmn} is the outer product of matrix {\tt m} and matrix {\tt n}, -the above is equivalent to this. - -\spadcommand{trace(m) * n} -$$ -\left[ -\begin{array}{cc} -{12} & {18} \\ -0 & 6 -\end{array} -\right] -$$ -\returnType{Type: SquareMatrix(2,Integer)} - -In this and the next few examples, we show all possible contractions -of {\tt Tmn} and their matrix algebra equivalents. - -\spadcommand{contract(Tmn,1,2) = trace(m) * n} -$$ -{\left[ -\begin{array}{cc} -{12} & {18} \\ -0 & 6 -\end{array} -\right]}={\left[ -\begin{array}{cc} -{12} & {18} \\ -0 & 6 -\end{array} -\right]} -$$ -\returnType{Type: Equation CartesianTensor(1,2,Integer)} - -\spadcommand{contract(Tmn,1,3) = transpose(m) * n} -$$ -{\left[ -\begin{array}{cc} -2 & 7 \\ -4 & {11} -\end{array} -\right]}={\left[ -\begin{array}{cc} -2 & 7 \\ -4 & {11} -\end{array} -\right]} -$$ -\returnType{Type: Equation CartesianTensor(1,2,Integer)} - -\spadcommand{contract(Tmn,1,4) = transpose(m) * transpose(n)} -$$ -{\left[ -\begin{array}{cc} -{14} & 4 \\ -{19} & 5 -\end{array} -\right]}={\left[ -\begin{array}{cc} -{14} & 4 \\ -{19} & 5 -\end{array} -\right]} -$$ -\returnType{Type: Equation CartesianTensor(1,2,Integer)} - -\spadcommand{contract(Tmn,2,3) = m * n} -$$ -{\left[ -\begin{array}{cc} -2 & 5 \\ -8 & {17} -\end{array} -\right]}={\left[ -\begin{array}{cc} -2 & 5 \\ -8 & {17} -\end{array} -\right]} -$$ -\returnType{Type: Equation CartesianTensor(1,2,Integer)} - -\spadcommand{contract(Tmn,2,4) = m * transpose(n)} -$$ -{\left[ -\begin{array}{cc} -8 & 2 \\ -{23} & 5 -\end{array} -\right]}={\left[ -\begin{array}{cc} -8 & 2 \\ -{23} & 5 -\end{array} -\right]} -$$ -\returnType{Type: Equation CartesianTensor(1,2,Integer)} - -\spadcommand{contract(Tmn,3,4) = trace(n) * m} -$$ -{\left[ -\begin{array}{cc} -3 & 6 \\ -{12} & {15} -\end{array} -\right]}={\left[ -\begin{array}{cc} -3 & 6 \\ -{12} & {15} -\end{array} -\right]} -$$ -\returnType{Type: Equation CartesianTensor(1,2,Integer)} - -\subsubsection{Transpositions} - -You can exchange any desired pair of indices using the -\spadfunFrom{transpose}{CartesianTensor} operation. - -Here the indices in positions one and three are exchanged, that is, -$tT_{mn}(i,j,k,l) = T_{mn}(k,j,i,l).$ - -\spadcommand{tTmn := transpose(Tmn,1,3)} -$$ -\left[ -\begin{array}{cc} -{\left[ -\begin{array}{cc} -2 & 3 \\ -8 & {12} -\end{array} -\right]} -& {\left[ -\begin{array}{cc} -4 & 6 \\ -{10} & {15} -\end{array} -\right]} -\\ -{\left[ -\begin{array}{cc} -0 & 1 \\ -0 & 4 -\end{array} -\right]} -& {\left[ -\begin{array}{cc} -0 & 2 \\ -0 & 5 -\end{array} -\right]} -\end{array} -\right] -$$ -\returnType{Type: CartesianTensor(1,2,Integer)} - -If no indices are specified, the first and last index are exchanged. - -\spadcommand{transpose Tmn} -$$ -\left[ -\begin{array}{cc} -{\left[ -\begin{array}{cc} -2 & 8 \\ -0 & 0 -\end{array} -\right]} -& {\left[ -\begin{array}{cc} -4 & {10} \\ -0 & 0 -\end{array} -\right]} -\\ -{\left[ -\begin{array}{cc} -3 & {12} \\ -1 & 4 -\end{array} -\right]} -& {\left[ -\begin{array}{cc} -6 & {15} \\ -2 & 5 -\end{array} -\right]} -\end{array} -\right] -$$ -\returnType{Type: CartesianTensor(1,2,Integer)} - -This is consistent with the matrix transpose. - -\spadcommand{transpose Tm = transpose m} -$$ -{\left[ -\begin{array}{cc} -1 & 4 \\ -2 & 5 -\end{array} -\right]}={\left[ -\begin{array}{cc} -1 & 4 \\ -2 & 5 -\end{array} -\right]} -$$ -\returnType{Type: Equation CartesianTensor(1,2,Integer)} - -If a more complicated reordering of the indices is required, then the -\spadfunFrom{reindex}{CartesianTensor} operation can be used. -This operation allows the indices to be arbitrarily permuted. - -This defines $rT_{mn}(i,j,k,l) = \allowbreak T_{mn}(i,l,j,k).$ - -\spadcommand{rTmn := reindex(Tmn, [1,4,2,3])} -$$ -\left[ -\begin{array}{cc} -{\left[ -\begin{array}{cc} -2 & 0 \\ -4 & 0 -\end{array} -\right]} -& {\left[ -\begin{array}{cc} -3 & 1 \\ -6 & 2 -\end{array} -\right]} -\\ -{\left[ -\begin{array}{cc} -8 & 0 \\ -{10} & 0 -\end{array} -\right]} -& {\left[ -\begin{array}{cc} -{12} & 4 \\ -{15} & 5 -\end{array} -\right]} -\end{array} -\right] -$$ -\returnType{Type: CartesianTensor(1,2,Integer)} - -\subsubsection{Arithmetic} - -Tensors of equal rank can be added or subtracted so arithmetic -expressions can be used to produce new tensors. - -\spadcommand{tt := transpose(Tm)*Tn - Tn*transpose(Tm)} -$$ -\left[ -\begin{array}{cc} --6 & -{16} \\ -2 & 6 -\end{array} -\right] -$$ -\returnType{Type: CartesianTensor(1,2,Integer)} - -\spadcommand{Tv*(tt+Tn)} -$$ -\left[ --4, -{11} -\right] -$$ -\returnType{Type: CartesianTensor(1,2,Integer)} - -\spadcommand{reindex(product(Tn,Tn),[4,3,2,1])+3*Tn*product(Tm,Tm)} -$$ -\left[ -\begin{array}{cc} -{\left[ -\begin{array}{cc} -{46} & {84} \\ -{174} & {212} -\end{array} -\right]} -& {\left[ -\begin{array}{cc} -{57} & {114} \\ -{228} & {285} -\end{array} -\right]} -\\ -{\left[ -\begin{array}{cc} -{18} & {24} \\ -{57} & {63} -\end{array} -\right]} -& {\left[ -\begin{array}{cc} -{17} & {30} \\ -{63} & {76} -\end{array} -\right]} -\end{array} -\right] -$$ -\returnType{Type: CartesianTensor(1,2,Integer)} - -\subsubsection{Specific Tensors} - -Two specific tensors have properties which depend only on the -dimension. - -The Kronecker delta satisfies -\begin{verbatim} - +- - | 1 if i = j -delta(i,j) = | - | 0 if i ^= j - +- -\end{verbatim} - -\spadcommand{delta: CT := kroneckerDelta()} -$$ -\left[ -\begin{array}{cc} -1 & 0 \\ -0 & 1 -\end{array} -\right] -$$ -\returnType{Type: CartesianTensor(1,2,Integer)} - -This can be used to reindex via contraction. - -\spadcommand{contract(Tmn, 2, delta, 1) = reindex(Tmn, [1,3,4,2])} -$$ -{\left[ -\begin{array}{cc} -{\left[ -\begin{array}{cc} -2 & 4 \\ -3 & 6 -\end{array} -\right]} -& {\left[ -\begin{array}{cc} -0 & 0 \\ -1 & 2 -\end{array} -\right]} -\\ -{\left[ -\begin{array}{cc} -8 & {10} \\ -{12} & {15} -\end{array} -\right]} -& {\left[ -\begin{array}{cc} -0 & 0 \\ -4 & 5 -\end{array} -\right]} -\end{array} -\right]}={\left[ -\begin{array}{cc} -{\left[ -\begin{array}{cc} -2 & 4 \\ -3 & 6 -\end{array} -\right]} -& {\left[ -\begin{array}{cc} -0 & 0 \\ -1 & 2 -\end{array} -\right]} -\\ -{\left[ -\begin{array}{cc} -8 & {10} \\ -{12} & {15} -\end{array} -\right]} -& {\left[ -\begin{array}{cc} -0 & 0 \\ -4 & 5 -\end{array} -\right]} -\end{array} -\right]} -$$ -\returnType{Type: Equation CartesianTensor(1,2,Integer)} - -The Levi Civita symbol determines the sign of a permutation of indices. - -\spadcommand{epsilon:CT := leviCivitaSymbol()} -$$ -\left[ -\begin{array}{cc} -0 & 1 \\ --1 & 0 -\end{array} -\right] -$$ -\returnType{Type: CartesianTensor(1,2,Integer)} - -Here we have: -\begin{verbatim} -epsilon(i1,...,idim) - = +1 if i1,...,idim is an even permutation of i0,...,i0+dim-1 - = -1 if i1,...,idim is an odd permutation of i0,...,i0+dim-1 - = 0 if i1,...,idim is not a permutation of i0,...,i0+dim-1 -\end{verbatim} - -This property can be used to form determinants. - -\spadcommand{contract(epsilon*Tm*epsilon, 1,2) = 2 * determinant m} -$$ --6=-6 -$$ -\returnType{Type: Equation CartesianTensor(1,2,Integer)} - -\subsubsection{Properties of the CartesianTensor domain} - -{\tt GradedModule(R,E)} denotes ``{\tt E}-graded {\tt R}-module'', -that is, a collection of {\tt R}-modules indexed by an abelian monoid -{\tt E.} An element {\tt g} of {\tt G[s]} for some specific {\tt s} -in {\tt E} is said to be an element of {\tt G} with -\spadfunFrom{degree}{GradedModule} {\tt s}. Sums are defined in each -module {\tt G[s]} so two elements of {\tt G} can be added if they have -the same degree. Morphisms can be defined and composed by degree to -give the mathematical category of graded modules. - -{\tt GradedAlgebra(R,E)} denotes ``{\tt E}-graded {\tt R}-algebra.'' -A graded algebra is a graded module together with a degree preserving -{\tt R}-bilinear map, called the \spadfunFrom{product}{GradedAlgebra}. - -\begin{verbatim} -degree(product(a,b)) = degree(a) + degree(b) - -product(r*a,b) = product(a,r*b) = r*product(a,b) -product(a1+a2,b) = product(a1,b) + product(a2,b) -product(a,b1+b2) = product(a,b1) + product(a,b2) -product(a,product(b,c)) = product(product(a,b),c) -\end{verbatim} - -The domain {\tt CartesianTensor(i0, dim, R)} belongs to the category -{\tt GradedAlgebra(R, NonNegativeInteger)}. The non-negative integer -\spadfunFrom{degree}{GradedAlgebra} is the tensor rank and the graded -algebra \spadfunFrom{product}{GradedAlgebra} is the tensor outer -product. The graded module addition captures the notion that only -tensors of equal rank can be added. - -If {\tt V} is a vector space of dimension {\tt dim} over {\tt R}, -then the tensor module {\tt T[k](V)} is defined as -\begin{verbatim} -T[0](V) = R -T[k](V) = T[k-1](V) * V -\end{verbatim} -where {\tt *} denotes the {\tt R}-module tensor -\spadfunFrom{product}{GradedAlgebra}. {\tt CartesianTensor(i0,dim,R)} -is the graded algebra in which the degree {\tt k} module is {\tt -T[k](V)}. - -\subsubsection{Tensor Calculus} - -It should be noted here that often tensors are used in the context of -tensor-valued manifold maps. This leads to the notion of covariant -and contravariant bases with tensor component functions transforming -in specific ways under a change of coordinates on the manifold. This -is no more directly supported by the {\tt CartesianTensor} domain than -it is by the {\tt Vector} domain. However, it is possible to have the -components implicitly represent component maps by choosing a -polynomial or expression type for the components. In this case, it is -up to the user to satisfy any constraints which arise on the basis of -this interpretation. - -\section{Character} -\label{CharacterXmpPage} -The members of the domain {\tt Character} are values -representing letters, numerals and other text elements. -For more information on related topics, see -\ref{CharacterClassXmpPage} on page~\pageref{CharacterClassXmpPage} and -\ref{StringXmpPage} on page~\pageref{StringXmpPage}. - -Characters can be obtained using {\tt String} notation. - -\spadcommand{chars := [char "a", char "A", char "X", char "8", char "+"]} -$$ -\left[ -a, A, X, 8, + -\right] -$$ -\returnType{Type: List Character} - -Certain characters are available by name. -This is the blank character. - -\spadcommand{space()} -$$ -\ -$$ -\returnType{Type: Character} - -This is the quote that is used in strings. - -\spadcommand{quote()} -$$ -\mbox{\tt "} -$$ -\returnType{Type: Character} - -This is the escape character that allows quotes and other characters -within strings. - -\spadcommand{escape()} -$$ -\_ -$$ -\returnType{Type: Character} - -Characters are represented as integers in a machine-dependent way. -The integer value can be obtained using the -\spadfunFrom{ord}{Character} operation. It is always true that {\tt -char(ord c) = c} and {\tt ord(char i) = i}, provided that {\tt i} is -in the range {\tt 0..size()\$Character-1}. - -\spadcommand{[ord c for c in chars]} -$$ -\left[ -{97}, {65}, {88}, {56}, {43} -\right] -$$ -\returnType{Type: List Integer} - -The \spadfunFrom{lowerCase}{Character} operation converts an upper -case letter to the corresponding lower case letter. If the argument -is not an upper case letter, then it is returned unchanged. - -\spadcommand{[upperCase c for c in chars]} -$$ -\left[ -A, A, X, 8, + -\right] -$$ -\returnType{Type: List Character} - -Likewise, the \spadfunFrom{upperCase}{Character} operation converts lower -case letters to upper case. - -\spadcommand{[lowerCase c for c in chars] } -$$ -\left[ -a, a, x, 8, + -\right] -$$ -\returnType{Type: List Character} - -A number of tests are available to determine whether characters -belong to certain families. - -\spadcommand{[alphabetic? c for c in chars] } -$$ -\left[ -{\tt true}, {\tt true}, {\tt true}, {\tt false}, {\tt false} -\right] -$$ -\returnType{Type: List Boolean} - -\spadcommand{[upperCase? c for c in chars] } -$$ -\left[ -{\tt false}, {\tt true}, {\tt true}, {\tt false}, {\tt false} -\right] -$$ -\returnType{Type: List Boolean} - -\spadcommand{[lowerCase? c for c in chars] } -$$ -\left[ -{\tt true}, {\tt false}, {\tt false}, {\tt false}, {\tt false} -\right] -$$ -\returnType{Type: List Boolean} - -\spadcommand{[digit? c for c in chars] } -$$ -\left[ -{\tt false}, {\tt false}, {\tt false}, {\tt true}, {\tt false} -\right] -$$ -\returnType{Type: List Boolean} - -\spadcommand{[hexDigit? c for c in chars] } -$$ -\left[ -{\tt true}, {\tt true}, {\tt false}, {\tt true}, {\tt false} -\right] -$$ -\returnType{Type: List Boolean} - -\spadcommand{[alphanumeric? c for c in chars] } -$$ -\left[ -{\tt true}, {\tt true}, {\tt true}, {\tt true}, {\tt false} -\right] -$$ -\returnType{Type: List Boolean} - -\section{CharacterClass} -\label{CharacterClassXmpPage} -The {\tt CharacterClass} domain allows classes of characters to be -defined and manipulated efficiently. - -Character classes can be created by giving either a string or a list -of characters. - -\spadcommand{cl1 := charClass [char "a", char "e", char "i", char "o", char "u", char "y"] } -$$ -\mbox{\tt "aeiouy"} -$$ -\returnType{Type: CharacterClass} - -\spadcommand{cl2 := charClass "bcdfghjklmnpqrstvwxyz" } -$$ -\mbox{\tt "bcdfghjklmnpqrstvwxyz"} -$$ -\returnType{Type: CharacterClass} - -A number of character classes are predefined for convenience. - -\spadcommand{digit()} -$$ -\mbox{\tt "0123456789"} -$$ -\returnType{Type: CharacterClass} - -\spadcommand{hexDigit()} -$$ -\mbox{\tt "0123456789ABCDEFabcdef"} -$$ -\returnType{Type: CharacterClass} - -\spadcommand{upperCase()} -$$ -\mbox{\tt "ABCDEFGHIJKLMNOPQRSTUVWXYZ"} -$$ -\returnType{Type: CharacterClass} - -\spadcommand{lowerCase()} -$$ -\mbox{\tt "abcdefghijklmnopqrstuvwxyz"} -$$ -\returnType{Type: CharacterClass} - -\spadcommand{alphabetic()} -$$ -\mbox{\tt "ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz"} -$$ -\returnType{Type: CharacterClass} - -\spadcommand{alphanumeric()} -$$ -\mbox{\tt "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz"} -$$ -\returnType{Type: CharacterClass} - -You can quickly test whether a character belongs to a class. - -\spadcommand{member?(char "a", cl1) } -$$ -{\tt true} -$$ -\returnType{Type: Boolean} - -\spadcommand{member?(char "a", cl2) } -$$ -{\tt false} -$$ -\returnType{Type: Boolean} - -Classes have the usual set operations because the {\tt CharacterClass} -domain belongs to the category {\tt FiniteSetAggregate(Character)}. - -\spadcommand{intersect(cl1, cl2) } -$$ -\mbox{\tt "y"} -$$ -\returnType{Type: CharacterClass} - -\spadcommand{union(cl1,cl2) } -$$ -\mbox{\tt "abcdefghijklmnopqrstuvwxyz"} -$$ -\returnType{Type: CharacterClass} - -\spadcommand{difference(cl1,cl2) } -$$ -\mbox{\tt "aeiou"} -$$ -\returnType{Type: CharacterClass} - -\spadcommand{intersect(complement(cl1),cl2) } -$$ -\mbox{\tt "bcdfghjklmnpqrstvwxz"} -$$ -\returnType{Type: CharacterClass} - -You can modify character classes by adding or removing characters. - -\spadcommand{insert!(char "a", cl2) } -$$ -\mbox{\tt "abcdfghjklmnpqrstvwxyz"} -$$ -\returnType{Type: CharacterClass} - -\spadcommand{remove!(char "b", cl2) } -$$ -\mbox{\tt "acdfghjklmnpqrstvwxyz"} -$$ -\returnType{Type: CharacterClass} - -For more information on related topics, see -\ref{CharacterXmpPage} on page~\pageref{CharacterXmpPage} and -\ref{StringXmpPage} on page~\pageref{StringXmpPage}. - -\section{CliffordAlgebra} -\label{CliffordAlgebraXmpPage} -\noindent - -{\tt CliffordAlgebra(n,K,Q)} defines a vector space of dimension $2^n$ -over the field $K$ with a given quadratic form {\tt Q}. If $\{e_1, -\ldots, e_n\}$ is a basis for $K^n$ then -\begin{verbatim} -{ 1, - e(i) 1 <= i <= n, - e(i1)*e(i2) 1 <= i1 < i2 <=n, - ..., - e(1)*e(2)*...*e(n) } -\end{verbatim} -is a basis for the Clifford algebra. The algebra is defined by the relations -\begin{verbatim} -e(i)*e(i) = Q(e(i)) -e(i)*e(j) = -e(j)*e(i), i ^= j -\end{verbatim} -Examples of Clifford Algebras are -gaussians (complex numbers), quaternions, -exterior algebras and spin algebras. - -\subsection{The Complex Numbers as a Clifford Algebra} - -This is the field over which we will work, rational functions with -integer coefficients. - -\spadcommand{K := Fraction Polynomial Integer } -$$ -\mbox{\rm Fraction Polynomial Integer} -$$ -\returnType{Type: Domain} - -We use this matrix for the quadratic form. - -\spadcommand{m := matrix [ [-1] ] } -$$ -\left[ -\begin{array}{c} --1 -\end{array} -\right] -$$ -\returnType{Type: Matrix Integer} - -We get complex arithmetic by using this domain. - -\spadcommand{C := CliffordAlgebra(1, K, quadraticForm m) } -$$ -\mbox{\rm CliffordAlgebra(1,Fraction Polynomial Integer,MATRIX)} -$$ -\returnType{Type: Domain} - -Here is {\tt i}, the usual square root of {\tt -1.} - -\spadcommand{i: C := e(1) } -$$ -e \sb {1} -$$ -\returnType{Type: CliffordAlgebra(1,Fraction Polynomial Integer,MATRIX)} - -Here are some examples of the arithmetic. - -\spadcommand{x := a + b * i } -$$ -a+{b \ {e \sb {1}}} -$$ -\returnType{Type: CliffordAlgebra(1,Fraction Polynomial Integer,MATRIX)} - -\spadcommand{y := c + d * i } -$$ -c+{d \ {e \sb {1}}} -$$ -\returnType{Type: CliffordAlgebra(1,Fraction Polynomial Integer,MATRIX)} - -See \ref{ComplexXmpPage} on page~\pageref{ComplexXmpPage} -for examples of Axiom's constructor implementing complex numbers. - -\spadcommand{x * y } -$$ --{b \ d}+{a \ c}+{{\left( {a \ d}+{b \ c} -\right)} -\ {e \sb {1}}} -$$ -\returnType{Type: CliffordAlgebra(1,Fraction Polynomial Integer,MATRIX)} - -\subsection{The Quaternion Numbers as a Clifford Algebra} - -This is the field over which we will work, rational functions with -integer coefficients. - -\spadcommand{K := Fraction Polynomial Integer } -$$ -\mbox{\rm Fraction Polynomial Integer} -$$ -\returnType{Type: Domain} - -We use this matrix for the quadratic form. - -\spadcommand{m := matrix [ [-1,0],[0,-1] ] } -$$ -\left[ -\begin{array}{cc} --1 & 0 \\ -0 & -1 -\end{array} -\right] -$$ -\returnType{Type: Matrix Integer} - -The resulting domain is the quaternions. - -\spadcommand{H := CliffordAlgebra(2, K, quadraticForm m) } -$$ -\mbox{\rm CliffordAlgebra(2,Fraction Polynomial Integer,MATRIX)} -$$ -\returnType{Type: Domain} - -We use Hamilton's notation for {\tt i},{\tt j},{\tt k}. - -\spadcommand{i: H := e(1) } -$$ -e \sb {1} -$$ -\returnType{Type: CliffordAlgebra(2,Fraction Polynomial Integer,MATRIX)} - -\spadcommand{j: H := e(2) } -$$ -e \sb {2} -$$ -\returnType{Type: CliffordAlgebra(2,Fraction Polynomial Integer,MATRIX)} - -\spadcommand{k: H := i * j } -$$ -{e \sb {1}} \ {e \sb {2}} -$$ -\returnType{Type: CliffordAlgebra(2,Fraction Polynomial Integer,MATRIX)} - -\spadcommand{x := a + b * i + c * j + d * k } -$$ -a+{b \ {e \sb {1}}}+{c \ {e \sb {2}}}+{d \ {e \sb {1}} \ {e \sb {2}}} -$$ -\returnType{Type: CliffordAlgebra(2,Fraction Polynomial Integer,MATRIX)} - -\spadcommand{y := e + f * i + g * j + h * k } -$$ -e+{f \ {e \sb {1}}}+{g \ {e \sb {2}}}+{h \ {e \sb {1}} \ {e \sb {2}}} -$$ -\returnType{Type: CliffordAlgebra(2,Fraction Polynomial Integer,MATRIX)} - -\spadcommand{x + y } -$$ -e+a+{{\left( f+b -\right)} -\ {e \sb {1}}}+{{\left( g+c -\right)} -\ {e \sb {2}}}+{{\left( h+d -\right)} -\ {e \sb {1}} \ {e \sb {2}}} -$$ -\returnType{Type: CliffordAlgebra(2,Fraction Polynomial Integer,MATRIX)} - -\spadcommand{x * y } -$$ -\begin{array}{@{}l} --{d \ h} - -{c \ g} - -{b \ f}+ -{a \ e}+ -{{\left( {c \ h} -{d \ g}+{a \ f}+{b \ e} \right)}\ {e \sb {1}}}+ -\\ -\\ -\displaystyle -{{\left( -{b \ h}+{a \ g}+{d \ f}+{c \ e} \right)}\ {e \sb {2}}}+ -{{\left( {a \ h}+{b \ g} -{c \ f}+{d \ e} \right)} -\ {e \sb {1}} \ {e \sb {2}}} -\end{array} -$$ -\returnType{Type: CliffordAlgebra(2,Fraction Polynomial Integer,MATRIX)} - -See \ref{QuaternionXmpPage} on page~\pageref{QuaternionXmpPage} -for examples of Axiom's constructor implementing quaternions. - -\spadcommand{y * x } -$$ -\begin{array}{@{}l} --{d \ h} -{c \ g} -{b \ f}+{a \ e}+ -{{\left( -{c \ h}+{d \ g}+{a \ f}+{b \ e} \right)}\ {e \sb {1}}}+ -\\ -\\ -\displaystyle -{{\left( {b \ h}+{a \ g} -{d \ f}+{c \ e} \right)}\ {e \sb {2}}}+ -{{\left( {a \ h} - -{b \ g}+{c \ f}+{d \ e} \right)}\ {e \sb {1}} \ {e \sb {2}}} -\end{array} -$$ -\returnType{Type: CliffordAlgebra(2,Fraction Polynomial Integer,MATRIX)} - -\subsection{The Exterior Algebra on a Three Space} - -This is the field over which we will work, rational functions with -integer coefficients. - -\spadcommand{K := Fraction Polynomial Integer } -$$ -\mbox{\rm Fraction Polynomial Integer} -$$ -\returnType{Type: Domain} - -If we chose the three by three zero quadratic form, we obtain -the exterior algebra on {\tt e(1),e(2),e(3)}. - -\spadcommand{Ext := CliffordAlgebra(3, K, quadraticForm 0) } -$$ -\mbox{\rm CliffordAlgebra(3,Fraction Polynomial Integer,MATRIX)} -$$ -\returnType{Type: Domain} - -This is a three dimensional vector algebra. -We define {\tt i}, {\tt j}, {\tt k} as the unit vectors. - -\spadcommand{i: Ext := e(1) } -$$ -e \sb {1} -$$ -\returnType{Type: CliffordAlgebra(3,Fraction Polynomial Integer,MATRIX)} - -\spadcommand{j: Ext := e(2) } -$$ -e \sb {2} -$$ -\returnType{Type: CliffordAlgebra(3,Fraction Polynomial Integer,MATRIX)} - -\spadcommand{k: Ext := e(3) } -$$ -e \sb {3} -$$ -\returnType{Type: CliffordAlgebra(3,Fraction Polynomial Integer,MATRIX)} - -Now it is possible to do arithmetic. - -\spadcommand{x := x1*i + x2*j + x3*k } -$$ -{x1 \ {e \sb {1}}}+{x2 \ {e \sb {2}}}+{x3 \ {e \sb {3}}} -$$ -\returnType{Type: CliffordAlgebra(3,Fraction Polynomial Integer,MATRIX)} - -\spadcommand{y := y1*i + y2*j + y3*k } -$$ -{y1 \ {e \sb {1}}}+{y2 \ {e \sb {2}}}+{y3 \ {e \sb {3}}} -$$ -\returnType{Type: CliffordAlgebra(3,Fraction Polynomial Integer,MATRIX)} - -\spadcommand{x + y } -$$ -{{\left( y1+x1 -\right)} -\ {e \sb {1}}}+{{\left( y2+x2 -\right)} -\ {e \sb {2}}}+{{\left( y3+x3 -\right)} -\ {e \sb {3}}} -$$ -\returnType{Type: CliffordAlgebra(3,Fraction Polynomial Integer,MATRIX)} - -\spadcommand{x * y + y * x } -$$ -0 -$$ -\returnType{Type: CliffordAlgebra(3,Fraction Polynomial Integer,MATRIX)} - -On an {\tt n} space, a grade {\tt p} form has a dual {\tt n-p} form. -In particular, in three space the dual of a grade two element identifies -{\tt e1*e2->e3, e2*e3->e1, e3*e1->e2}. - -\spadcommand{dual2 a == coefficient(a,[2,3]) * i + coefficient(a,[3,1]) * j + coefficient(a,[1,2]) * k } -\returnType{Type: Void} - -The vector cross product is then given by this. - -\spadcommand{dual2(x*y) } -\begin{verbatim} - Compiling function dual2 with type CliffordAlgebra(3,Fraction - Polynomial Integer,MATRIX) -> CliffordAlgebra(3,Fraction - Polynomial Integer,MATRIX) -\end{verbatim} -$$ -{{\left( {x2 \ y3} -{x3 \ y2} -\right)} -\ {e \sb {1}}}+{{\left( -{x1 \ y3}+{x3 \ y1} -\right)} -\ {e \sb {2}}}+{{\left( {x1 \ y2} -{x2 \ y1} -\right)} -\ {e \sb {3}}} -$$ -\returnType{Type: CliffordAlgebra(3,Fraction Polynomial Integer,MATRIX)} - -\subsection{The Dirac Spin Algebra} - -In this section we will work over the field of rational numbers. - -\spadcommand{K := Fraction Integer } -$$ -\mbox{\rm Fraction Integer} -$$ -\returnType{Type: Domain} - -We define the quadratic form to be the Minkowski space-time metric. - -\spadcommand{g := matrix [ [1,0,0,0], [0,-1,0,0], [0,0,-1,0], [0,0,0,-1] ] } -$$ -\left[ -\begin{array}{cccc} -1 & 0 & 0 & 0 \\ -0 & -1 & 0 & 0 \\ -0 & 0 & -1 & 0 \\ -0 & 0 & 0 & -1 -\end{array} -\right] -$$ -\returnType{Type: Matrix Integer} - -We obtain the Dirac spin algebra used in Relativistic Quantum Field Theory. - -\spadcommand{D := CliffordAlgebra(4,K, quadraticForm g) } -$$ -\mbox{\rm CliffordAlgebra(4,Fraction Integer,MATRIX)} -$$ -\returnType{Type: Domain} - -The usual notation for the basis is $\gamma$ with a superscript. For -Axiom input we will use {\tt gam(i)}: - -\spadcommand{gam := [e(i)\$D for i in 1..4] } -$$ -\left[ -{e \sb {1}}, {e \sb {2}}, {e \sb {3}}, {e \sb {4}} -\right] -$$ -\returnType{Type: List CliffordAlgebra(4,Fraction Integer,MATRIX)} - -\noindent -There are various contraction identities of the form -\begin{verbatim} -g(l,t)*gam(l)*gam(m)*gam(n)*gam(r)*gam(s)*gam(t) = - 2*(gam(s)gam(m)gam(n)gam(r) + gam(r)*gam(n)*gam(m)*gam(s)) -\end{verbatim} -where a sum over {\tt l} and {\tt t} is implied. - -Verify this identity for particular values of {\tt m,n,r,s}. - -\spadcommand{m := 1; n:= 2; r := 3; s := 4; } -\returnType{Type: PositiveInteger} - -\spadcommand{lhs := reduce(+, [reduce(+, [ g(l,t)*gam(l)*gam(m)*gam(n)*gam(r)*gam(s)*gam(t) for l in 1..4]) for t in 1..4]) } -$$ --{4 \ {e \sb {1}} \ {e \sb {2}} \ {e \sb {3}} \ {e \sb {4}}} -$$ -\returnType{Type: CliffordAlgebra(4,Fraction Integer,MATRIX)} - -\spadcommand{rhs := 2*(gam s * gam m*gam n*gam r + gam r*gam n*gam m*gam s) } -$$ --{4 \ {e \sb {1}} \ {e \sb {2}} \ {e \sb {3}} \ {e \sb {4}}} -$$ -\returnType{Type: CliffordAlgebra(4,Fraction Integer,MATRIX)} - -\section{Complex} -\label{ComplexXmpPage} - -The {\tt Complex} constructor implements complex objects over a -commutative ring {\tt R}. Typically, the ring {\tt R} is {\tt Integer}, -{\tt Fraction Integer}, {\tt Float} or {\tt DoubleFloat}. -{\tt R} can also be a symbolic type, like {\tt Polynomial Integer}. -For more information about the numerical and graphical aspects of -complex numbers, see \ref{ugProblemNumeric} on page~\pageref{ugProblemNumeric}. - -Complex objects are created by the \spadfunFrom{complex}{Complex} operation. - -\spadcommand{a := complex(4/3,5/2) } -$$ -{4 \over 3}+{{5 \over 2} \ i} -$$ -\returnType{Type: Complex Fraction Integer} - -\spadcommand{b := complex(4/3,-5/2) } -$$ -{4 \over 3} -{{5 \over 2} \ i} -$$ -\returnType{Type: Complex Fraction Integer} - -The standard arithmetic operations are available. - -\spadcommand{a + b } -$$ -8 \over 3 -$$ -\returnType{Type: Complex Fraction Integer} - -\spadcommand{a - b } -$$ -5 \ i -$$ -\returnType{Type: Complex Fraction Integer} - -\spadcommand{a * b } -$$ -{289} \over {36} -$$ -\returnType{Type: Complex Fraction Integer} - -If {\tt R} is a field, you can also divide the complex objects. - -\spadcommand{a / b } -$$ --{{161} \over {289}}+{{{240} \over {289}} \ i} -$$ -\returnType{Type: Complex Fraction Integer} - -Use a conversion (\ref{ugTypesConvertPage} on -page~\pageref{ugTypesConvertPage} in Section -\ref{ugTypesConvertNumber} on page~\pageref{ugTypesConvertNumber}) -to view the last object as a fraction of complex integers. - -\spadcommand{\% :: Fraction Complex Integer } -$$ -{-{15}+{8 \ i}} \over {{15}+{8 \ i}} -$$ -\returnType{Type: Fraction Complex Integer} - -The predefined macro {\tt \%i} is defined to be {\tt complex(0,1)}. - -\spadcommand{3.4 + 6.7 * \%i} -$$ -{3.4}+{{6.7} \ i} -$$ -\returnType{Type: Complex Float} - -You can also compute the \spadfunFrom{conjugate}{Complex} and -\spadfunFrom{norm}{Complex} of a complex number. - -\spadcommand{conjugate a } -$$ -{4 \over 3} -{{5 \over 2} \ i} -$$ -\returnType{Type: Complex Fraction Integer} - -\spadcommand{norm a } -$$ -{289} \over {36} -$$ -\returnType{Type: Fraction Integer} - -The \spadfunFrom{real}{Complex} and \spadfunFrom{imag}{Complex} operations -are provided to extract the real and imaginary parts, respectively. - -\spadcommand{real a } -$$ -4 \over 3 -$$ -\returnType{Type: Fraction Integer} - -\spadcommand{imag a } -$$ -5 \over 2 -$$ -\returnType{Type: Fraction Integer} - -The domain {\tt Complex Integer} is also called the Gaussian integers. -If {\tt R} is the integers (or, more generally, a {\tt EuclideanDomain}), -you can compute greatest common divisors. - -\spadcommand{gcd(13 - 13*\%i,31 + 27*\%i)} -$$ -5+i -$$ -\returnType{Type: Complex Integer} - -You can also compute least common multiples. - -\spadcommand{lcm(13 - 13*\%i,31 + 27*\%i)} -$$ -{143} -{{39} \ i} -$$ -\returnType{Type: Complex Integer} - -You can \spadfunFrom{factor}{Complex} Gaussian integers. - -\spadcommand{factor(13 - 13*\%i)} -$$ --{{\left( 1+i -\right)} -\ {\left( 2+{3 \ i} -\right)} -\ {\left( 3+{2 \ i} -\right)}} -$$ -\returnType{Type: Factored Complex Integer} - -\spadcommand{factor complex(2,0)} -$$ --{i \ {{\left( 1+i -\right)} -\sp 2}} -$$ -\returnType{Type: Factored Complex Integer} - -\section{ContinuedFraction} -\label{ContinuedFractionXmpPage} - -Continued fractions have been a fascinating and useful tool in -mathematics for well over three hundred years. Axiom implements -continued fractions for fractions of any Euclidean domain. In -practice, this usually means rational numbers. In this section we -demonstrate some of the operations available for manipulating both -finite and infinite continued fractions. It may be helpful if you -review \ref{StreamXmpPage} on page~\pageref{StreamXmpPage} to remind -yourself of some of the operations with streams. - -The {\tt ContinuedFraction} domain is a field and therefore you can -add, subtract, multiply and divide the fractions. - -The \spadfunFrom{continuedFraction}{ContinuedFraction} operation -converts its fractional argument to a continued fraction. - -\spadcommand{c := continuedFraction(314159/100000) } -$$ -3+ \zag{1}{7}+ \zag{1}{{15}}+ \zag{1}{1}+ \zag{1}{{25}}+ \zag{1}{1}+ -\zag{1}{7}+ \zag{1}{4} -$$ -\returnType{Type: ContinuedFraction Integer} - -This display is a compact form of the bulkier -\begin{verbatim} - 3 + 1 - ------------------------------- - 7 + 1 - --------------------------- - 15 + 1 - ---------------------- - 1 + 1 - ------------------ - 25 + 1 - ------------- - 1 + 1 - --------- - 7 + 1 - ----- - 4 -\end{verbatim} - -You can write any rational number in a similar form. The fraction -will be finite and you can always take the ``numerators'' to be {\tt 1}. -That is, any rational number can be written as a simple, finite -continued fraction of the form - -\begin{verbatim} - a(1) + 1 - ------------------------- - a(2) + 1 - -------------------- - a(3) + - . - . - . - 1 - ------------- - a(n-1) + 1 - ---- - a(n) -\end{verbatim} - -The $a_i$ are called partial quotients and the operation -\spadfunFrom{partialQuotients}{ContinuedFraction} creates a stream of them. - -\spadcommand{partialQuotients c } -$$ -\left[ -3, 7, {15}, 1, {25}, 1, 7, \ldots -\right] -$$ -\returnType{Type: Stream Integer} - -By considering more and more of the fraction, you get the -\spadfunFrom{convergents}{ContinuedFraction}. For example, the first -convergent is $a_1$, the second is $a_1 + 1/a_2$ and so on. - -\spadcommand{convergents c } -$$ -\left[ -3, {{22} \over 7}, {{333} \over {106}}, {{355} \over {113}}, -{{9208} \over {2931}}, {{9563} \over {3044}}, {{76149} \over {24239}}, -\ldots -\right] -$$ -\returnType{Type: Stream Fraction Integer} - -Since this is a finite continued fraction, the last convergent is the -original rational number, in reduced form. The result of -\spadfunFrom{approximants}{ContinuedFraction} is always an infinite -stream, though it may just repeat the ``last'' value. - -\spadcommand{approximants c } -$$ -\left[ -3, {{22} \over 7}, {{333} \over {106}}, {{355} \over {113}}, -{{9208} \over {2931}}, {{9563} \over {3044}}, {{76149} \over {24239}}, -\ldots -\right] -$$ -\returnType{Type: Stream Fraction Integer} - -Inverting {\tt c} only changes the partial quotients of its fraction -by inserting a {\tt 0} at the beginning of the list. - -\spadcommand{pq := partialQuotients(1/c) } -$$ -\left[ -0, 3, 7, {15}, 1, {25}, 1, \ldots -\right] -$$ -\returnType{Type: Stream Integer} - -Do this to recover the original continued fraction from this list of -partial quotients. The three-argument form of the -\spadfunFrom{continuedFraction}{ContinuedFraction} operation takes an -element which is the whole part of the fraction, a stream of elements -which are the numerators of the fraction, and a stream of elements -which are the denominators of the fraction. - -\spadcommand{continuedFraction(first pq,repeating [1],rest pq) } -$$ -\zag{1}{3}+ \zag{1}{7}+ \zag{1}{{15}}+ \zag{1}{1}+ \zag{1}{{25}}+ \zag{1}{1}+ -\zag{1}{7}+\ldots -$$ -\returnType{Type: ContinuedFraction Integer} - -The streams need not be finite for -\spadfunFrom{continuedFraction}{ContinuedFraction}. Can you guess -which irrational number has the following continued fraction? See the -end of this section for the answer. - -\spadcommand{z:=continuedFraction(3,repeating [1],repeating [3,6]) } -$$ -3+ \zag{1}{3}+ \zag{1}{6}+ \zag{1}{3}+ \zag{1}{6}+ \zag{1}{3}+ \zag{1}{6}+ -\zag{1}{3}+\ldots -$$ -\returnType{Type: ContinuedFraction Integer} - -In 1737 Euler discovered the infinite continued fraction expansion -\begin{verbatim} - e - 1 1 - ----- = --------------------- - 2 1 + 1 - ----------------- - 6 + 1 - ------------- - 10 + 1 - -------- - 14 + ... -\end{verbatim} - -We use this expansion to compute rational and floating point -approximations of {\tt e}.\footnote{For this and other interesting -expansions, see C. D. Olds, {\it Continued Fractions,} New -Mathematical Library, (New York: Random House, 1963), pp. 134--139.} - -By looking at the above expansion, we see that the whole part is {\tt 0} -and the numerators are all equal to {\tt 1}. This constructs the -stream of denominators. - -\spadcommand{dens:Stream Integer := cons(1,generate((x+->x+4),6)) } -$$ -\left[ -1, 6, {10}, {14}, {18}, {22}, {26}, \ldots -\right] -$$ -\returnType{Type: Stream Integer} - -Therefore this is the continued fraction expansion for -$(e - 1) / 2$. - -\spadcommand{cf := continuedFraction(0,repeating [1],dens) } -$$ -\zag{1}{1}+ \zag{1}{6}+ \zag{1}{{10}}+ \zag{1}{{14}}+ \zag{1}{{18}}+ -\zag{1}{{22}}+ \zag{1}{{26}}+\ldots -$$ -\returnType{Type: ContinuedFraction Integer} - -These are the rational number convergents. - -\spadcommand{ccf := convergents cf } -$$ -\left[ -0, 1, {6 \over 7}, {{61} \over {71}}, {{860} \over {1001}}, -{{15541} \over {18089}}, {{342762} \over {398959}}, \ldots -\right] -$$ -\returnType{Type: Stream Fraction Integer} - -You can get rational convergents for {\tt e} by multiplying by {\tt 2} and -adding {\tt 1}. - -\spadcommand{eConvergents := [2*e + 1 for e in ccf] } -$$ -\left[ -1, 3, {{19} \over 7}, {{193} \over {71}}, {{2721} \over {1001}}, -{{49171} \over {18089}}, {{1084483} \over {398959}}, \ldots -\right] -$$ -\returnType{Type: Stream Fraction Integer} - -You can also compute the floating point approximations to these convergents. - -\spadcommand{eConvergents :: Stream Float } -$$ -\begin{array}{@{}l} -\left[ -{1.0}, {3.0}, {2.7142857142 857142857}, {2.7183098591 549295775}, -\right. -\\ -\\ -\displaystyle -{2.7182817182 817182817}, {2.7182818287 356957267}, -\\ -\\ -\displaystyle -\left. -{2.7182818284\ 585634113}, \ldots -\right] -\end{array} -$$ -\returnType{Type: Stream Float} - -Compare this to the value of {\tt e} computed by the -\spadfunFrom{exp}{Float} operation in {\tt Float}. - -\spadcommand{exp 1.0} -$$ -2.7182818284\ 590452354 -$$ -\returnType{Type: Float} - -In about 1658, Lord Brouncker established the following expansion -for $4 / \pi$, -\begin{verbatim} - 1 + 1 - ----------------------- - 2 + 9 - ------------------- - 2 + 25 - --------------- - 2 + 49 - ----------- - 2 + 81 - ------- - 2 + ... -\end{verbatim} - -Let's use this expansion to compute rational and floating point -approximations for $\pi$. - -\spadcommand{cf := continuedFraction(1,[(2*i+1)**2 for i in 0..],repeating [2])} -$$ -1+ \zag{1}{2}+ \zag{9}{2}+ \zag{{25}}{2}+ \zag{{49}}{2}+ \zag{{81}}{2}+ -\zag{{121}}{2}+ \zag{{169}}{2}+\ldots -$$ -\returnType{Type: ContinuedFraction Integer} - -\spadcommand{ccf := convergents cf } -$$ -\left[ -1, {3 \over 2}, {{15} \over {13}}, {{105} \over {76}}, {{315} -\over {263}}, {{3465} \over {2578}}, {{45045} \over {36979}}, \ldots -\right] -$$ -\returnType{Type: Stream Fraction Integer} - -\spadcommand{piConvergents := [4/p for p in ccf] } -$$ -\left[ -4, {8 \over 3}, {{52} \over {15}}, {{304} \over {105}}, {{1052} -\over {315}}, {{10312} \over {3465}}, {{147916} \over {45045}}, -\ldots -\right] -$$ -\returnType{Type: Stream Fraction Integer} - -As you can see, the values are converging to -$\pi = 3.14159265358979323846...$, -but not very quickly. - -\spadcommand{piConvergents :: Stream Float } -$$ -\begin{array}{@{}l} -\left[ -{4.0}, {2.6666666666\ 666666667}, {3.4666666666\ 666666667}, -\right. -\\ -\\ -\displaystyle -{2.8952380952\ 380952381}, {3.3396825396\ 825396825}, -\\ -\\ -\displaystyle -\left. -{2.9760461760\ 461760462}, {3.2837384837\ 384837385}, \ldots -\right] -\end{array} -$$ -\returnType{Type: Stream Float} - -You need not restrict yourself to continued fractions of integers. -Here is an expansion for a quotient of Gaussian integers. - -\spadcommand{continuedFraction((- 122 + 597*\%i)/(4 - 4*\%i))} -$$ --{90}+{{59} \ i}+ \zag{1}{{1 -{2 \ i}}}+ \zag{1}{{-1+{2 \ i}}} -$$ -\returnType{Type: ContinuedFraction Complex Integer} - -This is an expansion for a quotient of polynomials in one variable -with rational number coefficients. - -\spadcommand{r : Fraction UnivariatePolynomial(x,Fraction Integer) } -\returnType{Type: Void} - -\spadcommand{r := ((x - 1) * (x - 2)) / ((x-3) * (x-4)) } -$$ -{{x \sp 2} -{3 \ x}+2} \over {{x \sp 2} -{7 \ x}+{12}} -$$ -\returnType{Type: Fraction UnivariatePolynomial(x,Fraction Integer)} - -\spadcommand{continuedFraction r } -$$ -1+ \zag{1}{{{{1 \over 4} \ x} -{9 \over 8}}}+ \zag{1}{{{{{16} \over 3} \ x} --{{40} \over 3}}} -$$ -\returnType{Type: ContinuedFraction UnivariatePolynomial(x,Fraction Integer)} - -To conclude this section, we give you evidence that -\begin{verbatim} - z = 3 + 1 - ----------------------- - 3 + 1 - ------------------- - 6 + 1 - --------------- - 3 + 1 - ----------- - 6 + 1 - ------- - 3 + ... -\end{verbatim} - -is the expansion of $\sqrt{11}$. - -\spadcommand{[i*i for i in convergents(z) :: Stream Float] } -$$ -\begin{array}{@{}l} -\left[ -{9.0}, {11.1111111111\ 11111111}, {10.9944598337\ 9501385}, -\right. -\\ -\\ -\displaystyle -{11.0002777777\ 77777778}, {10.9999860763\ 98799786}, -\\ -\\ -\displaystyle -\left. -{11.0000006979\ 29731039}, {10.9999999650\ 15834446}, \ldots -\right] -\end{array} -$$ -\returnType{Type: Stream Float} - -\section{CycleIndicators} -\label{CycleIndicatorsXmpPage} - -This section is based upon the paper J. H. Redfield, ``The Theory of -Group-Reduced Distributions'', American J. Math.,49 (1927) 433-455, -and is an application of group theory to enumeration problems. It is -a development of the work by P. A. MacMahon on the application of -symmetric functions and Hammond operators to combinatorial theory. - -The theory is based upon the power sum symmetric functions -$s_i$ which are the sum of the $i$-th powers of the -variables. The cycle index of a permutation is an expression that -specifies the sizes of the cycles of a permutation, and may be -represented as a partition. A partition of a non-negative integer -{\tt n} is a collection of positive integers called its parts whose -sum is {\tt n}. For example, the partition $(3^2 \ 2 \ 1^2)$ will be -used to represent $s^2_3 s_2 s^2_1$ and will indicate that the -permutation has two cycles of length 3, one of length 2 and two of -length 1. The cycle index of a permutation group is the sum of the -cycle indices of its permutations divided by the number of -permutations. The cycle indices of certain groups are provided. - -The operation {\tt complete} returns the cycle index of the -symmetric group of order {\tt n} for argument {\tt n}. -Alternatively, it is the $n$-th complete homogeneous symmetric -function expressed in terms of power sum symmetric functions. - -\spadcommand{complete 1} -$$ -\left( -1 -\right) -$$ -\returnType{Type: SymmetricPolynomial Fraction Integer} - -\spadcommand{complete 2} -$$ -{{1 \over 2} \ {\left( 2 -\right)}}+{{1 -\over 2} \ {\left( 1 \sp 2 -\right)}} -$$ -\returnType{Type: SymmetricPolynomial Fraction Integer} - -\spadcommand{complete 3} -$$ -{{1 \over 3} \ {\left( 3 -\right)}}+{{1 -\over 2} \ {\left( {2 \sp {\ }} \ 1 -\right)}}+{{1 -\over 6} \ {\left( 1 \sp 3 -\right)}} -$$ -\returnType{Type: SymmetricPolynomial Fraction Integer} - -\spadcommand{complete 7} -$$ -\begin{array}{@{}l} -{{1 \over 7} \ {\left( 7 \right)}}+ -{{1\over 6} \ {\left( {6 \sp {\ }} \ 1 \right)}}+ -{{1\over {10}} \ {\left( {5 \sp {\ }} \ 2 \right)}}+ -{{1\over {10}} \ {\left( {5 \sp {\ }} \ {1 \sp 2} \right)}}+ -{{1\over {12}} \ {\left( {4 \sp {\ }} \ 3 \right)}}+ -{{1\over 8} \ {\left( {4 \sp {\ }} \ {2 \sp {\ }} \ 1 \right)}}+ -\\ -\\ -\displaystyle -{{1\over {24}} \ {\left( {4 \sp {\ }} \ {1 \sp 3} \right)}}+ -{{1\over {18}} \ {\left( {3 \sp 2} \ 1 \right)}}+ -{{1\over {24}} \ {\left( {3 \sp {\ }} \ {2 \sp 2} \right)}}+ -{{1\over {12}} \ {\left( {3 \sp {\ }} \ {2 \sp {\ }} \ {1 \sp 2} \right)}}+ -{{1\over {72}} \ {\left( {3 \sp {\ }} \ {1 \sp 4} \right)}}+ -\\ -\\ -\displaystyle -{{1\over {48}} \ {\left( {2 \sp 3} \ 1 \right)}}+ -{{1\over {48}} \ {\left( {2 \sp 2} \ {1 \sp 3} \right)}}+ -{{1\over {240}} \ {\left( {2 \sp {\ }} \ {1 \sp 5} \right)}}+ -{{1\over {5040}} \ {\left( 1 \sp 7 \right)}} -\end{array} -$$ -\returnType{Type: SymmetricPolynomial Fraction Integer} - -The operation {\tt elementary} computes the $n$-th -elementary symmetric function for argument {\tt n.} - -\spadcommand{elementary 7} -$$ -\begin{array}{@{}l} -{{1 \over 7} \ {\left( 7 \right)}} --{{1 \over 6} \ {\left( {6 \sp {\ }} \ 1 \right)}} --{{1 \over {10}} \ {\left( {5 \sp {\ }} \ 2 \right)}}+ -{{1\over {10}} \ {\left( {5 \sp {\ }} \ {1 \sp 2} \right)}} --{{1 \over {12}} \ {\left( {4 \sp {\ }} \ 3 \right)}}+ -{{1\over 8} \ {\left( {4 \sp {\ }} \ {2 \sp {\ }} \ 1 \right)}} -\\ -\\ -\displaystyle --{{1 \over {24}} \ {\left( {4 \sp {\ }} \ {1 \sp 3} \right)}}+ -{{1\over {18}} \ {\left( {3 \sp 2} \ 1 \right)}}+ -{{1\over {24}} \ {\left( {3 \sp {\ }} \ {2 \sp 2} \right)}} --{{1 \over {12}} \ {\left( {3 \sp {\ }} \ {2 \sp {\ }} \ {1 \sp 2} \right)}} -+{{1\over {72}} \ {\left( {3 \sp {\ }} \ {1 \sp 4} \right)}} -\\ -\\ -\displaystyle --{{1 \over {48}} \ {\left( {2 \sp 3} \ 1 \right)}}+ -{{1\over {48}} \ {\left( {2 \sp 2} \ {1 \sp 3} \right)}} --{{1 \over {240}} \ {\left( {2 \sp {\ }} \ {1 \sp 5} \right)}}+ -{{1\over {5040}} \ {\left( 1 \sp 7 \right)}} -\end{array} -$$ -\returnType{Type: SymmetricPolynomial Fraction Integer} - -The operation {\tt alternating} returns the cycle index of the alternating -group having an even number of even parts in each cycle partition. - -\spadcommand{alternating 7} -$$ -\begin{array}{@{}l} -{{2 \over 7} \ {\left( 7 \right)}}+ -{{1\over 5} \ {\left( {5 \sp {\ }} \ {1 \sp 2} \right)}}+ -{{1\over 4} \ {\left( {4 \sp {\ }} \ {2 \sp {\ }} \ 1 \right)}}+ -{{1\over 9} \ {\left( {3 \sp 2} \ 1 \right)}}+ -{{1\over {12}} \ {\left( {3 \sp {\ }} \ {2 \sp 2} \right)}}+ -{{1\over {36}} \ {\left( {3 \sp {\ }} \ {1 \sp 4} \right)}}+ -\\ -\\ -\displaystyle -{{1\over {24}} \ {\left( {2 \sp 2} \ {1 \sp 3} \right)}}+ -{{1\over {2520}} \ {\left( 1 \sp 7 \right)}} -\end{array} -$$ -\returnType{Type: SymmetricPolynomial Fraction Integer} - -The operation {\tt cyclic} returns the cycle index of the cyclic group. - -\spadcommand{cyclic 7} -$$ -{{6 \over 7} \ {\left( 7 \right)}}+ -{{1\over 7} \ {\left( 1 \sp 7 \right)}} -$$ -\returnType{Type: SymmetricPolynomial Fraction Integer} - -The operation {\tt dihedral} is the cycle index of the -dihedral group. - -\spadcommand{dihedral 7} -$$ -{{3 \over 7} \ {\left( 7 \right)}}+ -{{1\over 2} \ {\left( {2 \sp 3} \ 1 \right)}}+ -{{1\over {14}} \ {\left( 1 \sp 7 \right)}} -$$ -\returnType{Type: SymmetricPolynomial Fraction Integer} - -The operation {\tt graphs} for argument {\tt n} returns the cycle -index of the group of permutations on the edges of the complete graph -with {\tt n} nodes induced by applying the symmetric group to the -nodes. - -\spadcommand{graphs 5} -$$ -\begin{array}{@{}l} -{{1 \over 6} \ {\left( {6 \sp {\ }} \ {3 \sp {\ }} \ 1 \right)}}+ -{{1\over 5} \ {\left( 5 \sp 2 \right)}}+ -{{1\over 4} \ {\left( {4 \sp 2} \ 2 \right)}}+ -{{1\over 6} \ {\left( {3 \sp 3} \ 1 \right)}}+ -{{1\over 8} \ {\left( {2 \sp 4} \ {1 \sp 2} \right)}}+ -\\ -\\ -\displaystyle -{{1\over {12}} \ {\left( {2 \sp 3} \ {1 \sp 4} \right)}}+ -{{1\over {120}} \ {\left( 1 \sp {10} \right)}} -\end{array} -$$ -\returnType{Type: SymmetricPolynomial Fraction Integer} - -The cycle index of a direct product of two groups is the product of -the cycle indices of the groups. Redfield provided two operations on -two cycle indices which will be called ``cup'' and ``cap'' here. The -{\tt cup} of two cycle indices is a kind of scalar product that -combines monomials for permutations with the same cycles. The {\tt -cap} operation provides the sum of the coefficients of the result of -the {\tt cup} operation which will be an integer that enumerates what -Redfield called group-reduced distributions. - -We can, for example, represent {\tt complete 2 * complete 2} as the -set of objects {\tt a a b b} and -{\tt complete 2 * complete 1 * complete 1} as {\tt c c d e.} - -This integer is the number of different sets of four pairs. - -\spadcommand{cap(complete 2**2, complete 2*complete 1**2)} -$$ -4 -$$ -\returnType{Type: Fraction Integer} - -For example, -\begin{verbatim} -a a b b a a b b a a b b a a b b -c c d e c d c e c e c d d e c c -\end{verbatim} - -This integer is the number of different sets of four pairs no two -pairs being equal. - -\spadcommand{cap(elementary 2**2, complete 2*complete 1**2)} -$$ -2 -$$ -\returnType{Type: Fraction Integer} - -For example, -\begin{verbatim} -a a b b a a b b -c d c e c e c d -\end{verbatim} -In this case the configurations enumerated are easily constructed, -however the theory merely enumerates them providing little help in -actually constructing them. - -Here are the number of 6-pairs, first from {\tt a a a b b c,} second -from {\tt d d e e f g.} - -\spadcommand{cap(complete 3*complete 2*complete 1,complete 2**2*complete 1**2)} -$$ -24 -$$ -\returnType{Type: Fraction Integer} - -Here it is again, but with no equal pairs. - -\spadcommand{cap(elementary 3*elementary 2*elementary 1,complete 2**2*complete 1**2)} -$$ -8 -$$ -\returnType{Type: Fraction Integer} - -\spadcommand{cap(complete 3*complete 2*complete 1,elementary 2**2*elementary 1**2)} -$$ -8 -$$ -\returnType{Type: Fraction Integer} - -The number of 6-triples, first from {\tt a a a b b c,} second from -{\tt d d e e f g,} third from {\tt h h i i j j.} - -\spadcommand{eval(cup(complete 3*complete 2*complete 1, cup(complete 2**2*complete 1**2,complete 2**3)))} -$$ -1500 -$$ -\returnType{Type: Fraction Integer} - -The cycle index of vertices of a square is dihedral 4. - -\spadcommand{square:=dihedral 4} -$$ -{{1 \over 4} \ {\left( 4 \right)}}+ -{{3\over 8} \ {\left( 2 \sp 2 \right)}}+ -{{1\over 4} \ {\left( {2 \sp {\ }} \ {1 \sp 2} \right)}}+ -{{1\over 8} \ {\left( 1 \sp 4 \right)}} -$$ -\returnType{Type: SymmetricPolynomial Fraction Integer} - -The number of different squares with 2 red vertices and 2 blue vertices. - -\spadcommand{cap(complete 2**2,square)} -$$ -2 -$$ -\returnType{Type: Fraction Integer} - -The number of necklaces with 3 red beads, 2 blue beads and 2 green beads. - -\spadcommand{cap(complete 3*complete 2**2,dihedral 7)} -$$ -18 -$$ -\returnType{Type: Fraction Integer} - -The number of graphs with 5 nodes and 7 edges. - -\spadcommand{cap(graphs 5,complete 7*complete 3)} -$$ -4 -$$ -\returnType{Type: Fraction Integer} - -The cycle index of rotations of vertices of a cube. - -\spadcommand{s(x) == powerSum(x)} -\returnType{Type: Void} - -\spadcommand{cube:=(1/24)*(s 1**8+9*s 2**4 + 8*s 3**2*s 1**2+6*s 4**2)} -\begin{verbatim} - Compiling function s with type PositiveInteger -> - SymmetricPolynomial Fraction Integer -\end{verbatim} -$$ -{{1 \over 4} \ {\left( 4 \sp 2 \right)}}+ -{{1\over 3} \ {\left( {3 \sp 2} \ {1 \sp 2} \right)}}+ -{{3\over 8} \ {\left( 2 \sp 4 \right)}}+ -{{1\over {24}} \ {\left( 1 \sp 8 \right)}} -$$ -\returnType{Type: SymmetricPolynomial Fraction Integer} - -The number of cubes with 4 red vertices and 4 blue vertices. - -\spadcommand{cap(complete 4**2,cube)} -$$ -7 -$$ -\returnType{Type: Fraction Integer} - -The number of labeled graphs with degree sequence {\tt 2 2 2 1 1} -with no loops or multiple edges. - -\spadcommand{cap(complete 2**3*complete 1**2,wreath(elementary 4,elementary 2))} -$$ -7 -$$ -\returnType{Type: Fraction Integer} - -Again, but with loops allowed but not multiple edges. - -\spadcommand{cap(complete 2**3*complete 1**2,wreath(elementary 4,complete 2))} -$$ -17 -$$ -\returnType{Type: Fraction Integer} - -Again, but with multiple edges allowed, but not loops - -\spadcommand{cap(complete 2**3*complete 1**2,wreath(complete 4,elementary 2))} -$$ -10 -$$ -\returnType{Type: Fraction Integer} - -Again, but with both multiple edges and loops allowed - -\spadcommand{cap(complete 2**3*complete 1**2,wreath(complete 4,complete 2))} -$$ -23 -$$ -\returnType{Type: Fraction Integer} - -Having constructed a cycle index for a configuration we are at liberty -to evaluate the $s_i$ components any way we please. For example we -can produce enumerating generating functions. This is done by -providing a function {\tt f} on an integer {\tt i} to the value -required of $s_i$, and then evaluating {\tt eval(f, cycleindex)}. - -\spadcommand{x: ULS(FRAC INT,'x,0) := 'x } -$$ -x -$$ -\returnType{Type: UnivariateLaurentSeries(Fraction Integer,x,0)} - -\spadcommand{ZeroOrOne: INT -> ULS(FRAC INT, 'x, 0) } -\returnType{Type: Void} - -\spadcommand{Integers: INT -> ULS(FRAC INT, 'x, 0) } -\returnType{Type: Void} - -For the integers 0 and 1, or two colors. - -\spadcommand{ZeroOrOne n == 1+x**n } -\returnType{Type: Void} - -\spadcommand{ZeroOrOne 5 } -\begin{verbatim} - Compiling function ZeroOrOne with type Integer -> - UnivariateLaurentSeries(Fraction Integer,x,0) -\end{verbatim} -$$ -1+{x \sp 5} -$$ -\returnType{Type: UnivariateLaurentSeries(Fraction Integer,x,0)} - -For the integers {\tt 0, 1, 2, ...} we have this. - -\spadcommand{Integers n == 1/(1-x**n) } -\returnType{Type: Void} - -\spadcommand{Integers 5 } -\begin{verbatim} - Compiling function Integers with type Integer -> - UnivariateLaurentSeries(Fraction Integer,x,0) -\end{verbatim} -$$ -1+{x \sp 5}+{O -\left( -{{x \sp 8}} -\right)} -$$ -\returnType{Type: UnivariateLaurentSeries(Fraction Integer,x,0)} - -The coefficient of $x^n$ is the number of graphs with 5 nodes and {\tt n} -edges. - -Note that there is an eval function that takes two arguments. It has the -signature: -\begin{verbatim} -((Integer -> D1),SymmetricPolynomial Fraction Integer) -> D1 - from EvaluateCycleIndicators D1 if D1 has ALGEBRA FRAC INT -\end{verbatim} -This function is not normally exposed (it will not normally be considered -in the list of eval functions) as it is only useful for this particular -domain. To use it we ask that it be considered thus: - -\spadcommand{)expose EVALCYC} - -and now we can use it: - -\spadcommand{eval(ZeroOrOne, graphs 5) } -$$ -1+x+ -{2 \ {x \sp 2}}+ -{4 \ {x \sp 3}}+ -{6 \ {x \sp 4}}+ -{6 \ {x \sp 5}}+ -{6 \ {x \sp 6}}+ -{4 \ {x \sp 7}}+ -{O \left({{x \sp 8}} \right)} -$$ -\returnType{Type: UnivariateLaurentSeries(Fraction Integer,x,0)} - -The coefficient of $x^n$ is the number of necklaces with -{\tt n} red beads and {\tt n-8} green beads. - -\spadcommand{eval(ZeroOrOne,dihedral 8) } -$$ -1+x+ -{4 \ {x \sp 2}}+ -{5 \ {x \sp 3}}+ -{8 \ {x \sp 4}}+ -{5 \ {x \sp 5}}+ -{4 \ {x \sp 6}}+ -{x \sp 7}+ -{O \left({{x \sp 8}} \right)} -$$ -\returnType{Type: UnivariateLaurentSeries(Fraction Integer,x,0)} - -The coefficient of $x^n$ is the number of partitions of {\tt n} into 4 -or fewer parts. - -\spadcommand{eval(Integers,complete 4) } -$$ -1+x+ -{2 \ {x \sp 2}}+ -{3 \ {x \sp 3}}+ -{5 \ {x \sp 4}}+ -{6 \ {x \sp 5}}+ -{9 \ {x \sp 6}}+ -{{11} \ {x \sp 7}}+ -{O \left({{x \sp 8}} \right)} -$$ -\returnType{Type: UnivariateLaurentSeries(Fraction Integer,x,0)} - -The coefficient of $x^n$ is the number of partitions of {\tt n} into 4 -boxes containing ordered distinct parts. - -\spadcommand{eval(Integers,elementary 4) } -$$ -{x \sp 6}+ -{x \sp 7}+ -{2 \ {x \sp 8}}+ -{3 \ {x \sp 9}}+ -{5 \ {x \sp {10}}}+ -{6 \ {x \sp {11}}}+ -{9 \ {x \sp {12}}}+ -{{11} \ {x \sp {13}}}+ -{O \left({{x \sp {14}}} \right)} -$$ -\returnType{Type: UnivariateLaurentSeries(Fraction Integer,x,0)} - -The coefficient of $x^n$ is the number of different cubes with {\tt n} -red vertices and {\tt 8-n} green ones. - -\spadcommand{eval(ZeroOrOne,cube) } -$$ -1+x+ -{3 \ {x \sp 2}}+ -{3 \ {x \sp 3}}+ -{7 \ {x \sp 4}}+ -{3 \ {x \sp 5}}+ -{3 \ {x \sp 6}}+ -{x \sp 7}+ -{O \left({{x \sp 8}} \right)} -$$ -\returnType{Type: UnivariateLaurentSeries(Fraction Integer,x,0)} - -The coefficient of $x^n$ is the number of different cubes with integers -on the vertices whose sum is {\tt n.} - -\spadcommand{eval(Integers,cube) } -$$ -1+x+ -{4 \ {x \sp 2}}+ -{7 \ {x \sp 3}}+ -{{21} \ {x \sp 4}}+ -{{37} \ {x \sp 5}}+ -{{85} \ {x \sp 6}}+ -{{151} \ {x \sp 7}}+ -{O \left({{x \sp 8}} \right)} -$$ -\returnType{Type: UnivariateLaurentSeries(Fraction Integer,x,0)} - -The coefficient of $x^n$ is the number of graphs with 5 nodes and with -integers on the edges whose sum is {\tt n.} In other words, the -enumeration is of multigraphs with 5 nodes and {\tt n} edges. - -\spadcommand{eval(Integers,graphs 5) } -$$ -1+x+ -{3 \ {x \sp 2}}+ -{7 \ {x \sp 3}}+ -{{17} \ {x \sp 4}}+ -{{35} \ {x \sp 5}}+ -{{76} \ {x \sp 6}}+ -{{149} \ {x \sp 7}}+ -{O \left({{x \sp 8}} \right)} -$$ -\returnType{Type: UnivariateLaurentSeries(Fraction Integer,x,0)} - -Graphs with 15 nodes enumerated with respect to number of edges. - -\spadcommand{eval(ZeroOrOne ,graphs 15) } -$$ -1+x+ -{2 \ {x \sp 2}}+ -{5 \ {x \sp 3}}+ -{{11} \ {x \sp 4}}+ -{{26} \ {x \sp 5}}+ -{{68} \ {x \sp 6}}+ -{{177} \ {x \sp 7}}+ -{O \left({{x \sp 8}} \right)} -$$ -\returnType{Type: UnivariateLaurentSeries(Fraction Integer,x,0)} - -Necklaces with 7 green beads, 8 white beads, 5 yellow beads and 10 -red beads. - -\spadcommand{cap(dihedral 30,complete 7*complete 8*complete 5*complete 10)} -$$ -49958972383320 -$$ -\returnType{Type: Fraction Integer} - -The operation {\tt SFunction} is the S-function or Schur function of a -partition written as a descending list of integers expressed in terms -of power sum symmetric functions. - -In this case the argument partition represents a tableau shape. For -example {\tt 3,2,2,1} represents a tableau with three boxes in the -first row, two boxes in the second and third rows, and one box in the -fourth row. {\tt SFunction [3,2,2,1]} counts the number of different -tableaux of shape {\tt 3, 2, 2, 1} filled with objects with an -ascending order in the columns and a non-descending order in the rows. - -\spadcommand{sf3221:= SFunction [3,2,2,1] } -$$ -\begin{array}{@{}l} -{{1 \over {12}} \ {\left( {6 \sp {\ }} \ 2 \right)}} --{{1 \over {12}} \ {\left( {6 \sp {\ }} \ {1 \sp 2} \right)}} --{{1 \over {16}} \ {\left( 4 \sp 2 \right)}}+ -{{1\over {12}} \ {\left( {4 \sp {\ }} \ {3 \sp {\ }} \ 1 \right)}}+ -{{1\over {24}} \ {\left( {4 \sp {\ }} \ {1 \sp 4} \right)}} --{{1 \over {36}} \ {\left( {3 \sp 2} \ 2 \right)}}+ -\\ -\\ -\displaystyle -{{1\over {36}} \ {\left( {3 \sp 2} \ {1 \sp 2} \right)}} --{{1 \over {24}} \ {\left( {3 \sp {\ }} \ {2 \sp 2} \ 1 \right)}} --{{1 \over {36}} \ {\left( {3 \sp {\ }} \ {2 \sp {\ }} \ {1 \sp 3} \right)}} --{{1 \over {72}} \ {\left( {3 \sp {\ }} \ {1 \sp 5} \right)}} --{{1 \over {192}} \ {\left( 2 \sp 4 \right)}}+ -\\ -\\ -\displaystyle -{{1\over {48}} \ {\left( {2 \sp 3} \ {1 \sp 2} \right)}}+ -{{1\over {96}} \ {\left( {2 \sp 2} \ {1 \sp 4} \right)}} --{{1 \over {144}} \ {\left( {2 \sp {\ }} \ {1 \sp 6} \right)}}+ -{{1\over {576}} \ {\left( 1 \sp 8 \right)}} -\end{array} -$$ -\returnType{Type: SymmetricPolynomial Fraction Integer} - -This is the number filled with {\tt a a b b c c d d.} - -\spadcommand{cap(sf3221,complete 2**4) } -$$ -3 -$$ -\returnType{Type: Fraction Integer} - -The configurations enumerated above are: -\begin{verbatim} -a a b a a c a a d -b c b b b b -c d c d c c -d d d -\end{verbatim} - -This is the number of tableaux filled with {\tt 1..8.} - -\spadcommand{cap(sf3221, powerSum 1**8)} -$$ -70 -$$ -\returnType{Type: Fraction Integer} - -The coefficient of $x^n$ is the number of column strict reverse plane -partitions of {\tt n} of shape {\tt 3 2 2 1.} - -\spadcommand{eval(Integers, sf3221)} -$$ -{x \sp 9}+ -{3 \ {x \sp {10}}}+ -{7 \ {x \sp {11}}}+ -{{14} \ {x \sp {12}}}+ -{{27} \ {x \sp {13}}}+ -{{47} \ {x \sp {14}}}+ -{O \left({{x \sp {15}}} \right)} -$$ -\returnType{Type: UnivariateLaurentSeries(Fraction Integer,x,0)} - -The smallest is -\begin{verbatim} -0 0 0 -1 1 -2 2 -3 -\end{verbatim} - -\section{DeRhamComplex} -\label{DeRhamComplexXmpPage} - -The domain constructor {\tt DeRhamComplex} creates the class of -differential forms of arbitrary degree over a coefficient ring. The -De Rham complex constructor takes two arguments: a ring, {\tt -coefRing,} and a list of coordinate variables. - -This is the ring of coefficients. - -\spadcommand{coefRing := Integer } -$$ -Integer -$$ -\returnType{Type: Domain} - -These are the coordinate variables. - -\spadcommand{lv : List Symbol := [x,y,z] } -$$ -\left[ -x, y, z -\right] -$$ -\returnType{Type: List Symbol} - -This is the De Rham complex of Euclidean three-space using coordinates -{\tt x, y} and {\tt z.} - -\spadcommand{der := DERHAM(coefRing,lv) } -$$ -DeRhamComplex(Integer,[x,y,z]) -$$ -\returnType{Type: Domain} - -This complex allows us to describe differential forms having -expressions of integers as coefficients. These coefficients can -involve any number of variables, for example, {\tt f(x,t,r,y,u,z).} -As we've chosen to work with ordinary Euclidean three-space, -expressions involving these forms are treated as functions of -{\tt x, y} and {\tt z} with the additional arguments {\tt t, r} -and {\tt u} regarded as symbolic constants. - -Here are some examples of coefficients. - -\spadcommand{R := Expression coefRing } -$$ -\mbox{\rm Expression Integer} -$$ -\returnType{Type: Domain} - -\spadcommand{f : R := x**2*y*z-5*x**3*y**2*z**5 } -$$ --{5 \ {x \sp 3} \ {y \sp 2} \ {z \sp 5}}+{{x \sp 2} \ y \ z} -$$ -\returnType{Type: Expression Integer} - -\spadcommand{g : R := z**2*y*cos(z)-7*sin(x**3*y**2)*z**2 } -$$ --{7 \ {z \sp 2} \ {\sin \left({{{x \sp 3} \ {y \sp 2}}} \right)}}+ -{y\ {z \sp 2} \ {\cos \left({z} \right)}} -$$ -\returnType{Type: Expression Integer} - -\spadcommand{h : R :=x*y*z-2*x**3*y*z**2 } -$$ --{2 \ {x \sp 3} \ y \ {z \sp 2}}+{x \ y \ z} -$$ -\returnType{Type: Expression Integer} - -We now define the multiplicative basis elements for the exterior -algebra over {\tt R}. - -\spadcommand{dx : der := generator(1) } -$$ -dx -$$ -\returnType{Type: DeRhamComplex(Integer,[x,y,z])} - -\spadcommand{dy : der := generator(2)} -$$ -dy -$$ -\returnType{Type: DeRhamComplex(Integer,[x,y,z])} - -\spadcommand{dz : der := generator(3)} -$$ -dz -$$ -\returnType{Type: DeRhamComplex(Integer,[x,y,z])} - -This is an alternative way to give the above assignments. - -\spadcommand{[dx,dy,dz] := [generator(i)\$der for i in 1..3] } -$$ -\left[ -dx, dy, dz -\right] -$$ -\returnType{Type: List DeRhamComplex(Integer,[x,y,z])} - -Now we define some one-forms. - -\spadcommand{alpha : der := f*dx + g*dy + h*dz } -$$ -\begin{array}{@{}l} -{{\left( -{2 \ {x \sp 3} \ y \ {z \sp 2}}+{x \ y \ z} \right)}\ dz}+ -\\ -\\ -\displaystyle -{{\left( --{7 \ {z \sp 2} \ {\sin \left({{{x \sp 3} \ {y \sp 2}}} \right)}}+ -{y\ {z \sp 2} \ {\cos \left({z} \right)}} -\right)}\ dy}+ -\\ -\\ -\displaystyle -{{\left( -{5 \ {x \sp 3} \ {y \sp 2} \ {z \sp 5}}+{{x \sp 2} \ y \ z} -\right)}\ dx} -\end{array} -$$ -\returnType{Type: DeRhamComplex(Integer,[x,y,z])} - -\spadcommand{beta : der := cos(tan(x*y*z)+x*y*z)*dx + x*dy } -$$ -{x \ dy}+ -{{\cos -\left({{{\tan \left({{x \ y \ z}} \right)}+{x\ y \ z}}} \right)} -\ dx} -$$ -\returnType{Type: DeRhamComplex(Integer,[x,y,z])} - -A well-known theorem states that the composition of -\spadfunFrom{exteriorDifferential}{DeRhamComplex} with itself is the -zero map for continuous forms. Let's verify this theorem for {\tt alpha}. - -\spadcommand{exteriorDifferential alpha } -$$ -\begin{array}{@{}l} -{{\left( -{y \ {z \sp 2} \ {\sin \left({z} \right)}}+ -{{14}\ z \ {\sin \left({{{x \sp 3} \ {y \sp 2}}} \right)}} --{2 \ y \ z \ {\cos \left({z} \right)}} --{2 \ {x \sp 3} \ {z \sp 2}}+{x \ z} \right)}\ dy \ dz}+ -\\ -\\ -\displaystyle -{{\left( -{{25} \ {x \sp 3} \ {y \sp 2} \ {z \sp 4}} - -{6 \ {x \sp 2} \ y \ {z \sp 2}}+ -{y \ z} - -{{x \sp 2} \ y} -\right)}\ dx \ dz}+ -\\ -\\ -\displaystyle -{{\left( --{{21} \ {x \sp 2} \ {y \sp 2} \ {z \sp 2} \ -{\cos \left({{{x \sp 3} \ {y \sp 2}}} \right)}}+ -{{10}\ {x \sp 3} \ y \ {z \sp 5}} -{{x \sp 2} \ z} \right)}\ dx \ dy} -\end{array} -$$ -\returnType{Type: DeRhamComplex(Integer,[x,y,z])} - -We see a lengthy output of the last expression, but nevertheless, the -composition is zero. - -\spadcommand{exteriorDifferential \% } -$$ -0 -$$ -\returnType{Type: DeRhamComplex(Integer,[x,y,z])} - -Now we check that \spadfunFrom{exteriorDifferential}{DeRhamComplex} -is a ``graded derivation'' {\tt D,} that is, {\tt D} satisfies: -\begin{verbatim} -D(a*b) = D(a)*b + (-1)**degree(a)*a*D(b) -\end{verbatim} - -\spadcommand{gamma := alpha * beta } -$$ -\begin{array}{@{}l} -{{\left( -{2 \ {x \sp 4} \ y \ {z \sp 2}} - -{{x \sp 2} \ y \ z} -\right)}\ dy \ dz}+ -\\ -\\ -\displaystyle -{{\left( -{2 \ {x \sp 3} \ y \ {z \sp 2}} -{x \ y \ z} -\right)} -\ {\cos -\left( -{{{\tan \left({{x \ y \ z}} \right)}+ -{x\ y \ z}}} -\right)}\ dx \ dz}+ -\\ -\\ -\displaystyle -\left( -{{\left( -{7 \ {z \sp 2} \ {\sin \left({{{x \sp 3} \ {y \sp 2}}} \right)}} --{y \ {z \sp 2} \ {\cos \left({z} \right)}} -\right)} -\ {\cos \left({{{\tan \left({{x \ y \ z}} \right)}+{x\ y \ z}}} \right)}}- -\right. -\\ -\\ -\left. -\displaystyle -{5 \ {x \sp 4} \ {y \sp 2} \ {z \sp 5}}+ -{{x \sp 3} \ y \ z} -\right)\ dx \ dy -\end{array} -$$ -\returnType{Type: DeRhamComplex(Integer,[x,y,z])} - -We try this for the one-forms {\tt alpha} and {\tt beta}. - -\spadcommand{exteriorDifferential(gamma) - (exteriorDifferential(alpha)*beta - alpha * exteriorDifferential(beta)) } -$$ -0 -$$ -\returnType{Type: DeRhamComplex(Integer,[x,y,z])} - -Now we define some ``basic operators'' (see -\ref{OperatorXmpPage} on page~\pageref{OperatorXmpPage}). - -\spadcommand{a : BOP := operator('a) } -$$ -a -$$ -\returnType{Type: BasicOperator} - -\spadcommand{b : BOP := operator('b) } -$$ -b -$$ -\returnType{Type: BasicOperator} - -\spadcommand{c : BOP := operator('c) } -$$ -c -$$ -\returnType{Type: BasicOperator} - -We also define some indeterminate one- and two-forms using these -operators. - -\spadcommand{sigma := a(x,y,z) * dx + b(x,y,z) * dy + c(x,y,z) * dz } -$$ -{{c \left({x, y, z} \right)}\ dz}+ -{{b \left({x, y, z} \right)}\ dy}+ -{{a \left({x, y, z} \right)}\ dx} -$$ -\returnType{Type: DeRhamComplex(Integer,[x,y,z])} - -\spadcommand{theta := a(x,y,z) * dx * dy + b(x,y,z) * dx * dz + c(x,y,z) * dy * dz } -$$ -{{c \left({x, y, z} \right)}\ dy \ dz}+ -{{b \left({x, y, z} \right)}\ dx \ dz}+ -{{a \left({x, y, z} \right)}\ dx \ dy} -$$ -\returnType{Type: DeRhamComplex(Integer,[x,y,z])} - -This allows us to get formal definitions for the ``gradient'' \ldots - -\spadcommand{totalDifferential(a(x,y,z))\$der } -$$ -{{{a \sb {{,3}}} \left({x, y, z} \right)}\ dz}+ -{{{a \sb {{,2}}} \left({x, y, z} \right)}\ dy}+ -{{{a \sb {{,1}}} \left({x, y, z} \right)}\ dx} -$$ -\returnType{Type: DeRhamComplex(Integer,[x,y,z])} - -the ``curl'' \ldots - -\spadcommand{exteriorDifferential sigma } -$$ -\begin{array}{@{}l} -{{\left( -{{c \sb {{,2}}} \left({x, y, z} \right)} --{{b \sb {{,3}}} \left({x, y, z} \right)} -\right)}\ dy \ dz}+ -\\ -\\ -\displaystyle -{{\left( -{{c \sb {{,1}}} \left({x, y, z} \right)} --{{a \sb {{,3}}} \left({x, y, z} \right)} -\right)}\ dx \ dz}+ -\\ -\\ -\displaystyle -{{\left( -{{b \sb {{,1}}} \left({x, y, z} \right)} --{{a \sb {{,2}}} \left({x, y, z} \right)} -\right)}\ dx \ dy} -\end{array} -$$ -\returnType{Type: DeRhamComplex(Integer,[x,y,z])} - -and the ``divergence.'' - -\spadcommand{exteriorDifferential theta } -$$ -{\left( -{{c \sb {{,1}}} \left({x, y, z} \right)} --{{b \sb {{,2}}} \left({x, y, z} \right)}+ -{{a\sb {{,3}}} \left({x, y, z} \right)} -\right)}\ dx \ dy \ dz -$$ -\returnType{Type: DeRhamComplex(Integer,[x,y,z])} - -Note that the De Rham complex is an algebra with unity. This element -{\tt 1} is the basis for elements for zero-forms, that is, functions -in our space. - -\spadcommand{one : der := 1 } -$$ -1 -$$ -\returnType{Type: DeRhamComplex(Integer,[x,y,z])} - -To convert a function to a function lying in the De Rham complex, -multiply the function by ``one.'' - -\spadcommand{g1 : der := a([x,t,y,u,v,z,e]) * one } -$$ -a -\left( -{x, t, y, u, v, z, e} -\right) -$$ -\returnType{Type: DeRhamComplex(Integer,[x,y,z])} - -A current limitation of Axiom forces you to write functions with more -than four arguments using square brackets in this way. - -\spadcommand{h1 : der := a([x,y,x,t,x,z,y,r,u,x]) * one } -$$ -a -\left( -{x, y, x, t, x, z, y, r, u, x} -\right) -$$ -\returnType{Type: DeRhamComplex(Integer,[x,y,z])} - -Now note how the system keeps track of where your coordinate functions -are located in expressions. - -\spadcommand{exteriorDifferential g1 } -$$ -\begin{array}{@{}l} -{{{a \sb {{,6}}} \left({x, t, y, u, v, z, e} \right)}\ dz}+ -\\ -\\ -\displaystyle -{{{a \sb {{,3}}} \left({x, t, y, u, v, z, e} \right)}\ dy}+ -\\ -\\ -\displaystyle -{{{a \sb {{,1}}} \left({x, t, y, u, v, z, e} \right)}\ dx} -\end{array} -$$ -\returnType{Type: DeRhamComplex(Integer,[x,y,z])} - -\spadcommand{exteriorDifferential h1 } -$$ -\begin{array}{@{}l} -{{{a \sb {{,6}}} -\left({x, y, x, t, x, z, y, r, u, x} \right)}\ dz}+ -\\ -\\ -\displaystyle -\begin{array}{@{}l} -\left( {{a \sb {{,7}}} -\left({x, y, x, t, x, z, y, r, u, x} \right)}+ -\right. -\\ -\\ -\displaystyle -\left. -{{a\sb {{,2}}} -\left({x, y, x, t, x, z, y, r, u, x} \right)} -\right)\ dy+ -\end{array} -\\ -\\ -\displaystyle -\begin{array}{@{}l} -\left( {{a \sb {{,{10}}}} -\left({x, y, x, t, x, z, y, r, u, x} \right)}+ -\right. -\\ -\\ -\displaystyle -{{a\sb {{,5}}} -\left({x, y, x, t, x, z, y, r, u, x} \right)}+ -\\ -\\ -\displaystyle -{{a\sb {{,3}}} -\left({x, y, x, t, x, z, y, r, u, x} \right)}+ -\\ -\\ -\displaystyle -\left. -{{a\sb {{,1}}} -\left({x, y, x, t, x, z, y, r, u, x} \right)} -\right)\ dx -\end{array} -\end{array} -$$ -\returnType{Type: DeRhamComplex(Integer,[x,y,z])} - -In this example of Euclidean three-space, the basis for the De Rham complex -consists of the eight forms: {\tt 1}, {\tt dx}, {\tt dy}, {\tt dz}, -{\tt dx*dy}, {\tt dx*dz}, {\tt dy*dz}, and {\tt dx*dy*dz}. - -\spadcommand{coefficient(gamma, dx*dy) } -$$ -\begin{array}{@{}l} -{{\left( {7 \ {z \sp 2} \ {\sin \left({{{x \sp 3} \ {y \sp 2}}} \right)}} --{y \ {z \sp 2} \ {\cos \left({z} \right)}}\right)}\ {\cos -\left( -{{{\tan \left({{x \ y \ z}} \right)}+ -{x\ y \ z}}} -\right)}} -\\ -\\ -\displaystyle --{5 \ {x \sp 4} \ {y \sp 2} \ {z \sp 5}}+ -{{x \sp 3} \ y \ z} -\end{array} -$$ -\returnType{Type: Expression Integer} - -\spadcommand{coefficient(gamma, one) } -$$ -0 -$$ -\returnType{Type: Expression Integer} - -\spadcommand{coefficient(g1,one) } -$$ -a -\left( -{x, t, y, u, v, z, e} -\right) -$$ -\returnType{Type: Expression Integer} - -\section{DecimalExpansion} -\label{DecimalExpansionXmpPage} - -All rationals have repeating decimal expansions. Operations to access -the individual digits of a decimal expansion can be obtained by -converting the value to {\tt RadixExpansion(10)}. More examples of -expansions are available in -\ref{BinaryExpansionXmpPage} on page~\pageref{BinaryExpansionXmpPage}, -\ref{HexadecimalExpansionXmpPage} on -page~\pageref{HexadecimalExpansionXmpPage}, and -\ref{RadixExpansionXmpPage} on page~\pageref{RadixExpansionXmpPage}. - -The operation \spadfunFrom{decimal}{DecimalExpansion} is used to create -this expansion of type {\tt DecimalExpansion}. - -\spadcommand{r := decimal(22/7) } -$$ -3.{\overline {142857}} -$$ -\returnType{Type: DecimalExpansion} - -Arithmetic is exact. - -\spadcommand{r + decimal(6/7) } -$$ -4 -$$ -\returnType{Type: DecimalExpansion} - -The period of the expansion can be short or long \ldots - -\spadcommand{[decimal(1/i) for i in 350..354] } -$$ -\begin{array}{@{}l} -\left[ -{0.{00}{\overline {285714}}}, -{0.{\overline {002849}}}, -{0.{00284}{\overline {09}}}, -{0.{\overline {00283286118980169971671388101983}}}, -\right. -\\ -\\ -\displaystyle -\left. -{0.0{\overline {0282485875706214689265536723163841807909604519774011299435}}} -\right] -\end{array} -$$ -\returnType{Type: List DecimalExpansion} - -or very long. - -\spadcommand{decimal(1/2049) } -$$ -\begin{array}{@{}l} -0.{\overline -{000488042947779404587603709126403123474865788189360663738408979990239}} -\\ -\\ -\displaystyle -\ \ {\overline{ -141044411908247925817471937530502684236212786725231820400195217179111}} -\\ -\\ -\displaystyle -\ \ {\overline{ -761835041483650561249389946315275744265495363591996095656417764763299}} -\\ -\\ -\displaystyle -\ \ {\overline{ -170326988775012201073694485114690092728160078086871644704734016593460}} -\\ -\\ -\displaystyle -\ \ {\overline{ -22449975597852611029770619814543679843826256710590531966813079551}} -\end{array} -$$ -\returnType{Type: DecimalExpansion} - -These numbers are bona fide algebraic objects. - -\spadcommand{p := decimal(1/4)*x**2 + decimal(2/3)*x + decimal(4/9) } -$$ -{{0.{25}} \ {x \sp 2}}+{{0.{\overline 6}} \ x}+{0.{\overline 4}} -$$ -\returnType{Type: Polynomial DecimalExpansion} - -\spadcommand{q := differentiate(p, x) } -$$ -{{0.5} \ x}+{0.{\overline 6}} -$$ -\returnType{Type: Polynomial DecimalExpansion} - -\spadcommand{g := gcd(p, q) } -$$ -x+{1.{\overline 3}} -$$ -\returnType{Type: Polynomial DecimalExpansion} - -\section{DistributedMultivariatePolynomial} -\label{DistributedMultivariatePolynomialXmpPage} - -\hyphenation{Homo-gen-eous-Dis-tributed-Multi-var-i-ate-Pol-y-nomial} - -{\tt DistributedMultivariatePolynomial} which is abbreviated as {\tt DMP} -and {\tt HomogeneousDistributedMultivariatePolynomial}, which is abbreviated -as {\tt HDMP}, are very similar to {\tt MultivariatePolynomial} except that -they are represented and displayed in a non-recursive manner. - -\spadcommand{(d1,d2,d3) : DMP([z,y,x],FRAC INT) } -\returnType{Type: Void} - -The constructor {\tt DMP} orders its monomials lexicographically while -{\tt HDMP} orders them by total order refined by reverse lexicographic -order. - -\spadcommand{d1 := -4*z + 4*y**2*x + 16*x**2 + 1 } -$$ --{4 \ z}+{4 \ {y \sp 2} \ x}+{{16} \ {x \sp 2}}+1 -$$ -\returnType{Type: DistributedMultivariatePolynomial([z,y,x],Fraction Integer)} - -\spadcommand{d2 := 2*z*y**2 + 4*x + 1 } -$$ -{2 \ z \ {y \sp 2}}+{4 \ x}+1 -$$ -\returnType{Type: DistributedMultivariatePolynomial([z,y,x],Fraction Integer)} - -\spadcommand{d3 := 2*z*x**2 - 2*y**2 - x } -$$ -{2 \ z \ {x \sp 2}} -{2 \ {y \sp 2}} -x -$$ -\returnType{Type: DistributedMultivariatePolynomial([z,y,x],Fraction Integer)} - -These constructors are mostly used in Gr\"{o}bner basis calculations. - -\spadcommand{groebner [d1,d2,d3] } -$$ -\begin{array}{@{}l} -\left[ -{z - -{{{1568} \over {2745}} \ {x \sp 6}} - -{{{1264} \over {305}} \ {x \sp 5}}+ -{{6 \over {305}} \ {x \sp 4}}+ -{{{182} \over {549}} \ {x \sp 3}} --{{{2047} \over {610}} \ {x \sp 2}} - -{{{103} \over {2745}} \ x} - -{{2857} \over {10980}}}, -\right. -\\ -\\ -\displaystyle -{{y \sp 2}+ -{{{112} \over {2745}} \ {x \sp 6}} - -{{{84} \over {305}} \ {x \sp 5}} - -{{{1264} \over {305}} \ {x \sp 4}} - -{{{13} \over {549}} \ {x \sp 3}}+ -{{{84} \over {305}} \ {x \sp 2}}+ -{{{1772} \over {2745}} \ x}+ -{2 \over {2745}}}, -\\ -\\ -\displaystyle -\left. -{{x \sp 7}+ -{{{29} \over 4} \ {x \sp 6}} - -{{{17} \over {16}} \ {x \sp 4}} - -{{{11} \over 8} \ {x \sp 3}}+ -{{1 \over {32}} \ {x \sp 2}}+ -{{{15} \over {16}} \ x}+ -{1 \over 4}} -\right] -\end{array} -$$ -\returnType{Type: List DistributedMultivariatePolynomial([z,y,x],Fraction Integer)} - -\spadcommand{(n1,n2,n3) : HDMP([z,y,x],FRAC INT) } -\returnType{Type: Void} - -\spadcommand{n1 := d1 } -$$ -{4 \ {y \sp 2} \ x}+{{16} \ {x \sp 2}} -{4 \ z}+1 -$$ -\returnType{Type: HomogeneousDistributedMultivariatePolynomial([z,y,x],Fraction Integer)} - -\spadcommand{n2 := d2 } -$$ -{2 \ z \ {y \sp 2}}+{4 \ x}+1 -$$ -\returnType{Type: HomogeneousDistributedMultivariatePolynomial([z,y,x],Fraction Integer)} - -\spadcommand{n3 := d3 } -$$ -{2 \ z \ {x \sp 2}} -{2 \ {y \sp 2}} -x -$$ -\returnType{Type: HomogeneousDistributedMultivariatePolynomial([z,y,x],Fraction Integer)} - -Note that we get a different Gr\"{o}bner basis when we use the -{\tt HDMP} polynomials, as expected. - -\spadcommand{groebner [n1,n2,n3] } -$$ -\begin{array}{@{}l} -\left[ -{{y \sp 4}+ -{2 \ {x \sp 3}} - -{{3 \over 2} \ {x \sp 2}}+ -{{1 \over 2} \ z} - -{1 \over 8}}, -\right. -\\ -\\ -\displaystyle -{{x \sp 4}+ -{{{29} \over 4} \ {x \sp 3}} - -{{1 \over 8} \ {y \sp 2}} - -{{7 \over 4} \ z \ x} - -{{9 \over {16}} \ x} - -{1 \over 4}}, -\\ -\\ -\displaystyle -{{z \ {y \sp 2}}+ -{2 \ x}+ -{1 \over 2}}, -\\ -\\ -\displaystyle -{{{y \sp 2} \ x}+ -{4 \ {x \sp 2}} - -z+ -{1 \over 4}}, -\\ -\\ -\displaystyle -{{z \ {x \sp 2}} - -{y \sp 2} - -{{1 \over 2} \ x}}, -\\ -\\ -\displaystyle -\left. -{{z \sp 2} - -{4 \ {y \sp 2}}+ -{2 \ {x \sp 2}} - -{{1 \over 4} \ z} - -{{3 \over 2} \ x}} -\right] -\end{array} -$$ -\returnType{Type: List HomogeneousDistributedMultivariatePolynomial([z,y,x],Fraction Integer)} - -{\tt GeneralDistributedMultivariatePolynomial} is somewhat -more flexible in the sense that as well as accepting a list of -variables to specify the variable ordering, it also takes a -predicate on exponent vectors to specify the term ordering. -With this polynomial type the user can experiment with the effect -of using completely arbitrary term orderings. -This flexibility is mostly important for algorithms such as -Gr\"{o}bner basis calculations which can be very -sensitive to term ordering. - -For more information on related topics, see -\ref{ugIntroVariablesPage} on page~\pageref{ugIntroVariablesPage} in Section -\ref{ugIntroVariablesNumber} on page~\pageref{ugIntroVariablesNumber}, -\ref{ugTypesConvertPage} on page~\pageref{ugTypesConvertPage} in Section -\ref{ugTypesConvertNumber} on page~\pageref{ugTypesConvertNumber}, -\ref{PolynomialXmpPage} on page~\pageref{PolynomialXmpPage}, -\ref{UnivariatePolynomialXmpPage} on -page~\pageref{UnivariatePolynomialXmpPage}, and -\ref{MultivariatePolynomialXmpPage} on -page~\pageref{MultivariatePolynomialXmpPage}. - -\section{DoubleFloat} -\label{DoubleFloatXmpPage} - -Axiom provides two kinds of floating point numbers. The domain -{\tt Float} (abbreviation {\tt FLOAT}) implements a model of arbitrary -precision floating point numbers. The domain {\tt DoubleFloat} -(abbreviation {\tt DFLOAT}) is intended to make available hardware -floating point arithmetic in Axiom. The actual model of floating -point {\tt DoubleFloat} that provides is system-dependent. For -example, on the IBM system 370 Axiom uses IBM double precision which -has fourteen hexadecimal digits of precision or roughly sixteen -decimal digits. Arbitrary precision floats allow the user to specify -the precision at which arithmetic operations are computed. Although -this is an attractive facility, it comes at a cost. Arbitrary-precision -floating-point arithmetic typically takes twenty to two hundred times -more time than hardware floating point. - -The usual arithmetic and elementary functions are available for -{\tt DoubleFloat}. Use {\tt )show DoubleFloat} to get a list of operations -or the HyperDoc browse facility to get more extensive documentation -about {\tt DoubleFloat}. - -By default, floating point numbers that you enter into Axiom are of -type {\tt Float}. - -\spadcommand{2.71828} -$$ -2.71828 -$$ -\returnType{Type: Float} - -You must therefore tell Axiom that you want to use {\tt DoubleFloat} -values and operations. The following are some conservative guidelines -for getting Axiom to use {\tt DoubleFloat}. - -To get a value of type {\tt DoubleFloat}, use a target with {\tt @}, \ldots - -\spadcommand{2.71828@DoubleFloat} -$$ -2.71828 -$$ -\returnType{Type: DoubleFloat} - -a conversion, \ldots - -\spadcommand{2.71828 :: DoubleFloat} -$$ -2.71828 -$$ -\returnType{Type: DoubleFloat} - -or an assignment to a declared variable. It is more efficient if you -use a target rather than an explicit or implicit conversion. - -\spadcommand{eApprox : DoubleFloat := 2.71828 } -$$ -2.71828 -$$ -\returnType{Type: DoubleFloat} - -You also need to declare functions that work with {\tt DoubleFloat}. - -\spadcommand{avg : List DoubleFloat -> DoubleFloat } -\returnType{Type: Void} - -\begin{verbatim} -avg l == - empty? l => 0 :: DoubleFloat - reduce(_+,l) / #l -\end{verbatim} -\returnType{Type: Void} - -%\spadcommand{avg [] } -% this complains but succeeds - -\spadcommand{avg [3.4,9.7,-6.8] } -\begin{verbatim} - Compiling function avg with type List Float -> DoubleFloat - -\end{verbatim} -$$ -2.1 -$$ -\returnType{Type: DoubleFloat} - -Use package-calling for operations from {\tt DoubleFloat} unless -the arguments themselves are already of type {\tt DoubleFloat}. - -\spadcommand{cos(3.1415926)\$DoubleFloat} -$$ --{0.999999999999999} -$$ -\returnType{Type: DoubleFloat} - -\spadcommand{cos(3.1415926 :: DoubleFloat)} -$$ --{0.999999999999999} -$$ -\returnType{Type: DoubleFloat} - -By far, the most common usage of {\tt DoubleFloat} is for functions to -be graphed. For more information about Axiom's numerical and -graphical facilities, see Section -\ref{ugGraph} on page~\pageref{ugGraph}, -\ref{ugProblemNumeric} on page~\pageref{ugProblemNumeric}, and -\ref{FloatXmpPage} on page~\pageref{FloatXmpPage}. - -\section{EqTable} -\label{EqTableXmpPage} - -The {\tt EqTable} domain provides tables where the keys are compared -using \spadfunFrom{eq?}{EqTable}. Keys are considered equal only if -they are the same instance of a structure. This is useful if the keys -are themselves updatable structures. Otherwise, all operations are -the same as for type {\tt Table}. See -\ref{TableXmpPage} on page~\pageref{TableXmpPage} for general -information about tables. - -The operation \spadfunFrom{table}{EqTable} is here used to create a table -where the keys are lists of integers. - -\spadcommand{e: EqTable(List Integer, Integer) := table() } -$$ -table() -$$ -\returnType{Type: EqTable(List Integer,Integer)} - -These two lists are equal according to \spadopFrom{=}{List}, but not -according to \spadfunFrom{eq?}{List}. - -\spadcommand{l1 := [1,2,3] } -$$ -\left[ -1, 2, 3 -\right] -$$ -\returnType{Type: List PositiveInteger} - -\spadcommand{l2 := [1,2,3] } -$$ -\left[ -1, 2, 3 -\right] -$$ -\returnType{Type: List PositiveInteger} - -Because the two lists are not \spadfunFrom{eq?}{List}, separate values -can be stored under each. - -\spadcommand{e.l1 := 111 } -$$ -111 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{e.l2 := 222 } -$$ -222 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{e.l1} -$$ -111 -$$ -\returnType{Type: PositiveInteger} - -\section{Equation} -\label{EquationXmpPage} - -The {\tt Equation} domain provides equations as mathematical objects. -These are used, for example, as the input to various -\spadfunFrom{solve}{TransSolvePackage} operations. - -Equations are created using the equals symbol, \spadopFrom{=}{Equation}. - -\spadcommand{eq1 := 3*x + 4*y = 5 } -$$ -{{4 \ y}+{3 \ x}}=5 -$$ -\returnType{Type: Equation Polynomial Integer} - -\spadcommand{eq2 := 2*x + 2*y = 3 } -$$ -{{2 \ y}+{2 \ x}}=3 -$$ -\returnType{Type: Equation Polynomial Integer} - -The left- and right-hand sides of an equation are accessible using -the operations \spadfunFrom{lhs}{Equation} and \spadfunFrom{rhs}{Equation}. - -\spadcommand{lhs eq1 } -$$ -{4 \ y}+{3 \ x} -$$ -\returnType{Type: Polynomial Integer} - -\spadcommand{rhs eq1 } -$$ -5 -$$ -\returnType{Type: Polynomial Integer} - -Arithmetic operations are supported and operate on both sides of the -equation. - -\spadcommand{eq1 + eq2 } -$$ -{{6 \ y}+{5 \ x}}=8 -$$ -\returnType{Type: Equation Polynomial Integer} - -\spadcommand{eq1 * eq2 } -$$ -{{8 \ {y \sp 2}}+{{14} \ x \ y}+{6 \ {x \sp 2}}}={15} -$$ -\returnType{Type: Equation Polynomial Integer} - -\spadcommand{2*eq2 - eq1 } -$$ -x=1 -$$ -\returnType{Type: Equation Polynomial Integer} - -Equations may be created for any type so the arithmetic operations -will be defined only when they make sense. For example, exponentiation -is not defined for equations involving non-square matrices. - -\spadcommand{eq1**2 } -$$ -{{{16} \ {y \sp 2}}+{{24} \ x \ y}+{9 \ {x \sp 2}}}={25} -$$ -\returnType{Type: Equation Polynomial Integer} - -Note that an equals symbol is also used to {\it test} for equality of -values in certain contexts. For example, {\tt x+1} and {\tt y} are -unequal as polynomials. - -\spadcommand{if x+1 = y then "equal" else "unequal"} -$$ -\mbox{\tt "unequal"} -$$ -\returnType{Type: String} - -\spadcommand{eqpol := x+1 = y } -$$ -{x+1}=y -$$ -\returnType{Type: Equation Polynomial Integer} - -If an equation is used where a {\tt Boolean} value is required, then -it is evaluated using the equality test from the operand type. - -\spadcommand{if eqpol then "equal" else "unequal" } -$$ -\mbox{\tt "unequal"} -$$ -\returnType{Type: String} - -If one wants a {\tt Boolean} value rather than an equation, all one -has to do is ask! - -\spadcommand{eqpol::Boolean } -$$ -{\tt false} -$$ -\returnType{Type: Boolean} - -\section{Exit} -\label{ExitXmpPage} - -A function that does not return directly to its caller has {\tt Exit} -as its return type. The operation {\tt error} is an example of one -which does not return to its caller. Instead, it causes a return to -top-level. - -\spadcommand{n := 0 } -$$ -0 -$$ -\returnType{Type: NonNegativeInteger} - -The function {\tt gasp} is given return type {\tt Exit} since it is -guaranteed never to return a value to its caller. - -\begin{verbatim} -gasp(): Exit == - free n - n := n + 1 - error "Oh no!" - -Function declaration gasp : () -> Exit has been added to workspace. - -\end{verbatim} -\returnType{Type: Void} - -The return type of {\tt half} is determined by resolving the types of -the two branches of the {\tt if}. - -\begin{verbatim} -half(k) == - if odd? k then gasp() - else k quo 2 -\end{verbatim} - -Because {\tt gasp} has the return type {\tt Exit}, the type of -{\tt if} in {\tt half} is resolved to be {\tt Integer}. - -\spadcommand{half 4 } -\begin{verbatim} - Compiling function gasp with type () -> Exit - Compiling function half with type PositiveInteger -> Integer -\end{verbatim} -$$ -2 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{half 3 } -\begin{verbatim} - Error signalled from user code in function gasp: - Oh no! -\end{verbatim} - -\spadcommand{n } -$$ -1 -$$ -\returnType{Type: NonNegativeInteger} - -For functions which return no value at all, use {\tt Void}. See -\ref{ugUserPage} on page~\pageref{ugUserPage} in Section -\ref{ugUserNumber} on page~\pageref{ugUserNumber} and -\ref{VoidXmpPage} on page~\pageref{VoidXmpPage} for -more information. - -\section{Expression} -\label{ExpressionXmpPage} - -{\tt Expression} is a constructor that creates domains whose objects -can have very general symbolic forms. Here are some examples: - -This is an object of type {\tt Expression Integer}. - -\spadcommand{sin(x) + 3*cos(x)**2} -$$ -{\sin \left({x} \right)}+{3\ {{\cos \left({x} \right)}\sp 2}} -$$ -\returnType{Type: Expression Integer} - -This is an object of type {\tt Expression Float}. - -\spadcommand{tan(x) - 3.45*x} -$$ -{\tan \left({x} \right)}-{{3.45} \ x} -$$ -\returnType{Type: Expression Float} - -This object contains symbolic function applications, sums, -products, square roots, and a quotient. - -\spadcommand{(tan sqrt 7 - sin sqrt 11)**2 / (4 - cos(x - y))} -$$ -{-{{\tan \left({{\sqrt {7}}} \right)}\sp 2}+ -{2 \ {\sin \left({{\sqrt {{11}}}} \right)} -\ {\tan \left({{\sqrt {7}}} \right)}} --{{\sin \left({{\sqrt {{11}}}} \right)}\sp 2}} -\over {{\cos \left({{y -x}} \right)} --4} -$$ -\returnType{Type: Expression Integer} - -As you can see, {\tt Expression} actually takes an argument domain. -The {\it coefficients} of the terms within the expression belong to -the argument domain. {\tt Integer} and {\tt Float}, along with -{\tt Complex Integer} and {\tt Complex Float} are the most common -coefficient domains. - -The choice of whether to use a {\tt Complex} coefficient domain or not -is important since Axiom can perform some simplifications on -real-valued objects - -\spadcommand{log(exp x)@Expression(Integer)} -$$ -x -$$ -\returnType{Type: Expression Integer} - -... which are not valid on complex ones. - -\spadcommand{log(exp x)@Expression(Complex Integer)} -$$ -\log \left({{e \sp x}} \right) -$$ -\returnType{Type: Expression Complex Integer} - -Many potential coefficient domains, such as {\tt AlgebraicNumber}, are -not usually used because {\tt Expression} can subsume them. - -\spadcommand{sqrt 3 + sqrt(2 + sqrt(-5)) } -$$ -{\sqrt {{{\sqrt {-5}}+2}}}+{\sqrt {3}} -$$ -\returnType{Type: AlgebraicNumber} - -\spadcommand{\% :: Expression Integer } -$$ -{\sqrt {{{\sqrt {-5}}+2}}}+{\sqrt {3}} -$$ -\returnType{Type: Expression Integer} - -Note that we sometimes talk about ``an object of type {\tt -Expression}.'' This is not really correct because we should say, for -example, ``an object of type {\tt Expression Integer}'' or ``an object -of type {\tt Expression Float}.'' By a similar abuse of language, -when we refer to an ``expression'' in this section we will mean an -object of type {\tt Expression R} for some domain {\tt R}. - -The Axiom documentation contains many examples of the use of {\tt -Expression}. For the rest of this section, we'll give you some -pointers to those examples plus give you some idea of how to -manipulate expressions. - -It is important for you to know that {\tt Expression} creates domains -that have category {\tt Field}. Thus you can invert any non-zero -expression and you shouldn't expect an operation like {\tt factor} to -give you much information. You can imagine expressions as being -represented as quotients of ``multivariate'' polynomials where the -``variables'' are kernels (see -\ref{KernelXmpPage} on page~\pageref{KernelXmpPage}). A kernel can -either be a symbol such as {\tt x} or a symbolic function application -like {\tt sin(x + 4)}. The second example is actually a nested kernel -since the argument to {\tt sin} contains the kernel {\tt x}. - -\spadcommand{height mainKernel sin(x + 4)} -$$ -2 -$$ -\returnType{Type: PositiveInteger} - -Actually, the argument to {\tt sin} is an expression, and so the -structure of {\tt Expression} is recursive. -\ref{KernelXmpPage} on page~\pageref{KernelXmpPage} -demonstrates how to extract the kernels in an expression. - -Use the HyperDoc Browse facility to see what operations are applicable -to expression. At the time of this writing, there were 262 operations -with 147 distinct name in {\tt Expression Integer}. For example, -\spadfunFrom{numer}{Expression} and \spadfunFrom{denom}{Expression} -extract the numerator and denominator of an expression. - -\spadcommand{e := (sin(x) - 4)**2 / ( 1 - 2*y*sqrt(- y) ) } -$$ -{-{{\sin \left({x} \right)}\sp 2}+ -{8 \ {\sin \left({x} \right)}} --{16}} -\over {{2 \ y \ {\sqrt {-y}}} -1} -$$ -\returnType{Type: Expression Integer} - -\spadcommand{numer e } -$$ --{{\sin \left({x} \right)}\sp 2}+ -{8 \ {\sin \left({x} \right)}} --{16} -$$ -\returnType{Type: -SparseMultivariatePolynomial(Integer,Kernel Expression Integer)} - -\spadcommand{denom e } -$$ -{2 \ y \ {\sqrt {-y}}} -1 -$$ -\returnType{Type: -SparseMultivariatePolynomial(Integer,Kernel Expression Integer)} - -Use \spadfunFrom{D}{Expression} to compute partial derivatives. - -\spadcommand{D(e, x) } -$$ -{{{\left( -{4 \ y \ {\cos \left({x} \right)}\ {\sin \left({x} \right)}}- -{{16} \ y \ {\cos \left({x} \right)}}\right)}\ {\sqrt {-y}}} - -{2 \ {\cos \left({x} \right)}\ {\sin \left({x} \right)}}+ -{8\ {\cos \left({x} \right)}}} -\over {{4 \ y \ {\sqrt {-y}}}+{4 \ {y \sp 3}} -1} -$$ -\returnType{Type: Expression Integer} - -See -\ref{ugIntroCalcDerivPage} on page~\pageref{ugIntroCalcDerivPage} in Section -\ref{ugIntroCalcDerivNumber} on page~\pageref{ugIntroCalcDerivNumber} -for more examples of expressions and derivatives. - -\spadcommand{D(e, [x, y], [1, 2]) } -$$ -\left( -\begin{array}{@{}l} -{{\left( {{\left( -{{2304} \ {y \sp 7}}+{{960} \ {y \sp 4}} \right)} -\ {\cos \left({x} \right)}\ {\sin \left({x} \right)}}+ -{{\left({{9216} \ {y \sp 7}} -{{3840} \ {y \sp 4}} \right)} -\ {\cos \left({x} \right)}}\right)}\ {\sqrt {-y}}}+ -\\ -\\ -\displaystyle -{{\left( -{{960} \ {y \sp 9}}+{{2160} \ {y \sp 6}} -{{180} \ {y \sp 3}} -3 -\right)}\ {\cos \left({x} \right)}\ {\sin \left({x} \right)}}+ -\\ -\\ -\displaystyle -{{\left( -{{3840} \ {y \sp 9}} -{{8640} \ {y \sp 6}}+{{720} \ {y \sp 3}}+{12} -\right)}\ {\cos \left({x} \right)}} -\end{array} -\right) -\over -\left( -\begin{array}{@{}l} -{{\left( {{256} \ {y \sp {12}}} -{{1792} \ {y \sp 9}}+{{1120} \ {y -\sp 6}} -{{112} \ {y \sp 3}}+1 \right)}\ {\sqrt {-y}}} - -\\ -\\ -\displaystyle -{{1024} \ {y \sp {11}}}+{{1792} \ {y \sp 8}} -{{448} \ -{y \sp 5}}+{{16} \ {y \sp 2}} -\end{array} -\right) -$$ -\returnType{Type: Expression Integer} - -See -\ref{ugIntroCalcLimitsPage} on page~\pageref{ugIntroCalcLimitsPage} in Section -\ref{ugIntroCalcLimitsNumber} on page~\pageref{ugIntroCalcLimitsNumber} and -\ref{ugIntroSeriesPage} on page~\pageref{ugIntroSeriesPage} in Section -\ref{ugIntroSeriesNumber} on page~\pageref{ugIntroSeriesNumber} -for more examples of expressions and -calculus. Differential equations involving expressions are discussed -in \ref{ugProblemDEQPage} on page~\pageref{ugProblemDEQPage} in Section -\ref{ugProblemDEQNumber} on page~\pageref{ugProblemDEQNumber}. -Chapter 8 has many advanced examples: see -\ref{ugProblemIntegrationPage} on page~\pageref{ugProblemIntegrationPage} -in Section -\ref{ugProblemIntegrationNumber} on page~\pageref{ugProblemIntegrationNumber} -for a discussion of Axiom's integration facilities. - -When an expression involves no ``symbol kernels'' (for example, -{\tt x}), it may be possible to numerically evaluate the expression. - -If you suspect the evaluation will create a complex number, use -{\tt complexNumeric}. - -\spadcommand{complexNumeric(cos(2 - 3*\%i))} -$$ --{4.1896256909\ 688072301}+{{9.1092278937\ 55336598} \ i} -$$ -\returnType{Type: Complex Float} - -If you know it will be real, use {\tt numeric}. - -\spadcommand{numeric(tan 3.8)} -$$ -0.7735560905\ 0312607286 -$$ -\returnType{Type: Float} - -The {\tt numeric} operation will display an error message if the -evaluation yields a calue with an non-zero imaginary part. Both of -these operations have an optional second argument {\tt n} which -specifies that the accuracy of the approximation be up to {\tt n} -decimal places. - -When an expression involves no ``symbolic application'' kernels, it -may be possible to convert it a polynomial or rational function in the -variables that are present. - -\spadcommand{e2 := cos(x**2 - y + 3) } -$$ -\cos \left({{y -{x \sp 2} -3}} \right) -$$ -\returnType{Type: Expression Integer} - -\spadcommand{e3 := asin(e2) - \%pi/2 } -$$ --y+{x \sp 2}+3 -$$ -\returnType{Type: Expression Integer} - -\spadcommand{e3 :: Polynomial Integer } -$$ --y+{x \sp 2}+3 -$$ -\returnType{Type: Polynomial Integer} - -This also works for the polynomial types where specific variables -and their ordering are given. - -\spadcommand{e3 :: DMP([x, y], Integer) } -$$ -{x \sp 2} -y+3 -$$ -\returnType{Type: DistributedMultivariatePolynomial([x,y],Integer)} - -Finally, a certain amount of simplication takes place as expressions -are constructed. - -\spadcommand{sin \%pi} -$$ -0 -$$ -\returnType{Type: Expression Integer} - -\spadcommand{cos(\%pi / 4)} -$$ -{\sqrt {2}} \over 2 -$$ -\returnType{Type: Expression Integer} - -For simplications that involve multiple terms of the expression, use -{\tt simplify}. - -\spadcommand{tan(x)**6 + 3*tan(x)**4 + 3*tan(x)**2 + 1 } -$$ -{{\tan \left({x} \right)}\sp 6}+ -{3 \ {{\tan \left({x} \right)}\sp 4}}+ -{3 \ {{\tan \left({x} \right)}\sp 2}}+1 -$$ -\returnType{Type: Expression Integer} - -\spadcommand{simplify \% } -$$ -1 \over {{\cos \left({x} \right)}\sp 6} -$$ -\returnType{Type: Expression Integer} - -See \ref{ugUserRulesPage} on page~\pageref{ugUserRulesPage} in Section -\ref{ugUserRulesNumber} on page~\pageref{ugUserRulesNumber} for -examples of how to write your own rewrite rules for expressions. - -\section{Factored} -\label{FactoredXmpPage} - -{\tt Factored} creates a domain whose objects are kept in factored -form as long as possible. Thus certain operations like -\spadopFrom{*}{Factored} (multiplication) and -\spadfunFrom{gcd}{Factored} are relatively easy to do. Others, such -as addition, require somewhat more work, and the result may not be -completely factored unless the argument domain {\tt R} provides a -\spadfunFrom{factor}{Factored} operation. Each object consists of a -unit and a list of factors, where each factor consists of a member of -{\tt R} (the {\em base}), an exponent, and a flag indicating what is -known about the base. A flag may be one of ``{\tt nil}'', ``{\tt sqfr}'', -``{\tt irred}'' or ``{\tt prime}'', which mean that nothing is known about -the base, it is square-free, it is irreducible, or it is prime, -respectively. The current restriction to factored objects of integral -domains allows simplification to be performed without worrying about -multiplication order. - -\subsection{Decomposing Factored Objects} - -In this section we will work with a factored integer. - -\spadcommand{g := factor(4312) } -$$ -{2 \sp 3} \ {7 \sp 2} \ {11} -$$ -\returnType{Type: Factored Integer} - -Let's begin by decomposing {\tt g} into pieces. The only possible -units for integers are {\tt 1} and {\tt -1}. - -\spadcommand{unit(g) } -$$ -1 -$$ -\returnType{Type: PositiveInteger} - -There are three factors. - -\spadcommand{numberOfFactors(g) } -$$ -3 -$$ -\returnType{Type: PositiveInteger} - -We can make a list of the bases, \ldots - -\spadcommand{[nthFactor(g,i) for i in 1..numberOfFactors(g)] } -$$ -\left[ -2, 7, {11} -\right] -$$ -\returnType{Type: List Integer} - -and the exponents, \ldots - -\spadcommand{[nthExponent(g,i) for i in 1..numberOfFactors(g)] } -$$ -\left[ -3, 2, 1 -\right] -$$ -\returnType{Type: List Integer} - -and the flags. You can see that all the bases (factors) are prime. - -\spadcommand{[nthFlag(g,i) for i in 1..numberOfFactors(g)] } -$$ -\left[ -\mbox{\tt "prime"} , \mbox{\tt "prime"} , \mbox{\tt "prime"} -\right] -$$ -\returnType{Type: List Union("nil","sqfr","irred","prime")} - -A useful operation for pulling apart a factored object into a list -of records of the components is \spadfunFrom{factorList}{Factored}. - -\spadcommand{factorList(g) } -$$ -\begin{array}{@{}l} -\left[ -{\left[ {flg= \mbox{\tt "prime"} }, {fctr=2}, {xpnt=3} \right]}, -\right. -\\ -\\ -\displaystyle -{\left[ {flg= \mbox{\tt "prime"} }, {fctr=7}, {xpnt=2} \right]}, -\\ -\\ -\left. -\displaystyle -{\left[ {flg= \mbox{\tt "prime"} }, {fctr={11}}, {xpnt=1} \right]} -\right] -\end{array} -$$ -\returnType{Type: -List Record(flg: Union("nil","sqfr","irred","prime"), -fctr: Integer,xpnt: Integer)} - -If you don't care about the flags, use \spadfunFrom{factors}{Factored}. - -\spadcommand{factors(g) } -$$ -\begin{array}{@{}l} -\left[ -{\left[ {factor=2}, {exponent=3} \right]}, -\right. -\\ -\\ -\displaystyle -{\left[ {factor=7}, {exponent=2} \right]}, -\\ -\\ -\displaystyle -\left. -{\left[ {factor={11}}, {exponent=1} \right]} -\right] -\end{array} -$$ -\returnType{Type: List Record(factor: Integer,exponent: Integer)} - -Neither of these operations returns the unit. - -\spadcommand{first(\%).factor } -$$ -2 -$$ -\returnType{Type: PositiveInteger} - -\subsection{Expanding Factored Objects} - -Recall that we are working with this factored integer. - -\spadcommand{g := factor(4312) } -$$ -{2 \sp 3} \ {7 \sp 2} \ {11} -$$ -\returnType{Type: Factored Integer} - -To multiply out the factors with their multiplicities, use -\spadfunFrom{expand}{Factored}. - -\spadcommand{expand(g) } -$$ -4312 -$$ -\returnType{Type: PositiveInteger} - -If you would like, say, the distinct factors multiplied together but -with multiplicity one, you could do it this way. - -\spadcommand{reduce(*,[t.factor for t in factors(g)]) } -$$ -154 -$$ -\returnType{Type: PositiveInteger} - -\subsection{Arithmetic with Factored Objects} - -We're still working with this factored integer. - -\spadcommand{g := factor(4312) } -$$ -{2 \sp 3} \ {7 \sp 2} \ {11} -$$ -\returnType{Type: Factored Integer} - -We'll also define this factored integer. - -\spadcommand{f := factor(246960) } -$$ -{2 \sp 4} \ {3 \sp 2} \ 5 \ {7 \sp 3} -$$ -\returnType{Type: Factored Integer} - -Operations involving multiplication and division are particularly -easy with factored objects. - -\spadcommand{f * g } -$$ -{2 \sp 7} \ {3 \sp 2} \ 5 \ {7 \sp 5} \ {11} -$$ -\returnType{Type: Factored Integer} - -\spadcommand{f**500 } -$$ -{2 \sp {2000}} \ {3 \sp {1000}} \ {5 \sp {500}} \ {7 \sp {1500}} -$$ -\returnType{Type: Factored Integer} - -\spadcommand{gcd(f,g) } -$$ -{2 \sp 3} \ {7 \sp 2} -$$ -\returnType{Type: Factored Integer} - -\spadcommand{lcm(f,g) } -$$ -{2 \sp 4} \ {3 \sp 2} \ 5 \ {7 \sp 3} \ {11} -$$ -\returnType{Type: Factored Integer} - -If we use addition and subtraction things can slow down because -we may need to compute greatest common divisors. - -\spadcommand{f + g } -$$ -{2 \sp 3} \ {7 \sp 2} \ {641} -$$ -\returnType{Type: Factored Integer} - -\spadcommand{f - g } -$$ -{2 \sp 3} \ {7 \sp 2} \ {619} -$$ -\returnType{Type: Factored Integer} - -Test for equality with {\tt 0} and {\tt 1} by using -\spadfunFrom{zero?}{Factored} and \spadfunFrom{one?}{Factored}, -respectively. - -\spadcommand{zero?(factor(0))} -$$ -{\tt true} -$$ -\returnType{Type: Boolean} - -\spadcommand{zero?(g) } -$$ -{\tt false} -$$ -\returnType{Type: Boolean} - -\spadcommand{one?(factor(1))} -$$ -{\tt true} -$$ -\returnType{Type: Boolean} - -\spadcommand{one?(f) } -$$ -{\tt false} -$$ -\returnType{Type: Boolean} - -Another way to get the zero and one factored objects is to use -package calling (see -\ref{ugTypesPkgCallPage} on page~\pageref{ugTypesPkgCallPage} in Section -\ref{ugTypesPkgCallNumber} on page~\pageref{ugTypesPkgCallNumber}). - -\spadcommand{0\$Factored(Integer)} -$$ -0 -$$ -\returnType{Type: Factored Integer} - -\spadcommand{1\$Factored(Integer)} -$$ -1 -$$ -\returnType{Type: Factored Integer} - -\subsection{Creating New Factored Objects} - -The \spadfunFrom{map}{Factored} operation is used to iterate across -the unit and bases of a factored object. See -\ref{FactoredFunctionsTwoXmpPage} on -page~\pageref{FactoredFunctionsTwoXmpPage} for a discussion of -\spadfunFrom{map}{Factored}. - -The following four operations take a base and an exponent and create a -factored object. They differ in handling the flag component. - -\spadcommand{nilFactor(24,2) } -$$ -{24} \sp 2 -$$ -\returnType{Type: Factored Integer} - -This factor has no associated information. - -\spadcommand{nthFlag(\%,1) } -$$ -\mbox{\tt "nil"} -$$ -\returnType{Type: Union("nil",...)} - -This factor is asserted to be square-free. - -\spadcommand{sqfrFactor(30,2) } -$$ -{30} \sp 2 -$$ -\returnType{Type: Factored Integer} - -This factor is asserted to be irreducible. - -\spadcommand{irreducibleFactor(13,10) } -$$ -{13} \sp {10} -$$ -\returnType{Type: Factored Integer} - -This factor is asserted to be prime. - -\spadcommand{primeFactor(11,5) } -$$ -{11} \sp 5 -$$ -\returnType{Type: Factored Integer} - -A partial inverse to \spadfunFrom{factorList}{Factored} is -\spadfunFrom{makeFR}{Factored}. - -\spadcommand{h := factor(-720) } -$$ --{{2 \sp 4} \ {3 \sp 2} \ 5} -$$ -\returnType{Type: Factored Integer} - -The first argument is the unit and the second is a list of records as -returned by \spadfunFrom{factorList}{Factored}. - -\spadcommand{h - makeFR(unit(h),factorList(h)) } -$$ -0 -$$ -\returnType{Type: Factored Integer} - -\subsection{Factored Objects with Variables} - -Some of the operations available for polynomials are also available -for factored polynomials. - -\spadcommand{p := (4*x*x-12*x+9)*y*y + (4*x*x-12*x+9)*y + 28*x*x - 84*x + 63 } -$$ -{{\left( {4 \ {x \sp 2}} -{{12} \ x}+9 -\right)} -\ {y \sp 2}}+{{\left( {4 \ {x \sp 2}} -{{12} \ x}+9 -\right)} -\ y}+{{28} \ {x \sp 2}} -{{84} \ x}+{63} -$$ -\returnType{Type: Polynomial Integer} - -\spadcommand{fp := factor(p) } -$$ -{{\left( {2 \ x} -3 -\right)} -\sp 2} \ {\left( {y \sp 2}+y+7 -\right)} -$$ -\returnType{Type: Factored Polynomial Integer} - -You can differentiate with respect to a variable. - -\spadcommand{D(p,x) } -$$ -{{\left( {8 \ x} -{12} -\right)} -\ {y \sp 2}}+{{\left( {8 \ x} -{12} -\right)} -\ y}+{{56} \ x} -{84} -$$ -\returnType{Type: Polynomial Integer} - -\spadcommand{D(fp,x) } -$$ -4 \ {\left( {2 \ x} -3 -\right)} -\ {\left( {y \sp 2}+y+7 -\right)} -$$ -\returnType{Type: Factored Polynomial Integer} - -\spadcommand{numberOfFactors(\%) } -$$ -3 -$$ -\returnType{Type: PositiveInteger} - -\section{FactoredFunctions2} -\label{FactoredFunctions2XmpPage} - -The {\tt FactoredFunctions2} package implements one operation, -\spadfunFrom{map}{FactoredFunctions2}, for applying an operation to every -base in a factored object and to the unit. - -\spadcommand{double(x) == x + x } -\returnType{Type: Void} - -\spadcommand{f := factor(720) } -$$ -{2 \sp 4} \ {3 \sp 2} \ 5 -$$ -\returnType{Type: Factored Integer} - -Actually, the \spadfunFrom{map}{FactoredFunctions2} operation used -in this example comes from {\tt Factored} itself, since {\tt double} -takes an integer argument and returns an integer result. - -\spadcommand{map(double,f) } -$$ -2 \ {4 \sp 4} \ {6 \sp 2} \ {10} -$$ -\returnType{Type: Factored Integer} - -If we want to use an operation that returns an object that has a type -different from the operation's argument, -the \spadfunFrom{map}{FactoredFunctions2} in {\tt Factored} -cannot be used and we use the one in {\tt FactoredFunctions2}. - -\spadcommand{makePoly(b) == x + b } -\returnType{Type: Void} - -In fact, the ``2'' in the name of the package means that we might -be using factored objects of two different types. - -\spadcommand{g := map(makePoly,f) } -$$ -{\left( x+1 \right)} -\ {{\left( x+2 \right)}\sp 4} -\ {{\left( x+3 \right)}\sp 2} -\ {\left( x+5 \right)} -$$ -\returnType{Type: Factored Polynomial Integer} - -It is important to note that both versions of -\spadfunFrom{map}{FactoredFunctions2} destroy any information known -about the bases (the fact that they are prime, for instance). - -The flags for each base are set to ``nil'' in the object returned -by \spadfunFrom{map}{FactoredFunctions2}. - -\spadcommand{nthFlag(g,1) } -$$ -\mbox{\tt "nil"} -$$ -\returnType{Type: Union("nil",...)} - -For more information about factored objects and their use, see -\ref{FactoredXmpPage} on page~\pageref{FactoredXmpPage} and -\ref{ugProblemGaloisPage} on page~\pageref{ugProblemGaloisPage} in Section -\ref{ugProblemGaloisNumber} on page~\pageref{ugProblemGaloisNumber}. - -\section{File} -\label{FileXmpPage} - -The {\tt File(S)} domain provides a basic interface to read and -write values of type {\tt S} in files. - -Before working with a file, it must be made accessible to Axiom with -the \spadfunFrom{open}{File} operation. - -\spadcommand{ifile:File List Integer:=open("/tmp/jazz1","output") } -$$ -\mbox{\tt "/tmp/jazz1"} -$$ -\returnType{Type: File List Integer} - -The \spadfunFrom{open}{File} function arguments are a {\tt FileName} -and a {\tt String} specifying the mode. If a full pathname is not -specified, the current default directory is assumed. The mode must be -one of ``{\tt input}'' or ``{\tt output}''. If it is not specified, -``{\tt input}'' is assumed. Once the file has been opened, you can read or -write data. - -The operations \spadfunFrom{read}{File} and \spadfunFrom{write}{File} are -provided. - -\spadcommand{write!(ifile, [-1,2,3])} -$$ -\left[ --1, 2, 3 -\right] -$$ -\returnType{Type: List Integer} - -\spadcommand{write!(ifile, [10,-10,0,111])} -$$ -\left[ -{10}, -{10}, 0, {111} -\right] -$$ -\returnType{Type: List Integer} - -\spadcommand{write!(ifile, [7])} -$$ -\left[ -7 -\right] -$$ -\returnType{Type: List Integer} - -You can change from writing to reading (or vice versa) by reopening a file. - -\spadcommand{reopen!(ifile, "input")} -$$ -\mbox{\tt "/tmp/jazz1"} -$$ -\returnType{Type: File List Integer} - -\spadcommand{read! ifile} -$$ -\left[ --1, 2, 3 -\right] -$$ -\returnType{Type: List Integer} - -\spadcommand{read! ifile} -$$ -\left[ -{10}, -{10}, 0, {111} -\right] -$$ -\returnType{Type: List Integer} - -The \spadfunFrom{read}{File} operation can cause an error if one tries -to read more data than is in the file. To guard against this -possibility the \spadfunFrom{readIfCan}{File} operation should be -used. - -\spadcommand{readIfCan! ifile } -$$ -\left[ -7 -\right] -$$ -\returnType{Type: Union(List Integer,...)} - -\spadcommand{readIfCan! ifile } -$$ -\mbox{\tt "failed"} -$$ -\returnType{Type: Union("failed",...)} - -You can find the current mode of the file, and the file's name. - -\spadcommand{iomode ifile} -$$ -\mbox{\tt "input"} -$$ -\returnType{Type: String} - -\spadcommand{name ifile} -$$ -\mbox{\tt "/tmp/jazz1"} -$$ -\returnType{Type: FileName} - -When you are finished with a file, you should close it. - -\spadcommand{close! ifile} -$$ -\mbox{\tt "/tmp/jazz1"} -$$ -\returnType{Type: File List Integer} - -\spadcommand{)system rm /tmp/jazz1} - -A limitation of the underlying LISP system is that not all values can -be represented in a file. In particular, delayed values containing -compiled functions cannot be saved. - -For more information on related topics, see -\ref{TextFileXmpPage} on page~\pageref{TextFileXmpPage}, -\ref{KeyedAccessFileXmpPage} on page~\pageref{KeyedAccessFileXmpPage}, -\ref{LibraryXmpPage} on page~\pageref{LibraryXmpPage}, and -\ref{FileNameXmpPage} on page~\pageref{FileNameXmpPage}. - -\section{FileName} -\label{FileNameXmpPage} - -The {\tt FileName} domain provides an interface to the computer's file -system. Functions are provided to manipulate file names and to test -properties of files. - -The simplest way to use file names in the Axiom interpreter is to rely -on conversion to and from strings. The syntax of these strings -depends on the operating system. - -\spadcommand{fn: FileName } -\returnType{Type: Void} - -On Linux, this is a proper file syntax: - -\spadcommand{fn := "/tmp/fname.input" } -$$ -\mbox{\tt "/tmp/fname.input"} -$$ -\returnType{Type: FileName} - -Although it is very convenient to be able to use string notation -for file names in the interpreter, it is desirable to have a portable -way of creating and manipulating file names from within programs. - -A measure of portability is obtained by considering a file name -to consist of three parts: the {\it directory}, the {\it name}, -and the {\it extension}. - -\spadcommand{directory fn } -$$ -\mbox{\tt "/tmp"} -$$ -\returnType{Type: String} - -\spadcommand{name fn } -$$ -\mbox{\tt "fname"} -$$ -\returnType{Type: String} - -\spadcommand{extension fn } -$$ -\mbox{\tt "input"} -$$ -\returnType{Type: String} - -The meaning of these three parts depends on the operating system. -For example, on CMS the file ``{\tt SPADPROF INPUT M}'' -would have directory ``{\tt M}'', name ``{\tt SPADPROF}'' and -extension ``{\tt INPUT}''. - -It is possible to create a filename from its parts. - -\spadcommand{fn := filename("/u/smwatt/work", "fname", "input") } -$$ -\mbox{\tt "/u/smwatt/work/fname.input"} -$$ -\returnType{Type: FileName} - -When writing programs, it is helpful to refer to directories via -variables. - -\spadcommand{objdir := "/tmp" } -$$ -\mbox{\tt "/tmp"} -$$ -\returnType{Type: String} - -\spadcommand{fn := filename(objdir, "table", "spad") } -$$ -\mbox{\tt "/tmp/table.spad"} -$$ -\returnType{Type: FileName} - -If the directory or the extension is given as an empty string, then -a default is used. On AIX, the defaults are the current directory -and no extension. - -\spadcommand{fn := filename("", "letter", "") } -$$ -\mbox{\tt "letter"} -$$ -\returnType{Type: FileName} - -Three tests provide information about names in the file system. - -The \spadfunFrom{exists?}{FileName} operation tests whether the named -file exists. - -\spadcommand{exists? "/etc/passwd"} -$$ -{\tt true} -$$ -\returnType{Type: Boolean} - -The operation \spadfunFrom{readable?}{FileName} tells whether the named file -can be read. If the file does not exist, then it cannot be read. - -\spadcommand{readable? "/etc/passwd"} -$$ -{\tt true} -$$ -\returnType{Type: Boolean} - -\spadcommand{readable? "/etc/security/passwd"} -$$ -{\tt false} -$$ -\returnType{Type: Boolean} - -\spadcommand{readable? "/ect/passwd"} -$$ -{\tt false} -$$ -\returnType{Type: Boolean} - -Likewise, the operation \spadfunFrom{writable?}{FileName} tells -whether the named file can be written. If the file does not exist, -the test is determined by the properties of the directory. - -\spadcommand{writable? "/etc/passwd"} -$$ -{\tt false} -$$ -\returnType{Type: Boolean} - -\spadcommand{writable? "/dev/null"} -$$ -{\tt true} -$$ -\returnType{Type: Boolean} - -\spadcommand{writable? "/etc/DoesNotExist"} -$$ -{\tt false} -$$ -\returnType{Type: Boolean} - -\spadcommand{writable? "/tmp/DoesNotExist"} -$$ -{\tt true} -$$ -\returnType{Type: Boolean} - -The \spadfunFrom{new}{FileName} operation constructs the name of a new -writable file. The argument sequence is the same as for -\spadfunFrom{filename}{FileName}, except that the name part is -actually a prefix for a constructed unique name. - -The resulting file is in the specified directory -with the given extension, and the same defaults are used. - -\spadcommand{fn := new(objdir, "xxx", "yy") } -$$ -\mbox{\tt "/tmp/xxx82404.yy"} -$$ -\returnType{Type: FileName} - -\section{FlexibleArray} -\label{FlexibleArrayXmpPage} - -The {\tt FlexibleArray} domain constructor creates one-dimensional -arrays of elements of the same type. Flexible arrays are an attempt -to provide a data type that has the best features of both -one-dimensional arrays (fast, random access to elements) and lists -(flexibility). They are implemented by a fixed block of storage. -When necessary for expansion, a new, larger block of storage is -allocated and the elements from the old storage area are copied into -the new block. - -Flexible arrays have available most of the operations provided by -{\tt OneDimensionalArray} (see -\ref{OneDimensionalArrayXmpPage} on page~\pageref{OneDimensionalArrayXmpPage} -and \ref{VectorXmpPage} on page~\pageref{VectorXmpPage}). -Since flexible arrays are also of category -{\tt ExtensibleLinearAggregate}, they have operations {\tt concat!}, -{\tt delete!}, {\tt insert!}, {\tt merge!}, {\tt remove!}, -{\tt removeDuplicates!}, and {\tt select!}. In addition, the operations -{\tt physicalLength} and {\tt physicalLength!} provide user-control -over expansion and contraction. - -A convenient way to create a flexible array is to apply the operation -{\tt flexibleArray} to a list of values. - -\spadcommand{flexibleArray [i for i in 1..6]} -$$ -\left[ -1, 2, 3, 4, 5, 6 -\right] -$$ -\returnType{Type: FlexibleArray PositiveInteger} - -Create a flexible array of six zeroes. - -\spadcommand{f : FARRAY INT := new(6,0)} -$$ -\left[ -0, 0, 0, 0, 0, 0 -\right] -$$ -\returnType{Type: FlexibleArray Integer} - -For $i=1\ldots 6$ set the $i$-th element to $i$. Display {\tt f}. - -\spadcommand{for i in 1..6 repeat f.i := i; f} -$$ -\left[ -1, 2, 3, 4, 5, 6 -\right] -$$ -\returnType{Type: FlexibleArray Integer} - -Initially, the physical length is the same as the number of elements. - -\spadcommand{physicalLength f} -$$ -6 -$$ -\returnType{Type: PositiveInteger} - -Add an element to the end of {\tt f}. - -\spadcommand{concat!(f,11)} -$$ -\left[ -1, 2, 3, 4, 5, 6, {11} -\right] -$$ -\returnType{Type: FlexibleArray Integer} - -See that its physical length has grown. - -\spadcommand{physicalLength f} -$$ -10 -$$ -\returnType{Type: PositiveInteger} - -Make {\tt f} grow to have room for {\tt 15} elements. - -\spadcommand{physicalLength!(f,15)} -$$ -\left[ -1, 2, 3, 4, 5, 6, {11} -\right] -$$ -\returnType{Type: FlexibleArray Integer} - -Concatenate the elements of {\tt f} to itself. The physical length -allows room for three more values at the end. - -\spadcommand{concat!(f,f)} -$$ -\left[ -1, 2, 3, 4, 5, 6, {11}, 1, 2, 3, 4, 5, 6, -{11} -\right] -$$ -\returnType{Type: FlexibleArray Integer} - -Use {\tt insert!} to add an element to the front of a flexible array. - -\spadcommand{insert!(22,f,1)} -$$ -\left[ -{22}, 1, 2, 3, 4, 5, 6, {11}, 1, 2, 3, 4, -5, 6, {11} -\right] -$$ -\returnType{Type: FlexibleArray Integer} - -Create a second flexible array from {\tt f} consisting of the elements -from index 10 forward. - -\spadcommand{g := f(10..)} -$$ -\left[ -2, 3, 4, 5, 6, {11} -\right] -$$ -\returnType{Type: FlexibleArray Integer} - -Insert this array at the front of {\tt f}. - -\spadcommand{insert!(g,f,1)} -$$ -\left[ -2, 3, 4, 5, 6, {11}, {22}, 1, 2, 3, 4, 5, -6, {11}, 1, 2, 3, 4, 5, 6, {11} -\right] -$$ -\returnType{Type: FlexibleArray Integer} - -Merge the flexible array {\tt f} into {\tt g} after sorting each in place. - -\spadcommand{merge!(sort! f, sort! g)} -$$ -\left[ -1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, -4, 5, 5, 5, 5, 6, 6, 6, 6, {11}, {11}, {11}, -{11}, {22} -\right] -$$ -\returnType{Type: FlexibleArray Integer} - -Remove duplicates in place. - -\spadcommand{removeDuplicates! f} -$$ -\left[ -1, 2, 3, 4, 5, 6, {11}, {22} -\right] -$$ -\returnType{Type: FlexibleArray Integer} - -Remove all odd integers. - -\spadcommand{select!(i +-> even? i,f)} -$$ -\left[ -2, 4, 6, {22} -\right] -$$ -\returnType{Type: FlexibleArray Integer} - -All these operations have shrunk the physical length of {\tt f}. - -\spadcommand{physicalLength f} -$$ -8 -$$ -\returnType{Type: PositiveInteger} - -To force Axiom not to shrink flexible arrays call the {\tt shrinkable} -operation with the argument {\tt false}. You must package call this -operation. The previous value is returned. - -\spadcommand{shrinkable(false)\$FlexibleArray(Integer)} -$$ -{\tt true} -$$ -\returnType{Type: Boolean} - -\section{Float} -\label{FloatXmpPage} - -Axiom provides two kinds of floating point numbers. The domain -{\tt Float} (abbreviation {\tt FLOAT}) implements a model of arbitrary -precision floating point numbers. The domain {\tt DoubleFloat} -(abbreviation {\tt DFLOAT}) is intended to make available hardware -floating point arithmetic in Axiom. The actual model of floating -point that {\tt DoubleFloat} provides is system-dependent. For -example, on the IBM system 370 Axiom uses IBM double precision which -has fourteen hexadecimal digits of precision or roughly sixteen -decimal digits. Arbitrary precision floats allow the user to specify -the precision at which arithmetic operations are computed. Although -this is an attractive facility, it comes at a cost. -Arbitrary-precision floating-point arithmetic typically takes twenty -to two hundred times more time than hardware floating point. - -For more information about Axiom's numeric and graphic facilities, see -\ref{ugGraphPage} on page~\pageref{ugGraphPage} in Section -\ref{ugGraphNumber} on page~\pageref{ugGraphNumber}, -\ref{ugProblemNumeric} on page~\pageref{ugProblemNumeric}, and -\ref{DoubleFloatXmpPage} on page~\pageref{DoubleFloatXmpPage}. - -\subsection{Introduction to Float} - -Scientific notation is supported for input and output of floating -point numbers. A floating point number is written as a string of -digits containing a decimal point optionally followed by the letter -``{\tt E}'', and then the exponent. - -We begin by doing some calculations using arbitrary precision floats. -The default precision is twenty decimal digits. - -\spadcommand{1.234} -$$ -1.234 -$$ -\returnType{Type: Float} - -A decimal base for the exponent is assumed, so the number -{\tt 1.234E2} denotes $1.234 \cdot 10^2$. - -\spadcommand{1.234E2} -$$ -123.4 -$$ -\returnType{Type: Float} - -The normal arithmetic operations are available for floating point numbers. - -\spadcommand{sqrt(1.2 + 2.3 / 3.4 ** 4.5)} -$$ -1.0996972790\ 671286226 -$$ -\returnType{Type: Float} - -\subsection{Conversion Functions} - -You can use conversion (\ref{ugTypesConvertPage} on -page~\pageref{ugTypesConvertPage} in Section -\ref{ugTypesConvertNumber} on page~\pageref{ugTypesConvertNumber}) to -go back and forth between {\tt Integer}, {\tt Fraction Integer} and -{\tt Float}, as appropriate. - -\spadcommand{i := 3 :: Float } -$$ -3.0 -$$ -\returnType{Type: Float} - -\spadcommand{i :: Integer } -$$ -3 -$$ -\returnType{Type: Integer} - -\spadcommand{i :: Fraction Integer } -$$ -3 -$$ -\returnType{Type: Fraction Integer} - -Since you are explicitly asking for a conversion, you must take -responsibility for any loss of exactness. - -\spadcommand{r := 3/7 :: Float } -$$ -0.4285714285\ 7142857143 -$$ -\returnType{Type: Float} - -\spadcommand{r :: Fraction Integer } -$$ -3 \over 7 -$$ -\returnType{Type: Fraction Integer} - -This conversion cannot be performed: use \spadfunFrom{truncate}{Float} -or \spadfunFrom{round}{Float} if that is what you intend. - -\spadcommand{r :: Integer } -\begin{verbatim} - Cannot convert from type Float to Integer for value - 0.4285714285 7142857143 -\end{verbatim} - -The operations \spadfunFrom{truncate}{Float} and \spadfunFrom{round}{Float} -truncate \ldots - -\spadcommand{truncate 3.6} -$$ -3.0 -$$ -\returnType{Type: Float} - -and round to the nearest integral {\tt Float} respectively. - -\spadcommand{round 3.6} -$$ -4.0 -$$ -\returnType{Type: Float} - -\spadcommand{truncate(-3.6)} -$$ --{3.0} -$$ -\returnType{Type: Float} - -\spadcommand{round(-3.6)} -$$ --{4.0} -$$ -\returnType{Type: Float} - -The operation \spadfunFrom{fractionPart}{Float} computes the -fractional part of {\tt x}, that is, {\tt x - truncate x}. - -\spadcommand{fractionPart 3.6} -$$ -0.6 -$$ -\returnType{Type: Float} - -The operation \spadfunFrom{digits}{Float} allows the user to set the -precision. It returns the previous value it was using. - -\spadcommand{digits 40 } -$$ -20 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{sqrt 0.2} -$$ -0.4472135954\ 9995793928\ 1834733746\ 2552470881 -$$ -\returnType{Type: Float} - -\spadcommand{pi()\$Float } -$$ -3.1415926535\ 8979323846\ 2643383279\ 502884197 -$$ -\returnType{Type: Float} - -The precision is only limited by the computer memory available. -Calculations at 500 or more digits of precision are not difficult. - -\spadcommand{digits 500 } -$$ -40 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{pi()\$Float } -$$ -\begin{array}{@{}l} -3.1415926535\ 8979323846\ 2643383279\ 5028841971\ 6939937510\ 5820974944 -\\ -\displaystyle -\ \ 5923078164\ 0628620899\ 8628034825\ 3421170679\ 8214808651\ 3282306647 -\\ -\displaystyle -\ \ 0938446095\ 5058223172\ 5359408128\ 4811174502\ 8410270193\ 8521105559 -\\ -\displaystyle -\ \ 6446229489\ 5493038196\ 4428810975\ 6659334461\ 2847564823\ 3786783165 -\\ -\displaystyle -\ \ 2712019091\ 4564856692\ 3460348610\ 4543266482\ 1339360726\ 0249141273 -\\ -\displaystyle -\ \ 7245870066\ 0631558817\ 4881520920\ 9628292540\ 9171536436\ 7892590360 -\\ -\displaystyle -\ \ 0113305305\ 4882046652\ 1384146951\ 9415116094\ 3305727036\ 5759591953 -\\ -\displaystyle -\ \ 0921861173\ 8193261179\ 3105118548\ 0744623799\ 6274956735\ 1885752724 -\\ -\displaystyle -\ \ 8912279381\ 830119491 -\end{array} -$$ -\returnType{Type: Float} - -Reset \spadfunFrom{digits}{Float} to its default value. - -\spadcommand{digits 20} -$$ -500 -$$ -\returnType{Type: PositiveInteger} - -Numbers of type {\tt Float} are represented as a record of two -integers, namely, the mantissa and the exponent where the base of the -exponent is binary. That is, the floating point number {\tt (m,e)} -represents the number $m \cdot 2^e$. A consequence of using a binary -base is that decimal numbers can not, in general, be represented -exactly. - -\subsection{Output Functions} - -A number of operations exist for specifying how numbers of type -{\tt Float} are to be displayed. By default, spaces are inserted every ten -digits in the output for readability.\footnote{Note that you cannot -include spaces in the input form of a floating point number, though -you can use underscores.} - -Output spacing can be modified with the \spadfunFrom{outputSpacing}{Float} -operation. This inserts no spaces and then displays the value of {\tt x}. - -\spadcommand{outputSpacing 0; x := sqrt 0.2 } -$$ -0.44721359549995793928 -$$ -\returnType{Type: Float} - -Issue this to have the spaces inserted every {\tt 5} digits. - -\spadcommand{outputSpacing 5; x } -$$ -0.44721\ 35954\ 99957\ 93928 -$$ -\returnType{Type: Float} - -By default, the system displays floats in either fixed format -or scientific format, depending on the magnitude of the number. - -\spadcommand{y := x/10**10 } -$$ -0.44721\ 35954\ 99957\ 93928\ {\rm E\ }-10 -$$ -\returnType{Type: Float} - -A particular format may be requested with the operations -\spadfunFrom{outputFloating}{Float} and \spadfunFrom{outputFixed}{Float}. - -\spadcommand{outputFloating(); x } -$$ -0.44721\ 35954\ 99957\ 93928\ {\rm E\ }0 -$$ -\returnType{Type: Float} - -\spadcommand{outputFixed(); y } -$$ -0.00000\ 00000\ 44721\ 35954\ 99957\ 93928 -$$ -\returnType{Type: Float} - -Additionally, you can ask for {\tt n} digits to be displayed after the -decimal point. - -\spadcommand{outputFloating 2; y } -$$ -0.45\ {\rm E\ } -10 -$$ -\returnType{Type: Float} - -\spadcommand{outputFixed 2; x } -$$ -0.45 -$$ -\returnType{Type: Float} - -This resets the output printing to the default behavior. - -\spadcommand{outputGeneral()} -\returnType{Type: Void} - -\subsection{An Example: Determinant of a Hilbert Matrix} - -Consider the problem of computing the determinant of a {\tt 10} by -{\tt 10} Hilbert matrix. The $(i,j)$-th entry of a Hilbert -matrix is given by {\tt 1/(i+j+1)}. - -First do the computation using rational numbers to obtain the -exact result. - -\spadcommand{a: Matrix Fraction Integer := matrix [ [1/(i+j+1) for j in 0..9] for i in 0..9] } -$$ -\left[ -\begin{array}{cccccccccc} -1 & {1 \over 2} & {1 \over 3} & {1 \over 4} & {1 \over 5} & {1 \over 6} & -{1 \over 7} & {1 \over 8} & {1 \over 9} & {1 \over {10}} \\ -{1 \over 2} & {1 \over 3} & {1 \over 4} & {1 \over 5} & {1 \over 6} & -{1 \over 7} & {1 \over 8} & {1 \over 9} & {1 \over {10}} & {1 \over {11}} \\ -{1 \over 3} & {1 \over 4} & {1 \over 5} & {1 \over 6} & {1 \over 7} & -{1 \over 8} & {1 \over 9} & {1 \over {10}} & {1 \over {11}} & -{1 \over {12}} \\ -{1 \over 4} & {1 \over 5} & {1 \over 6} & {1 \over 7} & {1 \over 8} & -{1 \over 9} & {1 \over {10}} & {1 \over {11}} & {1 \over {12}} & -{1 \over {13}} \\ -{1 \over 5} & {1 \over 6} & {1 \over 7} & {1 \over 8} & {1 \over 9} & -{1 \over {10}} & {1 \over {11}} & {1 \over {12}} & {1 \over {13}} & -{1 \over {14}} \\ -{1 \over 6} & {1 \over 7} & {1 \over 8} & {1 \over 9} & {1 \over {10}} & -{1 \over {11}} & {1 \over {12}} & {1 \over {13}} & {1 \over {14}} & -{1 \over {15}} \\ -{1 \over 7} & {1 \over 8} & {1 \over 9} & {1 \over {10}} & {1 \over {11}} & -{1 \over {12}} & {1 \over {13}} & {1 \over {14}} & {1 \over {15}} & -{1 \over {16}} \\ -{1 \over 8} & {1 \over 9} & {1 \over {10}} & {1 \over {11}} & {1 \over {12}} -& {1 \over {13}} & {1 \over {14}} & {1 \over {15}} & {1 \over {16}} & -{1 \over {17}} \\ -{1 \over 9} & {1 \over {10}} & {1 \over {11}} & {1 \over {12}} & -{1 \over {13}} & {1 \over {14}} & {1 \over {15}} & {1 \over {16}} & -{1 \over {17}} & {1 \over {18}} \\ -{1 \over {10}} & {1 \over {11}} & {1 \over {12}} & {1 \over {13}} & -{1 \over {14}} & {1 \over {15}} & {1 \over {16}} & {1 \over {17}} & -{1 \over {18}} & {1 \over {19}} -\end{array} -\right] -$$ -\returnType{Type: Matrix Fraction Integer} - -This version of \spadfunFrom{determinant}{Matrix} uses Gaussian elimination. - -\spadcommand{d:= determinant a } -$$ -1 \over {46206893947914691316295628839036278726983680000000000} -$$ -\returnType{Type: Fraction Integer} - -\spadcommand{d :: Float } -$$ -0.21641\ 79226\ 43149\ 18691\ {\rm E\ } -52 -$$ -\returnType{Type: Float} - -Now use hardware floats. Note that a semicolon (;) is used to prevent -the display of the matrix. - -\spadcommand{b: Matrix DoubleFloat := matrix [ [1/(i+j+1\$DoubleFloat) for j in 0..9] for i in 0..9]; } -\returnType{Type: Matrix DoubleFloat} - -The result given by hardware floats is correct only to four -significant digits of precision. In the jargon of numerical analysis, -the Hilbert matrix is said to be ``ill-conditioned.'' - -\spadcommand{determinant b } -$$ -2.1643677945721411e-53 -$$ -\returnType{Type: DoubleFloat} - -Now repeat the computation at a higher precision using {\tt Float}. - -\spadcommand{digits 40 } -$$ -20 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{c: Matrix Float := matrix [ [1/(i+j+1\$Float) for j in 0..9] for i in 0..9]; } -\returnType{Type: Matrix Float} - -\spadcommand{determinant c } -$$ -0.21641\ 79226\ 43149\ 18690\ 60594\ 98362\ 26174\ 36159\ {\rm E\ } -52 -$$ -\returnType{Type: Float} - -Reset \spadfunFrom{digits}{Float} to its default value. - -\spadcommand{digits 20} -$$ -40 -$$ -\returnType{Type: PositiveInteger} - -\section{Fraction} -\label{FractionXmpPage} - -The {\tt Fraction} domain implements quotients. The elements must -belong to a domain of category {\tt IntegralDomain}: multiplication -must be commutative and the product of two non-zero elements must not -be zero. This allows you to make fractions of most things you would -think of, but don't expect to create a fraction of two matrices! The -abbreviation for {\tt Fraction} is {\tt FRAC}. - -Use \spadopFrom{/}{Fraction} to create a fraction. - -\spadcommand{a := 11/12 } -$$ -{11} \over {12} -$$ -\returnType{Type: Fraction Integer} - -\spadcommand{b := 23/24 } -$$ -{23} \over {24} -$$ -\returnType{Type: Fraction Integer} - -The standard arithmetic operations are available. - -\spadcommand{3 - a*b**2 + a + b/a } -$$ -{313271} \over {76032} -$$ -\returnType{Type: Fraction Integer} - -Extract the numerator and denominator by using -\spadfunFrom{numer}{Fraction} and \spadfunFrom{denom}{Fraction}, -respectively. - -\spadcommand{numer(a) } -$$ -11 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{denom(b) } -$$ -24 -$$ -\returnType{Type: PositiveInteger} - -Operations like \spadfunFrom{max}{Fraction}, -\spadfunFrom{min}{Fraction}, \spadfunFrom{negative?}{Fraction}, -\spadfunFrom{positive?}{Fraction} and \spadfunFrom{zero?}{Fraction} -are all available if they are provided for the numerators and -denominators. -See \ref{IntegerXmpPage} on page~\pageref{IntegerXmpPage} for examples. - -Don't expect a useful answer from \spadfunFrom{factor}{Fraction}, -\spadfunFrom{gcd}{Fraction} or \spadfunFrom{lcm}{Fraction} if you apply -them to fractions. - -\spadcommand{r := (x**2 + 2*x + 1)/(x**2 - 2*x + 1) } -$$ -{{x \sp 2}+{2 \ x}+1} \over {{x \sp 2} -{2 \ x}+1} -$$ -\returnType{Type: Fraction Polynomial Integer} - -Since all non-zero fractions are invertible, these operations have trivial -definitions. - -\spadcommand{factor(r) } -$$ -{{x \sp 2}+{2 \ x}+1} \over {{x \sp 2} -{2 \ x}+1} -$$ -\returnType{Type: Factored Fraction Polynomial Integer} - -Use \spadfunFrom{map}{Fraction} to apply \spadfunFrom{factor}{Fraction} to -the numerator and denominator, which is probably what you mean. - -\spadcommand{map(factor,r) } -$$ -{{\left( x+1 -\right)} -\sp 2} \over {{\left( x -1 -\right)} -\sp 2} -$$ -\returnType{Type: Fraction Factored Polynomial Integer} - -Other forms of fractions are available. Use {\tt continuedFraction} -to create a continued fraction. - -\spadcommand{continuedFraction(7/12)} -$$ -\zag{1}{1}+ \zag{1}{1}+ \zag{1}{2}+ \zag{1}{2} -$$ -\returnType{Type: ContinuedFraction Integer} - -Use {\tt partialFraction} to create a partial fraction. -See -\ref{ContinuedFractionXmpPage} on page~\pageref{ContinuedFractionXmpPage} -and \ref{PartialFractionXmpPage} on page~\pageref{PartialFractionXmpPage} for -additional information and examples. - -\spadcommand{partialFraction(7,12)} -$$ -1 -{3 \over {2 \sp 2}}+{1 \over 3} -$$ -\returnType{Type: PartialFraction Integer} - -Use conversion to create alternative views of fractions with objects -moved in and out of the numerator and denominator. - -\spadcommand{g := 2/3 + 4/5*\%i } -$$ -{2 \over 3}+{{4 \over 5} \ i} -$$ -\returnType{Type: Complex Fraction Integer} - -Conversion is discussed in detail in -Section~\ref{ugTypesConvertPage} -on page~\pageref{ugTypesConvertPage}. - -\spadcommand{g :: FRAC COMPLEX INT } -$$ -{{10}+{{12} \ i}} \over {15} -$$ -\returnType{Type: Fraction Complex Integer} - -\section{FullPartialFractionExpansion} -\label{FullPartialFractionExpansionXmpPage} - -The domain {\tt FullPartialFractionExpansion} implements -factor-free conversion of quotients to full partial fractions. - -Our examples will all involve quotients of univariate polynomials -with rational number coefficients. - -\spadcommand{Fx := FRAC UP(x, FRAC INT) } -$$ -\mbox{\rm Fraction UnivariatePolynomial(x,Fraction Integer)} -$$ -\returnType{Type: Domain} - -Here is a simple-looking rational function. - -\spadcommand{f : Fx := 36 / (x**5-2*x**4-2*x**3+4*x**2+x-2) } -$$ -{36} \over {{x \sp 5} -{2 \ {x \sp 4}} -{2 \ {x \sp 3}}+{4 \ {x \sp 2}}+x --2} -$$ -\returnType{Type: Fraction UnivariatePolynomial(x,Fraction Integer)} - -We use \spadfunFrom{fullPartialFraction}{FullPartialFractionExpansion} -to convert it to an object of type {\tt FullPartialFractionExpansion}. - -\spadcommand{g := fullPartialFraction f } -$$ -{4 \over {x -2}} -{4 \over {x+1}}+ -{\sum \sb{\displaystyle {{{ \%A \sp 2} -1}=0}} -{{-{3 \ \%A} -6} \over {{\left( x - \%A \right)}\sp 2}}} -$$ -\returnType{Type: FullPartialFractionExpansion(Fraction Integer,UnivariatePolynomial(x,Fraction Integer))} - -Use a coercion to change it back into a quotient. - -\spadcommand{g :: Fx } -$$ -{36} \over {{x \sp 5} -{2 \ {x \sp 4}} -{2 \ {x \sp 3}}+{4 \ {x \sp 2}}+x --2} -$$ -\returnType{Type: Fraction UnivariatePolynomial(x,Fraction Integer)} - -Full partial fractions differentiate faster than rational functions. - -\spadcommand{g5 := D(g, 5) } -$$ --{{480} \over {{\left( x -2 \right)}\sp 6}}+ -{{480} \over {{\left( x+1 \right)}\sp 6}}+ -{\sum \sb{\displaystyle {{{ \%A \sp 2} -1}=0}} -{{{{2160} \ \%A}+{4320}} \over {{\left( x - \%A \right)}\sp 7}}} -$$ -\returnType{Type: FullPartialFractionExpansion(Fraction Integer,UnivariatePolynomial(x,Fraction Integer))} - -\spadcommand{f5 := D(f, 5) } -$$ -\left( -\begin{array}{@{}l} --{{544320} \ {x \sp {10}}}+ -{{4354560} \ {x \sp 9}} - -{{14696640} \ {x \sp 8}}+ -{{28615680} \ {x \sp 7}} - -\\ -\\ -\displaystyle -{{40085280} \ {x \sp 6}}+ -{{46656000} \ {x \sp 5}} - -{{39411360} \ {x \sp 4}}+ -{{18247680} \ {x \sp 3}} - -\\ -\\ -\displaystyle -{{5870880} \ {x \sp 2}}+ -{{3317760} \ x}+{246240} -\end{array} -\right) -\over -\left( -\begin{array}{@{}l} -{x \sp {20}} - -{{12} \ {x \sp {19}}}+ -{{53} \ {x \sp {18}}} - -{{76} \ {x \sp {17}}} - -{{159} \ {x \sp {16}}}+ -{{676} \ {x \sp {15}}} - -{{391} \ {x \sp {14}}} - -\\ -\\ -\displaystyle -{{1596} \ {x \sp {13}}}+ -{{2527} \ {x \sp {12}}}+ -{{1148} \ {x \sp {11}}} - -{{4977} \ {x \sp {10}}}+ -{{1372} \ {x \sp 9}}+ -\\ -\\ -\displaystyle -{{4907} \ {x \sp 8}} - -{{3444} \ {x \sp 7}} --{{2381} \ {x \sp 6}}+ -{{2924} \ {x \sp 5}}+ -{{276} \ {x \sp 4}} - -\\ -\\ -\displaystyle -{{1184} \ {x \sp 3}}+ -{{208} \ {x \sp 2}}+ -{{192} \ x} - -{64} -\end{array} -\right) -$$ -\returnType{Type: Fraction UnivariatePolynomial(x,Fraction Integer)} - -We can check that the two forms represent the same function. - -\spadcommand{g5::Fx - f5 } -$$ -0 -$$ -\returnType{Type: Fraction UnivariatePolynomial(x,Fraction Integer)} - -Here are some examples that are more complicated. - -\spadcommand{f : Fx := (x**5 * (x-1)) / ((x**2 + x + 1)**2 * (x-2)**3) } -$$ -{{x \sp 6} - -{x \sp 5}} -\over -{{x \sp 7} - -{4 \ {x \sp 6}}+ -{3 \ {x \sp 5}}+ -{9 \ {x \sp 3}} - -{6 \ {x \sp 2}} - -{4 \ x} - -8} -$$ -\returnType{Type: Fraction UnivariatePolynomial(x,Fraction Integer)} - -\spadcommand{g := fullPartialFraction f } -$$ -\begin{array}{@{}l} -{{{1952} \over {2401}} \over {x -2}}+ -{{{464} \over {343}} \over {{\left( x -2 \right)}\sp 2}}+ -{{{32} \over {49}} \over {{\left( x -2 \right)}\sp 3}}+ -\\ -\\ -\displaystyle -{\sum \sb{\displaystyle {{{ \%A \sp 2}+ \%A+1}=0}} -{{-{{{179} \over {2401}} \ \%A}+{{135} \over {2401}}} \over {x - \%A}}}+ -\\ -\\ -\displaystyle -{\sum \sb{\displaystyle {{{ \%A \sp 2}+ \%A+1}=0}} -{{{{{37} \over {1029}} \ \%A}+ -{{20} \over {1029}}} \over {{\left( x - \%A \right)}\sp 2}}} -\end{array} -$$ -\returnType{Type: FullPartialFractionExpansion(Fraction Integer,UnivariatePolynomial(x,Fraction Integer))} - -\spadcommand{g :: Fx - f } -$$ -0 -$$ -\returnType{Type: Fraction UnivariatePolynomial(x,Fraction Integer)} - -\spadcommand{f : Fx := (2*x**7-7*x**5+26*x**3+8*x) / (x**8-5*x**6+6*x**4+4*x**2-8) } -$$ -{{2 \ {x \sp 7}} -{7 \ {x \sp 5}}+{{26} \ {x \sp 3}}+{8 \ x}} -\over -{{x \sp 8} -{5 \ {x \sp 6}}+{6 \ {x \sp 4}}+{4 \ {x \sp 2}} -8} -$$ -\returnType{Type: Fraction UnivariatePolynomial(x,Fraction Integer)} - -\spadcommand{g := fullPartialFraction f } -$$ -\begin{array}{@{}l} -{\sum \sb{\displaystyle {{{ \%A \sp 2} -2}=0}} -{{1 \over 2} \over {x - \%A}}}+ -\\ -\\ -\displaystyle -{\sum \sb{\displaystyle {{{ \%A \sp 2} -2}=0}} -{1 \over {{\left( x - \%A \right)}\sp 3}}}+ -\\ -\\ -\displaystyle -{\sum \sb{\displaystyle {{{ \%A \sp 2}+1}=0}} -{{1 \over 2} \over {x - \%A}}} -\end{array} -$$ -\returnType{Type: FullPartialFractionExpansion(Fraction Integer,UnivariatePolynomial(x,Fraction Integer))} - -\spadcommand{g :: Fx - f } -$$ -0 -$$ -\returnType{Type: Fraction UnivariatePolynomial(x,Fraction Integer)} - -\spadcommand{f:Fx := x**3 / (x**21 + 2*x**20 + 4*x**19 + 7*x**18 + 10*x**17 + 17*x**16 + 22*x**15 + 30*x**14 + 36*x**13 + 40*x**12 + 47*x**11 + 46*x**10 + 49*x**9 + 43*x**8 + 38*x**7 + 32*x**6 + 23*x**5 + 19*x**4 + 10*x**3 + 7*x**2 + 2*x + 1)} -$$ -{x \sp 3} -\over -\left( -\begin{array}{@{}l} -{x \sp {21}}+ -{2 \ {x \sp {20}}}+ -{4 \ {x \sp {19}}}+ -{7 \ {x \sp {18}}}+ -{{10} \ {x \sp {17}}}+ -{{22} \ {x \sp {15}}}+ -{{30} \ {x \sp {14}}}+ -\\ -\\ -\displaystyle -{{36} \ {x \sp {13}}}+ -{{40} \ {x \sp {12}}}+ -{{47} \ {x \sp {11}}}+ -{{46} \ {x \sp {10}}}+ -{{49} \ {x \sp 9}}+ -{{43} \ {x \sp 8}}+ -{{38} \ {x \sp 7}}+ -\\ -\\ -\displaystyle -{{32} \ {x \sp 6}}+ -{{23} \ {x \sp 5}}+ -{{19} \ {x \sp 4}}+ -{{10} \ {x \sp 3}}+ -{7 \ {x \sp 2}}+ -{2 \ x}+ -1 -\end{array} -\right) -$$ -\returnType{Type: Fraction UnivariatePolynomial(x,Fraction Integer)} - -\spadcommand{g := fullPartialFraction f } -$$ -\begin{array}{@{}l} -{\sum \sb{\displaystyle {{{ \%A \sp 2}+1}=0}} -{{{1 \over 2} \ \%A} \over {x - \%A}}}+ -{\sum \sb{\displaystyle {{{ \%A \sp 2}+ \%A+1}=0}} -{{{{1 \over 9} \ \%A} -{{19} \over {27}}} \over {x - \%A}}}+ -\\ -\\ -\displaystyle -{\sum \sb{\displaystyle {{{ \%A \sp 2}+ \%A+1}=0}} -{{{{1 \over {27}} \ \%A} -{1 \over {27}}} -\over {{\left( x - \%A \right)}\sp 2}}}+ -\\ -\\ -\displaystyle -\sum \sb{\displaystyle {{{ \%A \sp 5}+{ \%A \sp 2}+1}=0}} -\left( -\begin{array}{@{}l} --{{{96556567040} \over {912390759099}} \ { \%A \sp 4}}+ -{{{420961732891} \over {912390759099}} \ { \%A \sp 3}} - -\\ -\\ -\displaystyle -{{{59101056149} \over {912390759099}} \ { \%A \sp 2}} - -{{{373545875923} \over {912390759099}} \ \%A}+ -\\ -\\ -\displaystyle -{{529673492498} \over {912390759099}} -\end{array} -\right) -\over {x - \%A}+ -\\ -\\ -\displaystyle -\sum \sb{\displaystyle {{{ \%A \sp 5}+{ \%A \sp 2}+1}=0}} -\left( -\begin{array}{@{}l} --{{{5580868} \over {94070601}} \ { \%A \sp 4}} - -{{{2024443} \over {94070601}} \ { \%A \sp 3}}+ -{{{4321919} \over {94070601}} \ { \%A \sp 2}} - -\\ -\\ -\displaystyle -{{{84614} \over {1542141}} \ \%A} - -{{5070620} \over {94070601}} -\end{array} -\right) -\over {{\left( x - \%A \right)}\sp 2}+ -\\ -\\ -\displaystyle -\sum \sb{\displaystyle {{{ \%A \sp 5}+{ \%A \sp 2}+1}=0}} -\left( -\begin{array}{@{}l} -{{{1610957} \over {94070601}} \ { \%A \sp 4}}+ -{{{2763014} \over {94070601}} \ { \%A \sp 3}} - -{{{2016775} \over {94070601}} \ { \%A \sp 2}}+ -\\ -\\ -\displaystyle -{{{266953} \over {94070601}} \ \%A}+ -{{4529359} \over {94070601}} -\end{array} -\right) -\over {{\left( x - \%A \right)}\sp 3} -\end{array} -$$ -\returnType{Type: FullPartialFractionExpansion(Fraction Integer,UnivariatePolynomial(x,Fraction Integer))} - -This verification takes much longer than the conversion to -partial fractions. - -\spadcommand{g :: Fx - f } -$$ -0 -$$ -\returnType{Type: Fraction UnivariatePolynomial(x,Fraction Integer)} - -For more information, see the paper: Bronstein, M and Salvy, B. -``Full Partial Fraction Decomposition of Rational Functions,'' -{\it Proceedings of ISSAC'93, Kiev}, ACM Press. Also see -\ref{PartialFractionXmpPage} on page~\pageref{PartialFractionXmpPage} -for standard partial fraction decompositions. - -\section{GeneralSparseTable} -\label{GeneralSparseTableXmpPage} - -Sometimes when working with tables there is a natural value to use as -the entry in all but a few cases. The {\tt GeneralSparseTable} -constructor can be used to provide any table type with a default value -for entries. See \ref{TableXmpPage} on page~\pageref{TableXmpPage} -for general information about tables. - -Suppose we launched a fund-raising campaign to raise fifty thousand dollars. -To record the contributions, we want a table with strings as keys -(for the names) and integer entries (for the amount). -In a data base of cash contributions, unless someone -has been explicitly entered, it is reasonable to assume they have made -a zero dollar contribution. - -This creates a keyed access file with default entry {\tt 0}. - -\spadcommand{patrons: GeneralSparseTable(String, Integer, KeyedAccessFile(Integer), 0) := table() ; } -\returnType{Type: GeneralSparseTable(String,Integer,KeyedAccessFile Integer,0)} - - -Now {\tt patrons} can be used just as any other table. -Here we record two gifts. - -\spadcommand{patrons."Smith" := 10500 } -$$ -10500 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{patrons."Jones" := 22000 } -$$ -22000 -$$ -\returnType{Type: PositiveInteger} - -Now let us look up the size of the contributions from Jones and Stingy. - -\spadcommand{patrons."Jones" } -$$ -22000 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{patrons."Stingy" } -$$ -0 -$$ -\returnType{Type: NonNegativeInteger} - -Have we met our seventy thousand dollar goal? - -\spadcommand{reduce(+, entries patrons) } -$$ -32500 -$$ -\returnType{Type: PositiveInteger} - -So the project is cancelled and we can delete the data base: - -\spadcommand{)system rm -r kaf*.sdata } - -\section{GroebnerFactorizationPackage} -\label{GroebnerFactorizationPackageXmpPage} - -Solving systems of polynomial equations with the Gr\"{o}bner basis -algorithm can often be very time consuming because, in general, the -algorithm has exponential run-time. These systems, which often come -from concrete applications, frequently have symmetries which are not -taken advantage of by the algorithm. However, it often happens in -this case that the polynomials which occur during the Gr\"{o}bner -calculations are reducible. Since Axiom has an excellent polynomial -factorization algorithm, it is very natural to combine the Gr\"{o}bner -and factorization algorithms. - -{\tt GroebnerFactorizationPackage} exports the -\spadfunFrom{groebnerFactorize}{GroebnerFactorizationPackage} -operation which implements a modified Gr\"{o}bner basis algorithm. In -this algorithm, each polynomial that is to be put into the partial -list of the basis is first factored. The remaining calculation is -split into as many parts as there are irreducible factors. Call these -factors $p_1, \ldots,p_n.$ In the branches corresponding to $p_2, -\ldots,p_n,$ the factor $p_1$ can be divided out, and so on. This -package also contains operations that allow you to specify the -polynomials that are not zero on the common roots of the final -Gr\"{o}bner basis. - -Here is an example from chemistry. In a theoretical model of the -cyclohexan ${\rm C}_6{\rm H}_{12}$, the six carbon atoms each sit in -the center of gravity of a tetrahedron that has two hydrogen atoms and -two carbon atoms at its corners. We first normalize and set the -length of each edge to 1. Hence, the distances of one fixed carbon -atom to each of its immediate neighbours is 1. We will denote the -distances to the other three carbon atoms by $x$, $y$ and $z$. - -A.~Dress developed a theory to decide whether a set of points -and distances between them can be realized in an $n$-dimensional space. -Here, of course, we have $n = 3$. - -\spadcommand{mfzn : SQMATRIX(6,DMP([x,y,z],Fraction INT)) := [ [0,1,1,1,1,1], [1,0,1,8/3,x,8/3], [1,1,0,1,8/3,y], [1,8/3,1,0,1,8/3], [1,x,8/3,1,0,1], [1,8/3,y,8/3,1,0] ] } -$$ -\left[ -\begin{array}{cccccc} -0 & 1 & 1 & 1 & 1 & 1 \\ -1 & 0 & 1 & {8 \over 3} & x & {8 \over 3} \\ -1 & 1 & 0 & 1 & {8 \over 3} & y \\ -1 & {8 \over 3} & 1 & 0 & 1 & {8 \over 3} \\ -1 & x & {8 \over 3} & 1 & 0 & 1 \\ -1 & {8 \over 3} & y & {8 \over 3} & 1 & 0 -\end{array} -\right] -$$ -\returnType{Type: SquareMatrix(6,DistributedMultivariatePolynomial([x,y,z],Fraction Integer))} - -For the cyclohexan, the distances have to satisfy this equation. - -\spadcommand{eq := determinant mfzn } -$$ -\begin{array}{@{}l} --{{x \sp 2} \ {y \sp 2}}+ -{{{22} \over 3} \ {x \sp 2} \ y} - -{{{25} \over 9} \ {x \sp 2}}+ -{{{22} \over 3} \ x \ {y \sp 2}} - -{{{388} \over 9} \ x \ y} - -\\ -\\ -\displaystyle -{{{250} \over {27}} \ x} - -{{{25} \over 9} \ {y \sp 2}} - -{{{250} \over {27}} \ y}+ -{{14575} \over {81}} -\end{array} -$$ -\returnType{Type: DistributedMultivariatePolynomial([x,y,z],Fraction Integer)} - -They also must satisfy the equations -given by cyclic shifts of the indeterminates. - -\spadcommand{groebnerFactorize [eq, eval(eq, [x,y,z], [y,z,x]), eval(eq, [x,y,z], [z,x,y])] } -$$ -\begin{array}{@{}l} -\left[ - -\begin{array}{@{}l} -\left[ -{x \ y}+ -{x \ z} - -{{{22} \over 3} \ x}+ -{y \ z} - -{{{22} \over 3} \ y} - -{{{22} \over 3} \ z}+ -{{121} \over 3}, -\right. -\\ -\\ -\displaystyle -{x \ {z \sp 2}} - -{{{22} \over 3} \ x \ z}+ -{{{25} \over 9} \ x}+ -{y \ {z \sp 2}} - -{{{22} \over 3} \ y \ z}+ -{{{25} \over 9} \ y} - -{{{22} \over 3} \ {z \sp 2}}+ -{{{388} \over 9} \ z}+ -{{250} \over {27}}, -\\ -\\ -\displaystyle -\left. -\begin{array}{@{}l} -{{y \sp 2} \ {z \sp 2}} - -{{{22} \over 3} \ {y \sp 2} \ z}+ -{{{25} \over 9} \ {y \sp 2}} - -{{{22} \over 3} \ y \ {z \sp 2}}+ -{{{388} \over 9} \ y \ z}+ -{{{250} \over {27}} \ y}+ -\\ -\\ -\displaystyle -{{{25} \over 9} \ {z \sp 2}}+ -{{{250} \over {27}} \ z} - -{{14575} \over {81}} -\end{array} -\right], -\end{array} -\right. -\\ -\\ -\displaystyle -{\left[ -{x+y -{{21994} \over {5625}}}, -{{y \sp 2} -{{{21994} \over {5625}} \ y}+{{4427} \over {675}}}, -{z -{{463} \over {87}}} -\right]}, -\\ -\\ -\displaystyle -{\left[ -{{x \sp 2} - -{{1 \over 2} \ x \ z} - -{{{11} \over 2} \ x} - -{{5 \over 6} \ z}+ -{{265} \over {18}}}, -{y -z}, -{{z \sp 2} -{{{38} \over 3} \ z}+{{265} \over 9}} -\right]}, -\\ -\\ -\displaystyle -{\left[ -{x -{{25} \over 9}}, -{y -{{11} \over 3}}, -{z -{{11} \over 3}} \right]}, -\\ -\\ -\displaystyle -{\left[ -{x -{{11} \over 3}}, -{y -{{11} \over 3}}, -{z -{{11} \over 3}} -\right]}, -\\ -\\ -\displaystyle -{\left[ -{x+{5 \over 3}}, -{y+{5 \over 3}}, -{z+{5 \over 3}} -\right]}, -\\ -\\ -\displaystyle -\left. -{\left[ -{x -{{19} \over 3}}, -{y+{5 \over 3}}, -{z+{5 \over 3}} -\right]} -\right] -\end{array} -$$ -\returnType{Type: List List -DistributedMultivariatePolynomial([x,y,z],Fraction Integer)} - -The union of the solutions of this list is the solution of our -original problem. If we impose positivity conditions, we get two -relevant ideals. One ideal is zero-dimensional, namely $x = y = z = 11/3$, -and this determines the ``boat'' form of the cyclohexan. The -other ideal is one-dimensional, which means that we have a solution -space given by one parameter. This gives the ``chair'' form of the -cyclohexan. The parameter describes the angle of the ``back of the -chair.'' - -\spadfunFrom{groebnerFactorize}{GroebnerFactorizationPackage} has an -optional {\tt Boolean}-valued second argument. When it is {\tt true} -partial results are displayed, since it may happen that the -calculation does not terminate in a reasonable time. See the source -code for {\tt GroebnerFactorizationPackage} in {\bf groebf.input} -for more details about the algorithms used. - -\section{Heap} -\label{HeapXmpPage} - -The domain {\tt Heap(S)} implements a priority queue of objects of -type {\tt S} such that the operation {\tt extract!} removes and -returns the maximum element. The implementation represents heaps as -flexible arrays (see -\ref{FlexibleArrayXmpPage} on page~\pageref{FlexibleArrayXmpPage}). -The representation and algorithms give complexity of $O(\log(n))$ -for insertion and extractions, and $O(n)$ for construction. - -Create a heap of six elements. - -\spadcommand{h := heap [-4,9,11,2,7,-7]} -$$ -\left[ -{11}, 7, 9, -4, 2, -7 -\right] -$$ -\returnType{Type: Heap Integer} - -Use {\tt insert!} to add an element. - -\spadcommand{insert!(3,h)} -$$ -\left[ -{11}, 7, 9, -4, 2, -7, 3 -\right] -$$ -\returnType{Type: Heap Integer} - -The operation {\tt extract!} removes and returns the maximum element. - -\spadcommand{extract! h} -$$ -11 -$$ -\returnType{Type: PositiveInteger} - -The internal structure of {\tt h} has been appropriately adjusted. - -\spadcommand{h} -$$ -\left[ -9, 7, 3, -4, 2, -7 -\right] -$$ -\returnType{Type: Heap Integer} - -Now {\tt extract!} elements repeatedly until none are left, collecting -the elements in a list. - -\spadcommand{[extract!(h) while not empty?(h)]} -$$ -\left[ -9, 7, 3, 2, -4, -7 -\right] -$$ -\returnType{Type: List Integer} - -Another way to produce the same result is by defining a {\tt heapsort} -function. - -\spadcommand{heapsort(x) == (empty? x => []; cons(extract!(x),heapsort x))} -\returnType{Void} - -Create another sample heap. - -\spadcommand{h1 := heap [17,-4,9,-11,2,7,-7]} -$$ -\left[ -{17}, 2, 9, -{11}, -4, 7, -7 -\right] -$$ -\returnType{Type: Heap Integer} - -Apply {\tt heapsort} to present elements in order. - -\spadcommand{heapsort h1} -$$ -\left[ -{17}, 9, 7, 2, -4, -7, -{11} -\right] -$$ -\returnType{Type: List Integer} - -\section{HexadecimalExpansion} -\label{HexadecimalExpansionXmpPage} - -All rationals have repeating hexadecimal expansions. The operation -\spadfunFrom{hex}{HexadecimalExpansion} returns these expansions of -type {\tt HexadecimalExpansion}. Operations to access the individual -numerals of a hexadecimal expansion can be obtained by converting the -value to {\tt RadixExpansion(16)}. More examples of expansions are -available in the -\ref{DecimalExpansionXmpPage} on page~\pageref{DecimalExpansionXmpPage}, -\ref{BinaryExpansionXmpPage} on page~\pageref{BinaryExpansionXmpPage}, and -\ref{RadixExpansionXmpPage} on page~\pageref{RadixExpansionXmpPage}. - -This is a hexadecimal expansion of a rational number. - -\spadcommand{r := hex(22/7) } -$$ -3.{\overline {249}} -$$ -\returnType{Type: HexadecimalExpansion} - -Arithmetic is exact. - -\spadcommand{r + hex(6/7) } -$$ -4 -$$ -\returnType{Type: HexadecimalExpansion} - -The period of the expansion can be short or long \ldots - -\spadcommand{[hex(1/i) for i in 350..354] } -$$ -\begin{array}{@{}l} -\left[ -{0.0{\overline {\rm 0BB3EE721A54D88}}}, -{0.{\overline {\rm 00BAB6561}}}, -{0.{00}{\overline {\rm BA2E8}}}, -\right. -\\ -\\ -\displaystyle -\left. -{0.{\overline {\rm 00B9A7862A0FF465879D5F}}}, -{0.0{\overline {\rm 0B92143FA36F5E02E4850FE8DBD78}}} -\right] -\end{array} -$$ -\returnType{Type: List HexadecimalExpansion} - -or very long! - -\spadcommand{hex(1/1007) } -$$ -\begin{array}{@{}l} -0.{\overline -{\rm 0041149783F0BF2C7D13933192AF6980619EE345E91EC2BB9D5CC}} -\\ -\displaystyle -\ \ {\overline -{\rm A5C071E40926E54E8DDAE24196C0B2F8A0AAD60DBA57F5D4C8}} -\\ -\displaystyle -\ \ {\overline -{\rm 536262210C74F1}} -\end{array} -$$ -\returnType{Type: HexadecimalExpansion} - -These numbers are bona fide algebraic objects. - -\spadcommand{p := hex(1/4)*x**2 + hex(2/3)*x + hex(4/9) } -$$ -{{0.4} \ {x \sp 2}}+{{0.{\overline {\rm A}}} \ x}+{0.{\overline {\rm 71C}}} -$$ -\returnType{Type: Polynomial HexadecimalExpansion} - -\spadcommand{q := D(p, x) } -$$ -{{0.8} \ x}+{0.{\overline {\rm A}}} -$$ -\returnType{Type: Polynomial HexadecimalExpansion} - -\spadcommand{g := gcd(p, q)} -$$ -x+{1.{\overline 5}} -$$ -\returnType{Type: Polynomial HexadecimalExpansion} - -\section{Integer} -\label{IntegerXmpPage} - -Axiom provides many operations for manipulating arbitrary precision -integers. In this section we will show some of those that come from -{\tt Integer} itself plus some that are implemented in other packages. -More examples of using integers are in the following sections: -\ref{ugIntroNumbersPage} on page~\pageref{ugIntroNumbersPage} in section -\ref{ugIntroNumbersNumber} on page~\pageref{ugIntroNumbersNumber} -\ref{IntegerNumberTheoryFunctionsXmpPage} on -page~\pageref{IntegerNumberTheoryFunctionsXmpPage}, -\ref{DecimalExpansionXmpPage} on page~\pageref{DecimalExpansionXmpPage}, -\ref{BinaryExpansionXmpPage} on page~\pageref{BinaryExpansionXmpPage}, -\ref{HexadecimalExpansionXmpPage} on -page~\pageref{HexadecimalExpansionXmpPage}, and -\ref{RadixExpansionXmpPage} on page~\pageref{RadixExpansionXmpPage}. - -\subsection{Basic Functions} - -The size of an integer in Axiom is only limited by the amount of -computer storage you have available. The usual arithmetic operations -are available. - -\spadcommand{2**(5678 - 4856 + 2 * 17)} -$$ -\begin{array}{@{}l} -48048107704350081471815409251259243912395261398716822634738556100 -\\ -\displaystyle -88084200076308293086342527091412083743074572278211496076276922026 -\\ -\displaystyle -43343568752733498024953930242542523045817764949544214392905306388 -\\ -\displaystyle -478705146745768073877141698859815495632935288783334250628775936 -\end{array} -$$ -\returnType{Type: PositiveInteger} - -There are a number of ways of working with the sign of an integer. -Let's use this {\tt x} as an example. - -\spadcommand{x := -101 } -$$ --{101} -$$ -\returnType{Type: Integer} - -First of all, there is the absolute value function. - -\spadcommand{abs(x) } -$$ -101 -$$ -\returnType{Type: PositiveInteger} - -The \spadfunFrom{sign}{Integer} operation returns {\tt -1} if its argument -is negative, {\tt 0} if zero and {\tt 1} if positive. - -\spadcommand{sign(x) } -$$ --1 -$$ -\returnType{Type: Integer} - -You can determine if an integer is negative in several other ways. - -\spadcommand{x < 0 } -$$ -{\tt true} -$$ -\returnType{Type: Boolean} - -\spadcommand{x <= -1 } -$$ -{\tt true} -$$ -\returnType{Type: Boolean} - -\spadcommand{negative?(x) } -$$ -{\tt true} -$$ -\returnType{Type: Boolean} - -Similarly, you can find out if it is positive. - -\spadcommand{x > 0 } -$$ -{\tt false} -$$ -\returnType{Type: Boolean} - -\spadcommand{x >= 1 } -$$ -{\tt false} -$$ -\returnType{Type: Boolean} - -\spadcommand{positive?(x) } -$$ -{\tt false} -$$ -\returnType{Type: Boolean} - -This is the recommended way of determining whether an integer is zero. - -\spadcommand{zero?(x) } -$$ -{\tt false} -$$ -\returnType{Type: Boolean} - -\boxed{4.6in}{ -\vskip 0.1cm -Use the \spadfunFrom{zero?}{Integer} operation whenever you are -testing any mathematical object for equality with zero. This is -usually more efficient that using {\tt =} (think of matrices: it is -easier to tell if a matrix is zero by just checking term by term than -constructing another ``zero'' matrix and comparing the two matrices -term by term) and also avoids the problem that {\tt =} is usually used -for creating equations.\\ -} - -This is the recommended way of determining whether an integer is equal -to one. - -\spadcommand{one?(x) } -$$ -{\tt false} -$$ -\returnType{Type: Boolean} - -This syntax is used to test equality using \spadopFrom{=}{Integer}. -It says that you want a {\tt Boolean} ({\tt true} or {\tt false}) -answer rather than an equation. - -\spadcommand{(x = -101)@Boolean } -$$ -{\tt true} -$$ -\returnType{Type: Boolean} - -The operations \spadfunFrom{odd?}{Integer} and -\spadfunFrom{even?}{Integer} determine whether an integer is odd or -even, respectively. They each return a {\tt Boolean} object. - -\spadcommand{odd?(x) } -$$ -{\tt true} -$$ -\returnType{Type: Boolean} - -\spadcommand{even?(x) } -$$ -{\tt false} -$$ -\returnType{Type: Boolean} - -The operation \spadfunFrom{gcd}{Integer} computes the greatest common -divisor of two integers. - -\spadcommand{gcd(56788,43688)} -$$ -4 -$$ -\returnType{Type: PositiveInteger} - -The operation \spadfunFrom{lcm}{Integer} computes their least common multiple. - -\spadcommand{lcm(56788,43688)} -$$ -620238536 -$$ -\returnType{Type: PositiveInteger} - -To determine the maximum of two integers, use \spadfunFrom{max}{Integer}. - -\spadcommand{max(678,567)} -$$ -678 -$$ -\returnType{Type: PositiveInteger} - -To determine the minimum, use \spadfunFrom{min}{Integer}. - -\spadcommand{min(678,567)} -$$ -567 -$$ -\returnType{Type: PositiveInteger} - -The {\tt reduce} operation is used to extend binary operations to more -than two arguments. For example, you can use {\tt reduce} to find the -maximum integer in a list or compute the least common multiple of all -integers in the list. - -\spadcommand{reduce(max,[2,45,-89,78,100,-45])} -$$ -100 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{reduce(min,[2,45,-89,78,100,-45])} -$$ --{89} -$$ -\returnType{Type: Integer} - -\spadcommand{reduce(gcd,[2,45,-89,78,100,-45])} -$$ -1 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{reduce(lcm,[2,45,-89,78,100,-45])} -$$ -1041300 -$$ -\returnType{Type: PositiveInteger} - -The infix operator ``/'' is {\it not} used to compute the quotient of -integers. Rather, it is used to create rational numbers as described -in \ref{FractionXmpPage} on page~\pageref{FractionXmpPage}. - -\spadcommand{13 / 4} -$$ -{13} \over 4 -$$ -\returnType{Type: Fraction Integer} - -The infix operation \spadfunFrom{quo}{Integer} computes the integer -quotient. - -\spadcommand{13 quo 4} -$$ -3 -$$ -\returnType{Type: PositiveInteger} - -The infix operation \spadfunFrom{rem}{Integer} computes the integer -remainder. - -\spadcommand{13 rem 4} -$$ -1 -$$ -\returnType{Type: PositiveInteger} - -One integer is evenly divisible by another if the remainder is zero. -The operation \spadfunFrom{exquo}{Integer} can also be used. See -\ref{ugTypesUnionsPage} on page~\pageref{ugTypesUnionsPage} in Section -\ref{ugTypesUnionsNumber} on page~\pageref{ugTypesUnionsNumber} for an -example. - -\spadcommand{zero?(167604736446952 rem 2003644)} -$$ -{\tt true} -$$ -\returnType{Type: Boolean} - -The operation \spadfunFrom{divide}{Integer} returns a record of the -quotient and remainder and thus is more efficient when both are needed. - -\spadcommand{d := divide(13,4) } -$$ -\left[ -{quotient=3}, {remainder=1} -\right] -$$ -\returnType{Type: Record(quotient: Integer,remainder: Integer)} - -\spadcommand{d.quotient } -$$ -3 -$$ -\returnType{Type: PositiveInteger} - -Records are discussed in detail in Section -\ref{ugTypesRecords} on page~\pageref{ugTypesRecords}. - -\spadcommand{d.remainder } -$$ -1 -$$ -\returnType{Type: PositiveInteger} - -\subsection{Primes and Factorization} - -Use the operation \spadfunFrom{factor}{Integer} to factor integers. -It returns an object of type {\tt Factored Integer}. -See \ref{FactoredXmpPage} on page~\pageref{FactoredXmpPage} -for a discussion of the manipulation of factored objects. - -\spadcommand{factor 102400} -$$ -{2 \sp {12}} \ {5 \sp 2} -$$ -\returnType{Type: Factored Integer} - -The operation \spadfunFrom{prime?}{Integer} returns {\tt true} or -{\tt false} depending on whether its argument is a prime. - -\spadcommand{prime? 7} -$$ -{\tt true} -$$ -\returnType{Type: Boolean} - -\spadcommand{prime? 8} -$$ -{\tt false} -$$ -\returnType{Type: Boolean} - -The operation \spadfunFrom{nextPrime}{IntegerPrimesPackage} returns the -least prime number greater than its argument. - -\spadcommand{nextPrime 100} -$$ -101 -$$ -\returnType{Type: PositiveInteger} - -The operation \spadfunFrom{prevPrime}{IntegerPrimesPackage} returns -the greatest prime number less than its argument. - -\spadcommand{prevPrime 100} -$$ -97 -$$ -\returnType{Type: PositiveInteger} - -To compute all primes between two integers (inclusively), use the -operation \spadfunFrom{primes}{IntegerPrimesPackage}. - -\spadcommand{primes(100,175)} -$$ -\left[ -{173}, {167}, {163}, {157}, {151}, {149}, {139}, {137}, -{131}, {127}, {113}, {109}, {107}, {103}, {101} -\right] -$$ -\returnType{Type: List Integer} - -You might sometimes want to see the factorization of an integer -when it is considered a {\it Gaussian integer}. -See \ref{ComplexXmpPage} on page~\pageref{ComplexXmpPage} for more details. - -\spadcommand{factor(2 :: Complex Integer)} -$$ --{i \ {{\left( 1+i -\right)} -\sp 2}} -$$ -\returnType{Type: Factored Complex Integer} - -\subsection{Some Number Theoretic Functions} - -Axiom provides several number theoretic operations for integers. -More examples are in \ref{IntegerNumberTheoryFunctionsXmpPage} on -page~\pageref{IntegerNumberTheoryFunctionsXmpPage}. - -The operation \spadfunFrom{fibonacci}{IntegerNumberTheoryFunctions} -computes the Fibonacci numbers. The algorithm has running time -$O\,(\log^3(n))$ for argument {\tt n}. - -\spadcommand{[fibonacci(k) for k in 0..]} -$$ -\left[ -0, 1, 1, 2, 3, 5, 8, {13}, {21}, {34}, \ldots -\right] -$$ -\returnType{Type: Stream Integer} - -The operation \spadfunFrom{legendre}{IntegerNumberTheoryFunctions} -computes the Legendre symbol for its two integer arguments where the -second one is prime. If you know the second argument to be prime, use -\spadfunFrom{jacobi}{IntegerNumberTheoryFunctions} instead where no -check is made. - -\spadcommand{[legendre(i,11) for i in 0..10]} -$$ -\left[ -0, 1, -1, 1, 1, 1, -1, -1, -1, 1, -1 -\right] -$$ -\returnType{Type: List Integer} - -The operation \spadfunFrom{jacobi}{IntegerNumberTheoryFunctions} -computes the Jacobi symbol for its two integer arguments. By -convention, {\tt 0} is returned if the greatest common divisor of the -numerator and denominator is not {\tt 1}. - -\spadcommand{[jacobi(i,15) for i in 0..9]} -$$ -\left[ -0, 1, 1, 0, 1, 0, 0, -1, 1, 0 -\right] -$$ -\returnType{Type: List Integer} - -The operation \spadfunFrom{eulerPhi}{IntegerNumberTheoryFunctions} -computes the values of Euler's $\phi$-function where $\phi(n)$ equals -the number of positive integers less than or equal to {\tt n} that are -relatively prime to the positive integer {\tt n}. - -\spadcommand{[eulerPhi i for i in 1..]} -$$ -\left[ -1, 1, 2, 2, 4, 2, 6, 4, 6, 4, \ldots -\right] -$$ -\returnType{Type: Stream Integer} - -The operation \spadfunFrom{moebiusMu}{IntegerNumberTheoryFunctions} -computes the M\"{o}bius $\mu$ function. - -\spadcommand{[moebiusMu i for i in 1..]} -$$ -\left[ -1, -1, -1, 0, -1, 1, -1, 0, 0, 1, \ldots -\right] -$$ -\returnType{Type: Stream Integer} - -Although they have somewhat limited utility, Axiom provides Roman numerals. - -\spadcommand{a := roman(78) } -$$ -{\rm LXXVIII } -$$ -\returnType{Type: RomanNumeral} - -\spadcommand{b := roman(87) } -$$ -{\rm LXXXVII } -$$ -\returnType{Type: RomanNumeral} - -\spadcommand{a + b } -$$ -{\rm CLXV } -$$ -\returnType{Type: RomanNumeral} - -\spadcommand{a * b } -$$ -{\rm MMMMMMDCCLXXXVI } -$$ -\returnType{Type: RomanNumeral} - -\spadcommand{b rem a } -$$ -{\rm IX } -$$ -\returnType{Type: RomanNumeral} - -\section{IntegerLinearDependence} -\label{IntegerLinearDependenceXmpPage} - -The elements $v_1, \dots,v_n$ of a module {\tt M} over a ring {\tt R} -are said to be {\it linearly dependent over {\tt R}} if there exist -$c_1,\dots,c_n$ in {\tt R}, not all $0$, such that $c_1 v_1 + -\dots c_n v_n = 0$. If such $c_i$'s exist, they form what is called a -{\it linear dependence relation over {\tt R}} for the $v_i$'s. - -The package {\tt IntegerLinearDependence} provides functions -for testing whether some elements of a module over the integers are -linearly dependent over the integers, and to find the linear -dependence relations, if any. - -Consider the domain of two by two square matrices with integer entries. - -\spadcommand{M := SQMATRIX(2,INT) } -$$ -SquareMatrix(2,Integer) -$$ -\returnType{Type: Domain} - -Now create three such matrices. - -\spadcommand{m1: M := squareMatrix matrix [ [1, 2], [0, -1] ] } -$$ -\left[ -\begin{array}{cc} -1 & 2 \\ -0 & -1 -\end{array} -\right] -$$ -\returnType{Type: SquareMatrix(2,Integer)} - -\spadcommand{m2: M := squareMatrix matrix [ [2, 3], [1, -2] ] } -$$ -\left[ -\begin{array}{cc} -2 & 3 \\ -1 & -2 -\end{array} -\right] -$$ -\returnType{Type: SquareMatrix(2,Integer)} - -\spadcommand{m3: M := squareMatrix matrix [ [3, 4], [2, -3] ] } -$$ -\left[ -\begin{array}{cc} -3 & 4 \\ -2 & -3 -\end{array} -\right] -$$ -\returnType{Type: SquareMatrix(2,Integer)} - -This tells you whether {\tt m1}, {\tt m2} and {\tt m3} are linearly -dependent over the integers. - -\spadcommand{linearlyDependentOverZ? vector [m1, m2, m3] } -$$ -{\tt true} -$$ -\returnType{Type: Boolean} - -Since they are linearly dependent, you can ask for the dependence relation. - -\spadcommand{c := linearDependenceOverZ vector [m1, m2, m3] } -$$ -\left[ -1, -2, 1 -\right] -$$ -\returnType{Type: Union(Vector Integer,...)} - -This means that the following linear combination should be {\tt 0}. - -\spadcommand{c.1 * m1 + c.2 * m2 + c.3 * m3 } -$$ -\left[ -\begin{array}{cc} -0 & 0 \\ -0 & 0 -\end{array} -\right] -$$ -\returnType{Type: SquareMatrix(2,Integer)} - -When a given set of elements are linearly dependent over {\tt R}, this -also means that at least one of them can be rewritten as a linear -combination of the others with coefficients in the quotient field of -{\tt R}. - -To express a given element in terms of other elements, use the operation -\spadfunFrom{solveLinearlyOverQ}{IntegerLinearDependence}. - -\spadcommand{solveLinearlyOverQ(vector [m1, m3], m2) } -$$ -\left[ -{1 \over 2}, {1 \over 2} -\right] -$$ -\returnType{Type: Union(Vector Fraction Integer,...)} - -\section{IntegerNumberTheoryFunctions} -\label{IntegerNumberTheoryFunctionsXmpPage} - -The {\tt IntegerNumberTheoryFunctions} package contains a variety of -operations of interest to number theorists. Many of these operations -deal with divisibility properties of integers. (Recall that an -integer {\tt a} divides an integer {\tt b} if there is an integer -{\tt c} such that {\tt b = a * c}.) - -The operation \spadfunFrom{divisors}{IntegerNumberTheoryFunctions} -returns a list of the divisors of an integer. - -\spadcommand{div144 := divisors(144) } -$$ -\left[ -1, 2, 3, 4, 6, 8, 9, {12}, {16}, {18}, {24}, -{36}, {48}, {72}, {144} -\right] -$$ -\returnType{Type: List Integer} - -You can now compute the number of divisors of {\tt 144} and the sum of -the divisors of {\tt 144} by counting and summing the elements of the -list we just created. - -\spadcommand{\#(div144) } -$$ -15 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{reduce(+,div144) } -$$ -403 -$$ -\returnType{Type: PositiveInteger} - -Of course, you can compute the number of divisors of an integer -{\tt n}, usually denoted {\tt d(n)}, and the sum of the divisors of an -integer {\tt n}, usually denoted {\tt $\sigma$(n)}, without ever -listing the divisors of {\tt n}. - -In Axiom, you can simply call the operations -\spadfunFrom{numberOfDivisors}{IntegerNumberTheoryFunctions} and -\spadfunFrom{sumOfDivisors}{IntegerNumberTheoryFunctions}. - -\spadcommand{numberOfDivisors(144)} -$$ -15 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{sumOfDivisors(144)} -$$ -403 -$$ -\returnType{Type: PositiveInteger} - -The key is that {\tt d(n)} and {\tt $\sigma$(n)} are ``multiplicative -functions.'' This means that when {\tt n} and {\tt m} are relatively -prime, that is, when {\tt n} and {\tt m} have no prime factor in -common, then {\tt d(nm) = d(n) d(m)} and {\tt $\sigma$(nm) = -$\sigma$(n) $\sigma$(m)}. Note that these functions are trivial to -compute when {\tt n} is a prime power and are computed for general -{\tt n} from the prime factorization of {\tt n}. Other examples of -multiplicative functions are {\tt $\sigma_k$(n)}, the sum of the -$k$-th powers of the divisors of {\tt n} and $\varphi(n)$, the -number of integers between 1 and {\tt n} which are prime to {\tt n}. -The corresponding Axiom operations are called -\spadfunFrom{sumOfKthPowerDivisors}{IntegerNumberTheoryFunctions} and -\spadfunFrom{eulerPhi}{IntegerNumberTheoryFunctions}. - -An interesting function is {\tt $\mu$(n)}, the M\"{o}bius $\mu$ -function, defined as follows: {\tt $\mu$(1) = 1}, {\tt $\mu$(n) = 0}, -when {\tt n} is divisible by a square, and {\tt $\mu = {(-1)}^k$, when -{\tt n} is the product of {\tt k} distinct primes. The corresponding -Axiom operation is \spadfunFrom{moebiusMu}{IntegerNumberTheoryFunctions}. -This function occurs in the following theorem: - -\noindent - -{\bf Theorem} (M\"{o}bius Inversion Formula): \newline Let {\tt f(n)} -be a function on the positive integers and let {\tt F(n)} be defined -by $${F(n) = \sum_{d \mid n} f(n)}$$ sum of {\tt f(n)} over -{\tt d | n}} where the sum is taken over the positive divisors of -{\tt n}. Then the values of {\tt f(n)} can be recovered from the values of -{\tt F(n)}: -$${f(n) = \sum_{d \mid n} \mu(n) F({{n}\over{d}})}$$ -where again the sum is taken over the positive divisors of {\tt n}. - -When {\tt f(n) = 1}, then {\tt F(n) = d(n)}. Thus, if you sum $\mu(d) -\cdot d(n/d)$ over the positive divisors {\tt d} of {\tt n}, you -should always get {\tt 1}. - -\spadcommand{f1(n) == reduce(+,[moebiusMu(d) * numberOfDivisors(quo(n,d)) for d in divisors(n)]) } -\returnType{Void} - -\spadcommand{f1(200) } -$$ -1 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{f1(846) } -$$ -1 -$$ -\returnType{Type: PositiveInteger} - -Similarly, when {\tt f(n) = n}, then {\tt F(n) = $\sigma$(n)}. Thus, -if you sum {\tt $\mu$(d) $\cdot$ $\sigma$(n/d)} over the positive -divisors {\tt d} of {\tt n}, you should always get {\tt n}. - -\spadcommand{f2(n) == reduce(+,[moebiusMu(d) * sumOfDivisors(quo(n,d)) for d in divisors(n)]) } -\returnType{Void} - -\spadcommand{f2(200) } -$$ -200 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{f2(846) } -$$ -846 -$$ -\returnType{Type: PositiveInteger} - -The Fibonacci numbers are defined by $F(1) = F(2) = 1$ and -$F(n) = F(n-1) + F(n-2)$ for $n = 3,4,\ldots$. - -The operation \spadfunFrom{fibonacci}{IntegerNumberTheoryFunctions} -computes the $n$-th Fibonacci number. - -\spadcommand{fibonacci(25)} -$$ -75025 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{[fibonacci(n) for n in 1..15]} -$$ -\left[ -1, 1, 2, 3, 5, 8, {13}, {21}, {34}, {55}, {89}, -{144}, {233}, {377}, {610} -\right] -$$ -\returnType{Type: List Integer} - -Fibonacci numbers can also be expressed as sums of binomial coefficients. - -\spadcommand{fib(n) == reduce(+,[binomial(n-1-k,k) for k in 0..quo(n-1,2)]) } -\returnType{Void} - -\spadcommand{fib(25) } -$$ -75025 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{[fib(n) for n in 1..15] } -$$ -\left[ -1, 1, 2, 3, 5, 8, {13}, {21}, {34}, {55}, {89}, -{144}, {233}, {377}, {610} -\right] -$$ -\returnType{Type: List Integer} - -Quadratic symbols can be computed with the operations -\spadfunFrom{legendre}{IntegerNumberTheoryFunctions} and -\spadfunFrom{jacobi}{IntegerNumberTheoryFunctions}. The Legendre -symbol $\left({a \over p}\right)$ is defined for integers $a$ and -$p$ with $p$ an odd prime number. By definition, -$\left({a\over p}\right)$ = +1, when $a$ is a square $({\rm mod\ }p)$, -$\left({a \over p}\right)$ = -1, when $a$ is not a square $({\rm mod\ }p)$, -and $\left({a \over p}\right)$ = 0, when $a$ is divisible by $p$. - -You compute $\left({a \over p}\right)$ via the command {\tt legendre(a,p)}. - -\spadcommand{legendre(3,5)} -$$ --1 -$$ -\returnType{Type: Integer} - -\spadcommand{legendre(23,691)} -$$ --1 -$$ -\returnType{Type: Integer} - -The Jacobi symbol $\left({a \over n}\right)$ is the usual extension of -the Legendre symbol, where {\tt n} is an arbitrary integer. The most -important property of the Jacobi symbol is the following: if {\tt K} -is a quadratic field with discriminant {\tt d} and quadratic character -$\chi$, then $\chi(n) = (d/n)$. Thus, you can use the Jacobi symbol -to compute, say, the class numbers of imaginary quadratic fields from -a standard class number formula. - -This function computes the class number of the imaginary quadratic -field with discriminant {\tt d}. - -\spadcommand{h(d) == quo(reduce(+, [jacobi(d,k) for k in 1..quo(-d, 2)]), 2 - jacobi(d,2)) } -\returnType{Void} - -\spadcommand{h(-163) } -$$ -1 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{h(-499) } -$$ -3 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{h(-1832) } -$$ -26 -$$ -\returnType{Type: PositiveInteger} - -\section{Kernel} -\label{KernelXmpPage} - -A {\it kernel} is a symbolic function application (such as {\tt sin(x+ y)}) -or a symbol (such as {\tt x}). More precisely, a non-symbol -kernel over a set {\it S} is an operator applied to a given list of -arguments from {\it S}. The operator has type {\tt BasicOperator} -(see \ref{BasicOperatorXmpPage} on page~\pageref{BasicOperatorXmpPage}) -and the kernel object is usually part of an expression object (see -\ref{ExpressionXmpPage} on page~\pageref{ExpressionXmpPage}). - -Kernels are created implicitly for you when you create expressions. - -\spadcommand{x :: Expression Integer} -$$ -x -$$ -\returnType{Type: Expression Integer} - -You can directly create a ``symbol'' kernel by using the -\spadfunFrom{kernel}{Kernel} operation. - -\spadcommand{kernel x} -$$ -x -$$ -\returnType{Type: Kernel Expression Integer} - -This expression has two different kernels. - -\spadcommand{sin(x) + cos(x) } -$$ -{\sin -\left( -{x} -\right)}+{\cos -\left( -{x} -\right)} -$$ -\returnType{Type: Expression Integer} - -The operator \spadfunFrom{kernels}{Expression} returns a list of the -kernels in an object of type {\tt Expression}. - -\spadcommand{kernels \% } -$$ -\left[ -{\sin -\left( -{x} -\right)}, -{\cos -\left( -{x} -\right)} -\right] -$$ -\returnType{Type: List Kernel Expression Integer} - -This expression also has two different kernels. - -\spadcommand{sin(x)**2 + sin(x) + cos(x) } -$$ -{{\sin -\left( -{x} -\right)} -\sp 2}+{\sin -\left( -{x} -\right)}+{\cos -\left( -{x} -\right)} -$$ -\returnType{Type: Expression Integer} - -The {\tt sin(x)} kernel is used twice. - -\spadcommand{kernels \% } -$$ -\left[ -{\sin -\left( -{x} -\right)}, -{\cos -\left( -{x} -\right)} -\right] -$$ -\returnType{Type: List Kernel Expression Integer} - -An expression need not contain any kernels. - -\spadcommand{kernels(1 :: Expression Integer)} -$$ -\left[\ -\right] -$$ -\returnType{Type: List Kernel Expression Integer} - -If one or more kernels are present, one of them is -designated the {\it main} kernel. - -\spadcommand{mainKernel(cos(x) + tan(x))} -$$ -\tan -\left( -{x} -\right) -$$ -\returnType{Type: Union(Kernel Expression Integer,...)} - -Kernels can be nested. Use \spadfunFrom{height}{Kernel} to determine -the nesting depth. - -\spadcommand{height kernel x} -$$ -1 -$$ -\returnType{Type: PositiveInteger} - -This has height 2 because the {\tt x} has height 1 and then we apply -an operator to that. - -\spadcommand{height mainKernel(sin x)} -$$ -2 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{height mainKernel(sin cos x)} -$$ -3 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{height mainKernel(sin cos (tan x + sin x))} -$$ -4 -$$ -\returnType{Type: PositiveInteger} - -Use the \spadfunFrom{operator}{Kernel} operation to extract the -operator component of the kernel. The operator has type {\tt BasicOperator}. - -\spadcommand{operator mainKernel(sin cos (tan x + sin x))} -$$ -\sin -$$ -\returnType{Type: BasicOperator} - -Use the \spadfunFrom{name}{Kernel} operation to extract the name of -the operator component of the kernel. The name has type {\tt Symbol}. -This is really just a shortcut for a two-step process of extracting -the operator and then calling \spadfunFrom{name}{BasicOperator} on -the operator. - -\spadcommand{name mainKernel(sin cos (tan x + sin x))} -$$ -\sin -$$ -\returnType{Type: Symbol} - -Axiom knows about functions such as {\tt sin}, {\tt cos} and so on and -can make kernels and then expressions using them. To create a kernel -and expression using an arbitrary operator, use -\spadfunFrom{operator}{BasicOperator}. - -Now {\tt f} can be used to create symbolic function applications. - -\spadcommand{f := operator 'f } -$$ -f -$$ -\returnType{Type: BasicOperator} - -\spadcommand{e := f(x, y, 10) } -$$ -f -\left( -{x, y, {10}} -\right) -$$ -\returnType{Type: Expression Integer} - -Use the \spadfunFrom{is?}{Kernel} operation to learn if the -operator component of a kernel is equal to a given operator. - -\spadcommand{is?(e, f) } -$$ -{\tt true} -$$ -\returnType{Type: Boolean} - -You can also use a symbol or a string as the second argument to -\spadfunFrom{is?}{Kernel}. - -\spadcommand{is?(e, 'f) } -$$ -{\tt true} -$$ -\returnType{Type: Boolean} - -Use the \spadfunFrom{argument}{Kernel} operation to get a list containing -the argument component of a kernel. - -\spadcommand{argument mainKernel e } -$$ -\left[ -x, y, {10} -\right] -$$ -\returnType{Type: List Expression Integer} - -Conceptually, an object of type {\tt Expression} can be thought of a -quotient of multivariate polynomials, where the ``variables'' are -kernels. The arguments of the kernels are again expressions and so -the structure recurses. See \ref{ExpressionXmpPage} on -page~\pageref{ExpressionXmpPage} for examples of -using kernels to take apart expression objects. - -\section{KeyedAccessFile} -\label{KeyedAccessFileXmpPage} - -The domain {\tt KeyedAccessFile(S)} provides files which can be used -as associative tables. Data values are stored in these files and can -be retrieved according to their keys. The keys must be strings so -this type behaves very much like the {\tt StringTable(S)} domain. The -difference is that keyed access files reside in secondary storage -while string tables are kept in memory. For more information on -table-oriented operations, see the description of {\tt Table}. - -Before a keyed access file can be used, it must first be opened. -A new file can be created by opening it for output. - -\spadcommand{ey: KeyedAccessFile(Integer) := open("/tmp/editor.year", "output") } -$$ -\mbox{\tt "/tmp/editor.year"} -$$ -\returnType{Type: KeyedAccessFile Integer} - -Just as for vectors, tables or lists, values are saved in a keyed access file -by setting elements. - -\spadcommand{ey."Char" := 1986 } -$$ -1986 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{ey."Caviness" := 1985 } -$$ -1985 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{ey."Fitch" := 1984 } -$$ -1984 -$$ -\returnType{Type: PositiveInteger} - -Values are retrieved using application, in any of its syntactic forms. - -\spadcommand{ey."Char"} -$$ -1986 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{ey("Char")} -$$ -1986 -$$ -\returnType{Type: PositiveInteger} - -\spadcommand{ey "Char"} -$$ -1986 -$$ -\returnType{Type: PositiveInteger} - -Attempting to retrieve a non-existent element in this way causes an error. -If it is not known whether a key exists, you should use the -\spadfunFrom{search}{KeyedAccessFile} operation. - -\spadcommand{search("Char", ey) } -$$ -1986 -$$ -\returnType{Type: Union(Integer,...)} - -\spadcommand{search("Smith", ey)} -$$ -\mbox{\tt "failed"} -$$ -\returnType{Type: Union("failed",...)} - -When an entry is no longer needed, it can be removed from the file. - -\spadcommand{remove!("Char", ey) } -$$ -1986 -$$ -\returnType{Type: Union(Integer,...)} - -The \spadfunFrom{keys}{KeyedAccessFile} operation returns a list of all the -keys for a given file. - -\spadcommand{keys ey } -$$ -\left[ -\mbox{\tt "Fitch"} , \mbox{\tt "Caviness"} -\right] -$$ -\returnType{Type: List String} - -The \spadfunFrom{\#}{KeyedAccessFile} operation gives the -number of entries. - -\spadcommand{\#ey} -$$ -2 -$$ -\returnType{Type: PositiveInteger} - -The table view of keyed access files provides safe operations. That -is, if the Axiom program is terminated between file operations, the -file is left in a consistent, current state. This means, however, -that the operations are somewhat costly. For example, after each -update the file is closed. - -Here we add several more items to the file, then check its contents. - -\spadcommand{KE := Record(key: String, entry: Integer) } -$$ -\mbox{\rm Record(key: String,entry: Integer)} -$$ -\returnType{Type: Domain} - -\spadcommand{reopen!(ey, "output") } -$$ -\mbox{\tt "/tmp/editor.year"} -$$ -\returnType{Type: KeyedAccessFile Integer} - -If many items are to be added to a file at the same time, then -it is more efficient to use the \spadfunFrom{write}{KeyedAccessFile} operation. - -\spadcommand{write!(ey, ["van Hulzen", 1983]\$KE) } -$$ -\left[ -{key= \mbox{\tt "van Hulzen"} }, {entry={1983}} -\right] -$$ -\returnType{Type: Record(key: String,entry: Integer)} - -\spadcommand{write!(ey, ["Calmet", 1982]\$KE)} -$$ -\left[ -{key= \mbox{\tt "Calmet"} }, {entry={1982}} -\right] -$$ -\returnType{Type: Record(key: String,entry: Integer)} - -\spadcommand{write!(ey, ["Wang", 1981]\$KE)} -$$ -\left[ -{key= \mbox{\tt "Wang"} }, {entry={1981}} -\right] -$$ -\returnType{Type: Record(key: String,entry: Integer)} - -\spadcommand{close! ey} -$$ -\mbox{\tt "/tmp/editor.year"} -$$ -\returnType{Type: KeyedAccessFile Integer} - -The \spadfunFrom{read}{KeyedAccessFile} operation is also available -from the file view, but it returns elements in a random order. It is -generally clearer and more efficient to use the -\spadfunFrom{keys}{KeyedAccessFile} operation and to extract elements -by key. - -\spadcommand{keys ey} -$$ -\left[ -\mbox{\tt "Wang"} , \mbox{\tt "Calmet"} , \mbox{\tt "van Hulzen"} , -\mbox{\tt "Fitch"} , \mbox{\tt "Caviness"} -\right] -$$ -\returnType{Type: List String} - -\spadcommand{members ey} -$$ -\left[ -{1981}, {1982}, {1983}, {1984}, {1985} -\right] -$$ -\returnType{Type: List Integer} - -\spadcommand{)system rm -r /tmp/editor.year} - -For more information on related topics, see -\ref{FileXmpPage} on page~\pageref{FileXmpPage}, -\ref{TextFileXmpPage} on page~\pageref{TextFileXmpPage}, and -\ref{LibraryXmpPage} on page~\pageref{LibraryXmpPage}. - -\section{LexTriangularPackage} -\label{LexTriangularPackageXmpPage} - -The {\tt LexTriangularPackage} package constructor provides an -implementation of the {\em lexTriangular} algorithm (D. Lazard -``Solving Zero-dimensional Algebraic Systems'', J. of Symbol. Comput., -1992). This algorithm decomposes a zero-dimensional variety into -zero-sets of regular triangular sets. Thus the input system must have -a finite number of complex solutions. Moreover, this system needs to -be a lexicographical Groebner basis. - -This package takes two arguments: the coefficient-ring {\bf R} of the -polynomials, which must be a {\tt GcdDomain} and their set of -variables given by {\bf ls} a {\tt List Symbol}. The type of the -input polynomials must be {\tt NewSparseMultivariatePolynomial(R,V)} -where {\bf V} is {\tt OrderedVariableList(ls)}. The abbreviation for -{\tt LexTriangularPackage} is {\tt LEXTRIPK}. The main operations are -\spadfunFrom{lexTriangular}{LexTriangularPackage} and -\spadfunFrom{squareFreeLexTriangular}{LexTriangularPackage}. The -later provide decompositions by means of square-free regular -triangular sets, built with the {\tt SREGSET} constructor, whereas the -former uses the {\tt REGSET} constructor. Note that these -constructors also implement another algorithm for solving algebraic -systems by means of regular triangular sets; in that case no -computations of Groebner bases are needed and the input system may -have any dimension (i.e. it may have an infinite number of solutions). - -The implementation of the {\em lexTriangular} algorithm provided in -the {\tt LexTriangularPackage} constructor differs from that reported -in ``Computations of gcd over algebraic towers of simple extensions'' by -M. Moreno Maza and R. Rioboo (in proceedings of AAECC11, Paris, 1995). -Indeed, the \spadfunFrom{squareFreeLexTriangular}{LexTriangularPackage} -operation removes all multiplicities of the solutions (i.e. the computed -solutions are pairwise different) and the -\spadfunFrom{lexTriangular}{LexTriangularPackage} operation may keep -some multiplicities; this later operation runs generally faster than -the former. - -The interest of the {\em lexTriangular} algorithm is due to the -following experimental remark. For some examples, a triangular -decomposition of a zero-dimensional variety can be computed faster via -a lexicographical Groebner basis computation than by using a direct -method (like that of {\tt SREGSET} and {\tt REGSET}). This happens -typically when the total degree of the system relies essentially on -its smallest variable (like in the {\em Katsura} systems). When this -is not the case, the direct method may give better timings (like in -the {\em Rose} system). - -Of course, the direct method can also be applied to a lexicographical -Groebner basis. However, the {\em lexTriangular} algorithm takes -advantage of the structure of this basis and avoids many unnecessary -computations which are performed by the direct method. - -For this purpose of solving algebraic systems with a finite number of -solutions, see also the {\tt ZeroDimensionalSolvePackage}. It allows -to use both strategies (the lexTriangular algorithm and the direct -method) for computing either the complex or real roots of a system. - -Note that the way of understanding triangular decompositions is -detailed in the example of the {\tt RegularTriangularSet} constructor. - -Since the {\tt LEXTRIPK} package constructor is limited to -zero-dimensional systems, it provides a -\spadfunFrom{zeroDimensional?}{LexTriangularPackage} operation to -check whether this requirement holds. There is also a -\spadfunFrom{groebner}{LexTriangularPackage} operation to compute the -lexicographical Groebner basis of a set of polynomials with type {\tt -NewSparseMultivariatePolynomial(R,V)}. The elimination ordering is -that given by {\bf ls} (the greatest variable being the first element -of {\bf ls}). This basis is computed by the {\em FLGM} algorithm -(Faugere et al. ``Efficient Computation of Zero-Dimensional Groebner -Bases by Change of Ordering'' , J. of Symbol. Comput., 1993) -implemented in the {\tt LinGroebnerPackage} package constructor. -Once a lexicographical Groebner basis is computed, -then one can call the operations -\spadfunFrom{lexTriangular}{LexTriangularPackage} -and \spadfunFrom{squareFreeLexTriangular}{LexTriangularPackage}. -Note that these operations admit an optional argument -to produce normalized triangular sets. -There is also a \spadfunFrom{zeroSetSplit}{LexTriangularPackage} operation -which does all the job from the input system; -an error is produced if this system is not zero-dimensional. - -Let us illustrate the facilities of the {\tt LEXTRIPK} constructor -by a famous example, the {\em cyclic-6 root} system. - -Define the coefficient ring. - -\spadcommand{R := Integer } -$$ -Integer -$$ -\returnType{Type: Domain} - -Define the list of variables, - -\spadcommand{ls : List Symbol := [a,b,c,d,e,f] } -$$ -\left[ -a, b, c, d, e, f -\right] -$$ -\returnType{Type: List Symbol} - -and make it an ordered set. - -\spadcommand{V := OVAR(ls) } -$$ -\mbox{\rm OrderedVariableList [a,b,c,d,e,f]} -$$ -\returnType{Type: Domain} - -Define the polynomial ring. - -\spadcommand{P := NSMP(R, V)} -$$ -\mbox{\rm NewSparseMultivariatePolynomial(Integer,OrderedVariableList -[a,b,c,d,e,f])} -$$ -\returnType{Type: Domain} - -Define the polynomials. - -\spadcommand{p1: P := a*b*c*d*e*f - 1 } -$$ -{f \ e \ d \ c \ b \ a} -1 -$$ -\returnType{Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [a,b,c,d,e,f])} - -\spadcommand{p2: P := a*b*c*d*e +a*b*c*d*f +a*b*c*e*f +a*b*d*e*f +a*c*d*e*f +b*c*d*e*f } -$$ -{{\left( {{\left( {{\left( {{\left( e+f -\right)} -\ d}+{f \ e} -\right)} -\ c}+{f \ e \ d} -\right)} -\ b}+{f \ e \ d \ c} -\right)} -\ a}+{f \ e \ d \ c \ b} -$$ -\returnType{Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [a,b,c,d,e,f])} - -\spadcommand{p3: P := a*b*c*d + a*b*c*f + a*b*e*f + a*d*e*f + b*c*d*e + c*d*e*f } -$$ -{{\left( {{\left( {{\left( d+f -\right)} -\ c}+{f \ e} -\right)} -\ b}+{f \ e \ d} -\right)} -\ a}+{e \ d \ c \ b}+{f \ e \ d \ c} -$$ -\returnType{Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [a,b,c,d,e,f])} - -\spadcommand{p4: P := a*b*c + a*b*f + a*e*f + b*c*d + c*d*e + d*e*f } -$$ -{{\left( {{\left( c+f -\right)} -\ b}+{f \ e} -\right)} -\ a}+{d \ c \ b}+{e \ d \ c}+{f \ e \ d} -$$ -\returnType{Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [a,b,c,d,e,f])} - -\spadcommand{p5: P := a*b + a*f + b*c + c*d + d*e + e*f } -$$ -{{\left( b+f -\right)} -\ a}+{c \ b}+{d \ c}+{e \ d}+{f \ e} -$$ -\returnType{Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [a,b,c,d,e,f])} - -\spadcommand{p6: P := a + b + c + d + e + f } -$$ -a+b+c+d+e+f -$$ -\returnType{Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [a,b,c,d,e,f])} - -\spadcommand{lp := [p1, p2, p3, p4, p5, p6]} -$$ -\begin{array}{@{}l} -\left[ -{{f \ e \ d \ c \ b \ a} -1}, -\right. -\\ -\\ -\displaystyle -{{{\left( -{{\left( -{{\left( -{{\left( e+f \right)} -\ d}+{f \ e} -\right)} -\ c}+{f \ e \ d} -\right)} -\ b}+{f \ e \ d \ c} -\right)} -\ a}+{f \ e \ d \ c \ b}}, -\\ -\\ -\displaystyle -{{{\left( -{{\left( -{{\left( d+f -\right)} -\ c}+{f \ e} -\right)} -\ b}+{f \ e \ d} -\right)} -\ a}+{e \ d \ c \ b}+{f \ e \ d \ c}}, -\\ -\\ -\displaystyle -{{{\left( -{{\left( c+f -\right)} -\ b}+{f \ e} -\right)} -\ a}+{d \ c \ b}+{e \ d \ c}+{f \ e \ d}}, -\\ -\\ -\displaystyle -{{{\left( b+f -\right)} -\ a}+{c \ b}+{d \ c}+{e \ d}+{f \ e}}, -\\ -\\ -\displaystyle -\left. -{a+b+c+d+e+f} -\right] -\end{array} -$$ -\returnType{Type: List NewSparseMultivariatePolynomial(Integer,OrderedVariableList [a,b,c,d,e,f])} - -Now call {\tt LEXTRIPK} . - -\spadcommand{lextripack := LEXTRIPK(R,ls)} -$$ -LexTriangularPackage(Integer,[a,b,c,d,e,f]) -$$ -\returnType{Type: Domain} - -Compute the lexicographical Groebner basis of the system. -This may take between 5 minutes and one hour, depending on your machine. - -\spadcommand{lg := groebner(lp)\$lextripack} - -$$ -\left[ -{a+b+c+d+e+f}, -\right. -$$ -$$ -\begin{array}{@{}l} -{{3968379498283200} \ {b \sp 2}}+ -{{15873517993132800} \ f \ b}+ -\\ -\displaystyle -{{3968379498283200} \ {d \sp 2}}+ -{{15873517993132800} \ f \ d}+ -\\ -\displaystyle -{{3968379498283200} \ {f \sp 3} \ {e \sp 5}} - -{{15873517993132800} \ {f \sp 4} \ {e \sp 4}}+ -\\ -\displaystyle -{{23810276989699200} \ {f \sp 5} \ {e \sp 3}}+ -\left( -{{206355733910726400} \ {f \sp 6}}+ -\right. -\\ -\displaystyle -\left. -{230166010900425600} -\right)\ {e \sp 2}+ -\left( --{{729705987316687} \ {f \sp {43}}}+ -\right. -\\ -\displaystyle -{{1863667496867205421} \ {f \sp {37}}}+ -{{291674853771731104461} \ {f \sp {31}}}+ -\\ -\displaystyle -{{365285994691106921745} \ {f \sp {25}}}+ -{{549961185828911895} \ {f \sp {19}}} - -\\ -\displaystyle -{{365048404038768439269} \ {f \sp {13}}} - -{{292382820431504027669} \ {f \sp 7}} - -\\ -\displaystyle -\left. -{{2271898467631865497} \ f} -\right)\ e - -{{3988812642545399} \ {f \sp {44}}}+ -\\ -\displaystyle -{{10187423878429609997} \ {f \sp {38}}}+ -{{1594377523424314053637} \ {f \sp {32}}}+ -\\ -\displaystyle -{{1994739308439916238065} \ {f \sp {26}}}+ -{{1596840088052642815} \ {f \sp {20}}} - -\\ -\displaystyle -{{1993494118301162145413} \ {f \sp {14}}} - -{{1596049742289689815053} \ {f \sp 8}} - -\\ -\displaystyle -{{11488171330159667449} \ {f \sp 2}}, \hbox{\hskip 8.0cm} -\end{array} -$$ -$$ -\begin{array}{@{}l} -\left( -{{23810276989699200} \ c} - -{{23810276989699200} \ f} -\right)\ b+ -\\ -\displaystyle -{{23810276989699200} \ {c \sp 2}}+ -{{71430830969097600} \ f \ c} - -\\ -\displaystyle -{{23810276989699200} \ {d \sp 2}} - -{{95241107958796800} \ f \ d} - -\\ -\displaystyle -{{55557312975964800} \ {f \sp 3} \ {e \sp 5}}+ -{{174608697924460800} \ {f \sp 4} \ {e \sp 4}} - -\\ -\displaystyle -{{174608697924460800} \ {f \sp 5} \ {e \sp 3}}+ -\left( --{{2428648252949318400} \ {f \sp 6}} - -\right. -\\ -\displaystyle -\left. -{2611193709870345600} -\right)\ {e \sp 2}+ -\left( -{{8305444561289527} \ {f \sp {43}}} - -\right. -\\ -\displaystyle -{{21212087151945459641} \ {f \sp {37}}} - -{{3319815883093451385381} \ {f \sp {31}}} - -\\ -\displaystyle -{{4157691646261657136445} \ {f \sp {25}}} - -{{6072721607510764095} \ {f \sp {19}}}+ -\\ -\displaystyle -{{4154986709036460221649} \ {f \sp {13}}}+ -{{3327761311138587096749} \ {f \sp 7}}+ -\\ -\displaystyle -\left. -{{25885340608290841637} \ f} -\right)\ e+ -{{45815897629010329} \ {f \sp {44}}} - -\\ -\displaystyle -{{117013765582151891207} \ {f \sp {38}}} - -{{18313166848970865074187} \ {f \sp {32}}}- -\\ -\displaystyle -{{22909971239649297438915} \ {f \sp {26}}} - -{{16133250761305157265} \ {f \sp {20}}}+ -\\ -\displaystyle -{{22897305857636178256623} \ {f \sp {14}}}+ -{{18329944781867242497923} \ {f \sp 8}}+ -\\ -\displaystyle -{{130258531002020420699} \ {f \sp 2}}, \hbox{\hskip 8.0cm} -\end{array} -$$ -$$ -\begin{array}{@{}l} -\left( -{{7936758996566400} \ d} - -{{7936758996566400} \ f} -\right)\ b - -\\ -\displaystyle -{{7936758996566400} \ f \ d} - -{{7936758996566400} \ {f \sp 3} \ {e \sp 5}}+ -\\ -\displaystyle -{{23810276989699200} \ {f \sp 4} \ {e \sp 4}} - -{{23810276989699200} \ {f \sp 5} \ {e \sp 3}}+ -\\ -\displaystyle -\left( --{{337312257354072000} \ {f \sp 6}} - -{369059293340337600} -\right)\ {e \sp 2}+ -\\ -\displaystyle -\left( -{{1176345388640471} \ {f \sp {43}}} - -{{3004383582891473073} \ {f \sp {37}}} - -\right. -\\ -\displaystyle -{{470203502707246105653} \ {f \sp {31}}} - -{{588858183402644348085} \ {f \sp {25}}} - -\\ -\displaystyle -{{856939308623513535} \ {f \sp {19}}}+ -{{588472674242340526377} \ {f \sp {13}}}+ -\\ -\displaystyle -\left. -{{471313241958371103517} \ {f \sp 7}}+ -{{3659742549078552381} \ f} -\right)\ e+ -\\ -\displaystyle -{{6423170513956901} \ {f \sp {44}}} - -{{16404772137036480803} \ {f \sp {38}}} - -\\ -\displaystyle -{{2567419165227528774463} \ {f \sp {32}}} - -{{3211938090825682172335} \ {f \sp {26}}} - -\\ -\displaystyle -{{2330490332697587485} \ {f \sp {20}}}+ -{{3210100109444754864587} \ {f \sp {14}}}+ -\\ -\displaystyle -{{2569858315395162617847} \ {f \sp 8}}+ -{{18326089487427735751} \ {f \sp 2}}, \hbox{\hskip 4.0cm} -\end{array} -$$ -$$ -\begin{array}{@{}l} -\left( -{{11905138494849600} \ e} - -{{11905138494849600} \ f} -\right)\ b - -\\ -\displaystyle -{{3968379498283200} \ {f \sp 3} \ {e \sp 5}}+ -{{15873517993132800} \ {f \sp 4} \ {e \sp 4}} - -\\ -\displaystyle -{{27778656487982400} \ {f \sp 5} \ {e \sp 3}}+ -\left( --{{208339923659868000} \ {f \sp 6}} - -\right. -\\ -\displaystyle -\left. -{240086959646133600} -\right)\ {e \sp 2}+ -\left( -{{786029984751110} \ {f \sp {43}}} - -\right. -\\ -\displaystyle -{{2007519008182245250} \ {f \sp {37}}} - -{{314188062908073807090} \ {f \sp {31}}} - -\\ -\displaystyle -{{393423667537929575250} \ {f \sp {25}}} - -{{550329120654394950} \ {f \sp {19}}}+ -\\ -\displaystyle -{{393196408728889612770} \ {f \sp {13}}}+ -{{314892372799176495730} \ {f \sp 7}}+ -\\ -\displaystyle -\left. -{{2409386515146668530} \ f} -\right)\ e+ -{{4177638546747827} \ {f \sp {44}}} - -\\ -\displaystyle -{{10669685294602576381} \ {f \sp {38}}} - -{{1669852980419949524601} \ {f \sp {32}}} - -\\ -\displaystyle -{{2089077057287904170745} \ {f \sp {26}}} - -{{1569899763580278795} \ {f \sp {20}}}+ -\\ -\displaystyle -{{2087864026859015573349} \ {f \sp {14}}}+ -{{1671496085945199577969} \ {f \sp 8}}+ -\\ -\displaystyle -{{11940257226216280177} \ {f \sp 2}}, \hbox{\hskip 8.0cm} -\end{array} -$$ -$$ -\begin{array}{@{}l} -\left( -{{11905138494849600} \ {f \sp 6}} -{11905138494849600} -\right)\ b - -\\ -\displaystyle -{{15873517993132800} \ {f \sp 2} \ {e \sp 5}}+ -{{39683794982832000} \ {f \sp 3} \ {e \sp 4}} - -\\ -\displaystyle -{{39683794982832000} \ {f \sp 4} \ {e \sp 3}}+ -\left( -{{686529653202993600} \ {f \sp {11}}} - -\right. -\\ -\displaystyle -\left. -{{607162063237329600} \ {f \sp 5}} -\right)\ {e \sp 2}+ -\\ -\displaystyle -\left( {{65144531306704} \ {f \sp {42}}} - -{{166381280901088652} \ {f \sp {36}}} - -\right. -\\ -\displaystyle -{{26033434502470283472} \ {f \sp {30}}} - -{{31696259583860650140} \ {f \sp {24}}}+ -\\ -\displaystyle -{{971492093167581360} \ {f \sp {18}}}+ -{{32220085033691389548} \ {f \sp {12}}}+ -\\ -\displaystyle -\left. -{{25526177666070529808} \ {f \sp 6}}+ -{138603268355749244} -\right)\ e+ -\\ -\displaystyle -{{167620036074811} \ {f \sp {43}}} - -{{428102417974791473} \ {f \sp {37}}} - -\\ -\displaystyle -{{66997243801231679313} \ {f \sp {31}}} - -{{83426716722148750485} \ {f \sp {25}}}+ -\\ -\displaystyle -{{203673895369980765} \ {f \sp {19}}}+ -{{83523056326010432457} \ {f \sp {13}}}+ -\\ -\displaystyle -{{66995789640238066937} \ {f \sp 7}}+ -{{478592855549587901} \ f}, \hbox{\hskip 4.0cm} -\end{array} -$$ -$$ -\begin{array}{@{}l} -{{801692827936} \ {c \sp 3}}+ -{{2405078483808} \ f \ {c \sp 2}} - -\\ -\displaystyle -{{2405078483808} \ {f \sp 2} \ c} - -{{13752945467} \ {f \sp {45}}}+ -\\ -\displaystyle -{{35125117815561} \ {f \sp {39}}}+ -{{5496946957826433} \ {f \sp {33}}}+ -\\ -\displaystyle -{{6834659447749117} \ {f \sp {27}}} - -{{44484880462461} \ {f \sp {21}}} - -\\ -\displaystyle -{{6873406230093057} \ {f \sp {15}}} - -{{5450844938762633} \ {f \sp 9}}+ -\\ -\displaystyle -{{1216586044571} \ {f \sp 3}},\hbox{\hskip 9.0cm} -\end{array} -$$ -$$ -\begin{array}{@{}l} -\left( {{23810276989699200} \ d} - -{{23810276989699200} \ f} -\right)\ c+ -\\ -\displaystyle -{{23810276989699200} \ {d \sp 2}}+ -{{71430830969097600} \ f \ d}+ -\\ -\displaystyle -{{7936758996566400} \ {f \sp 3} \ {e \sp 5}} - -{{31747035986265600} \ {f \sp 4} \ {e \sp 4}}+ -\\ -\displaystyle -{{31747035986265600} \ {f \sp 5} \ {e \sp 3}}+ -\left( {{404774708824886400} \ {f \sp 6}}+ -\right. -\\ -\displaystyle -\left. -{396837949828320000} -\right)\ {e \sp 2}+ -\left( --{{1247372229446701} \ {f \sp {43}}}+ -\right. -\\ -\displaystyle -{{3185785654596621203} \ {f \sp {37}}}+ -{{498594866849974751463} \ {f \sp {31}}}+ -\\ -\displaystyle -{{624542545845791047935} \ {f \sp {25}}}+ -{{931085755769682885} \ {f \sp {19}}} - -\\ -\displaystyle -{{624150663582417063387} \ {f \sp {13}}} - -{{499881859388360475647} \ {f \sp 7}} - -\\ -\displaystyle -\left. -{{3926885313819527351} \ f} -\right)\ e - -{{7026011547118141} \ {f \sp {44}}}+ -\\ -\displaystyle -{{17944427051950691243} \ {f \sp {38}}}+ -{{2808383522593986603543} \ {f \sp {32}}}+ -\\ -\displaystyle -{{3513624142354807530135} \ {f \sp {26}}}+ -{{2860757006705537685} \ {f \sp {20}}} - -\\ -\displaystyle -{{3511356735642190737267} \ {f \sp {14}}} - -{{2811332494697103819887} \ {f \sp 8}} - -\\ -\displaystyle -{{20315011631522847311} \ {f \sp 2}}, \hbox{\hskip 8.0cm} -\end{array} -$$ -$$ -\begin{array}{@{}l} -\left( -{{7936758996566400} \ e} -{{7936758996566400} \ f} -\right)\ c+ -\\ -\displaystyle -\left( --{{4418748183673} \ {f \sp {43}}}+ -\right. -\\ -\displaystyle -{{11285568707456559} \ {f \sp {37}}}+ -{{1765998617294451019} \ {f \sp {31}}}+ -\\ -\displaystyle -{{2173749283622606155} \ {f \sp {25}}} - -{{55788292195402895} \ {f \sp {19}}} - -\\ -\displaystyle -{{2215291421788292951} \ {f \sp {13}}} - -{{1718142665347430851} \ {f \sp 7}}+ -\\ -\displaystyle -\left. -{{30256569458230237} \ f} -\right)\ e+ -{{4418748183673} \ {f \sp {44}}} - -\\ -\displaystyle -{{11285568707456559} \ {f \sp {38}}} - -{{1765998617294451019} \ {f \sp {32}}} - -\\ -\displaystyle -{{2173749283622606155} \ {f \sp {26}}}+ -{{55788292195402895} \ {f \sp {20}}}+ -\\ -\displaystyle -{{2215291421788292951} \ {f \sp {14}}}+ -{{1718142665347430851} \ {f \sp 8}} - -\\ -\displaystyle -{{30256569458230237} \ {f \sp 2}}, \hbox{\hskip 9.0cm} -\end{array} -$$ -$$ -\begin{array}{@{}l} -\left( -{{72152354514240} \ {f \sp 6}} - -{72152354514240} -\right)\ c+ -\\ -\displaystyle -{{40950859449} \ {f \sp {43}}} - -{{104588980990367} \ {f \sp {37}}} - -\\ -\displaystyle -{{16367227395575307} \ {f \sp {31}}} - -{{20268523416527355} \ {f \sp {25}}}+ -\\ -\displaystyle -{{442205002259535} \ {f \sp {19}}}+ -{{20576059935789063} \ {f \sp {13}}}+ -\\ -\displaystyle -{{15997133796970563} \ {f \sp 7}} - -{{275099892785581} \ f}, \hbox{\hskip 5.0cm} -\end{array} -$$ -$$ -\begin{array}{@{}l} -{{1984189749141600} \ {d \sp 3}}+ -{{5952569247424800} \ f \ {d \sp 2}} - -\\ -\displaystyle -{{5952569247424800} \ {f \sp 2} \ d} - -{{3968379498283200} \ {f \sp 4} \ {e \sp 5}}+ -\\ -\displaystyle -{{15873517993132800} \ {f \sp 5} \ {e \sp 4}}+ -{{17857707742274400} \ {e \sp 3}}+ -\\ -\displaystyle -\left( --{{148814231185620000} \ {f \sp 7}} - -{{162703559429611200} \ f} -\right)\ {e \sp 2}+ -\\ -\displaystyle -\left( --{{390000914678878} \ {f \sp {44}}}+ -{{996062704593756434} \ {f \sp {38}}}+ -\right. -\\ -\displaystyle -{{155886323972034823914} \ {f \sp {32}}}+ -{{194745956143985421330} \ {f \sp {26}}}+ -\\ -\displaystyle -{{6205077595574430} \ {f \sp {20}}} - -{{194596512653299068786} \ {f \sp {14}}} - -\\ -\displaystyle -\left. -{{155796897940756922666} \ {f \sp 8}} - -{{1036375759077320978} \ {f \sp 2}} -\right)\ e - -\\ -\displaystyle -{{374998630035991} \ {f \sp {45}}}+ -{{957747106595453993} \ {f \sp {39}}}+ -\\ -\displaystyle -{{149889155566764891693} \ {f \sp {33}}}+ -{{187154171443494641685} \ {f \sp {27}}} - -\\ -\displaystyle -{{127129015426348065} \ {f \sp {21}}} - -{{187241533243115040417} \ {f \sp {15}}} - -\\ -\displaystyle -{{149719983567976534037} \ {f \sp 9}} - -{{836654081239648061} \ {f \sp 3}}, \hbox{\hskip 3.5cm} -\end{array} -$$ -$$ -\begin{array}{@{}l} -\left( -{{5952569247424800} \ e} - -{{5952569247424800} \ f} -\right)\ d - -\\ -\displaystyle -{{3968379498283200} \ {f \sp 3} \ {e \sp 5}}+ -{{9920948745708000} \ {f \sp 4} \ {e \sp 4}} - -\\ -\displaystyle -{{3968379498283200} \ {f \sp 5} \ {e \sp 3}}+ -\left( --{{148814231185620000} \ {f \sp 6}} - -\right. -\\ -\displaystyle -\left. -{150798420934761600} -\right)\ {e \sp 2}+ -\left( -{{492558110242553} \ {f \sp {43}}} - -\right. -\\ -\displaystyle -{{1257992359608074599} \ {f \sp {37}}} - -{{196883094539368513959} \ {f \sp {31}}} - -\\ -\displaystyle -{{246562115745735428055} \ {f \sp {25}}} - -{{325698701993885505} \ {f \sp {19}}}+ -\\ -\displaystyle -{{246417769883651808111} \ {f \sp {13}}}+ -{{197327352068200652911} \ {f \sp 7}}+ -\\ -\displaystyle -\left. -{{1523373796389332143} \ f} -\right)\ e+ -{{2679481081803026} \ {f \sp {44}}} - -\\ -\displaystyle -{{6843392695421906608} \ {f \sp {38}}} - -{{1071020459642646913578} \ {f \sp {32}}} - -\\ -\displaystyle -{{1339789169692041240060} \ {f \sp {26}}} - -{{852746750910750210} \ {f \sp {20}}}+ -\\ -\displaystyle -{{1339105101971878401312} \ {f \sp {14}}}+ -{{1071900289758712984762} \ {f \sp 8}}+ -\\ -\displaystyle -{{7555239072072727756} \ {f \sp 2}}, \hbox{\hskip 8.0cm} -\end{array} -$$ -$$ -\begin{array}{@{}l} -\left( -{{11905138494849600} \ {f \sp 6}} - -{11905138494849600} -\right)\ d - -\\ -\displaystyle -{{7936758996566400} \ {f \sp 2} \ {e \sp 5}}+ -{{31747035986265600} \ {f \sp 3} \ {e \sp 4}} - -\\ -\displaystyle -{{31747035986265600} \ {f \sp 4} \ {e \sp 3}}+ -\\ -\displaystyle -\left( -{{420648226818019200} \ {f \sp {11}}} - -{{404774708824886400} \ {f \sp 5}} -\right)\ {e \sp 2}+ -\\ -\displaystyle -\left( -{{15336187600889} \ {f \sp {42}}} - -{{39169739565161107} \ {f \sp {36}}} - -\right. -\\ -\displaystyle -{{6127176127489690827} \ {f \sp {30}}} - -{{7217708742310509615} \ {f \sp {24}}}+ -\\ -\displaystyle -{{538628483890722735} \ {f \sp {18}}}+ -{{7506804353843507643} \ {f \sp {12}}}+ -\\ -\displaystyle -\left. -{{5886160769782607203} \ {f \sp 6}}+ -{63576108396535879} -\right)\ e+ -\\ -\displaystyle -{{71737781777066} \ {f \sp {43}}} - -{{183218856207557938} \ {f \sp {37}}} - -\\ -\displaystyle -{{28672874271132276078} \ {f \sp {31}}} - -{{35625223686939812010} \ {f \sp {25}}}+ -\\ -\displaystyle -{{164831339634084390} \ {f \sp {19}}}+ -{{35724160423073052642} \ {f \sp {13}}}+ -\\ -\displaystyle -{{28627022578664910622} \ {f \sp 7}}+ -{{187459987029680506} \ f}, \hbox{\hskip 4.0cm} -\end{array} -$$ -$$ -\begin{array}{@{}l} -{{1322793166094400} \ {e \sp 6}} - -{{3968379498283200} \ f \ {e \sp 5}}+ -\\ -\displaystyle -{{3968379498283200} \ {f \sp 2} \ {e \sp 4}} - -{{5291172664377600} \ {f \sp 3} \ {e \sp 3}}+ -\\ -\displaystyle -\left( -{{230166010900425600} \ {f \sp {10}}} - -{{226197631402142400} \ {f \sp 4}} -\right)\ {e \sp 2}+ -\\ -\displaystyle -\left( --{{152375364610443885} \ {f \sp {47}}}+ -{{389166626064854890415} \ {f \sp {41}}}+ -\right. -\\ -\displaystyle -{{60906097841360558987335} \ {f \sp {35}}}+ -{{76167367934608798697275} \ {f \sp {29}}}+ -\\ -\displaystyle -{{27855066785995181125} \ {f \sp {23}}} - -{{76144952817052723145495} \ {f \sp {17}}} - -\\ -\displaystyle -\left. -{{60933629892463517546975} \ {f \sp {11}}} - -{{411415071682002547795} \ {f \sp 5}} -\right)\ e - -\\ -\displaystyle -{{209493533143822} \ {f \sp {42}}}+ -{{535045979490560586} \ {f \sp {36}}}+ -\\ -\displaystyle -{{83737947964973553146} \ {f \sp {30}}}+ -{{104889507084213371570} \ {f \sp {24}}}+ -\\ -\displaystyle -{{167117997269207870} \ {f \sp {18}}} - -{{104793725781390615514} \ {f \sp {12}}} - -\\ -\displaystyle -{{83842685189903180394} \ {f \sp 6}} - -{569978796672974242}, \hbox{\hskip 4.0cm} -\end{array} -$$ -$$ -\begin{array}{@{}l} -\left( {{25438330117200} \ {f \sp 6}}+ -{25438330117200} -\right)\ {e \sp 3}+ -\\ -\displaystyle -\left( {{76314990351600} \ {f \sp 7}}+ -{{76314990351600} \ f} -\right)\ {e \sp 2}+ -\\ -\displaystyle -\left( -{{1594966552735} \ {f \sp {44}}}+ -{{4073543370415745} \ {f \sp {38}}}+ -\right. -\\ -\displaystyle -{{637527159231148925} \ {f \sp {32}}}+ -{{797521176113606525} \ {f \sp {26}}}+ -\\ -\displaystyle -{{530440941097175} \ {f \sp {20}}} - -{{797160527306433145} \ {f \sp {14}}} - -\\ -\displaystyle -\left. -{{638132320196044965} \ {f \sp 8}} - -{{4510507167940725} \ {f \sp 2}} -\right)\ e - -\\ -\displaystyle -{{6036376800443} \ {f \sp {45}}}+ -{{15416903421476909} \ {f \sp {39}}}+ -\\ -\displaystyle -{{2412807646192304449} \ {f \sp {33}}}+ -{{3017679923028013705} \ {f \sp {27}}}+ -\\ -\displaystyle -{{1422320037411955} \ {f \sp {21}}} - -{{3016560402417843941} \ {f \sp {15}}} - -\\ -\displaystyle -{{2414249368183033161} \ {f \sp 9}} - -{{16561862361763873} \ {f \sp 3}}, \hbox{\hskip 4.0cm} -\end{array} -$$ -$$ -\begin{array}{@{}l} -\left( -{{1387545279120} \ {f \sp {12}}} - -{1387545279120} -\right)\ {e \sp 2}+ -\\ -\displaystyle -\left( {{4321823003} \ {f \sp {43}}} - -{{11037922310209} \ {f \sp {37}}} - -\right. -\\ -\displaystyle -{{1727510711947989} \ {f \sp {31}}} - -{{2165150991154425} \ {f \sp {25}}} - -\\ -\displaystyle -{{5114342560755} \ {f \sp {19}}}+ -{{2162682824948601} \ {f \sp {13}}}+ -\\ -\displaystyle -\left. -{{1732620732685741} \ {f \sp 7}}+ -{{13506088516033} \ f} -\right)\ e+ -\\ -\displaystyle -{{24177661775} \ {f \sp {44}}} - -{{61749727185325} \ {f \sp {38}}} - -\\ -\displaystyle -{{9664106795754225} \ {f \sp {32}}} - -{{12090487758628245} \ {f \sp {26}}} - -\\ -\displaystyle -{{8787672733575} \ {f \sp {20}}}+ -{{12083693383005045} \ {f \sp {14}}}+ -\\ -\displaystyle -{{9672870290826025} \ {f \sp 8}}+ -{{68544102808525} \ {f \sp 2}}, \hbox{\hskip 5.0cm} -\end{array} -$$ -$$ -\left. -\begin{array}{@{}l} -{f \sp {48}} - -{{2554} \ {f \sp {42}}} - -{{399710} \ {f \sp {36}}}- -{{499722} \ {f \sp {30}}}+ -\\ -\displaystyle -{{499722} \ {f \sp {18}}}+ -{{399710} \ {f \sp {12}}}+ -{{2554} \ {f \sp 6}} - -1 \hbox{\hskip 6.0cm} -\end{array} -\right] -$$ -\returnType{Type: List NewSparseMultivariatePolynomial(Integer,OrderedVariableList [a,b,c,d,e,f])} - -Apply lexTriangular to compute a decomposition into regular triangular sets. -This should not take more than 5 seconds. - -\spadcommand{lexTriangular(lg,false)\$lextripack} -$$ -\begin{array}{@{}l} -\left[ -\begin{array}{@{}l} -\left\{ -{{f \sp 6}+1}, -{{e \sp 6} - -{3 \ f \ {e \sp 5}}+ -{3 \ {f \sp 2} \ {e \sp 4}} - -{4 \ {f \sp 3} \ {e \sp 3}}+ -{3 \ {f \sp 4} \ {e \sp 2}} - -{3 \ {f \sp 5} \ e} -1}, -\right. -\\ -\displaystyle -{{3 \ d}+{{f \sp 2} \ {e \sp 5}} - -{4 \ {f \sp 3} \ {e \sp 4}}+ -{4 \ {f \sp 4} \ {e \sp 3}} - -{2 \ {f \sp 5} \ {e \sp 2}} - -{2 \ e}+{2 \ f}}, -{c+f}, -\\ -\displaystyle -{{3 \ b}+ -{2 \ {f \sp 2} \ {e \sp 5}} - -{5 \ {f \sp 3} \ {e \sp 4}}+ -{5 \ {f \sp 4} \ {e \sp 3}} - -{{10} \ {f \sp 5} \ {e \sp 2}} - -{4 \ e}+{7 \ f}}, -\\ -\displaystyle -\left. -{a -{{f \sp 2} \ {e \sp 5}}+ -{3 \ {f \sp 3} \ {e \sp 4}} - -{3 \ {f \sp 4} \ {e \sp 3}}+ -{4 \ {f \sp 5} \ {e \sp 2}}+ -{3 \ e} -{3 \ f}} -\right\}, -\end{array} -\right. -\\ -\\ -\displaystyle -\left\{ -{{f \sp 6} -1}, -{e -f}, -{d -f}, -{{c \sp 2}+ -{4 \ f \ c}+ -{f \sp 2}}, -{{{\left( -c -f -\right)}\ b} - -{f \ c} - -{5 \ {f \sp 2}}}, -{a+b+c+{3 \ f}} -\right\}, -\\ -\\ -\displaystyle -\left\{ -{{f \sp 6} -1}, -{e -f}, -{d -f}, -{c -f}, -{{b \sp 2}+ -{4 \ f \ b}+ -{f \sp 2}}, -{a+b+{4 \ f}} -\right\}, -\\ -\\ -\displaystyle -\left\{ -{{f \sp 6} -1}, -{e -f}, -{{d \sp 2}+ -{4 \ f \ d}+ -{f \sp 2}}, -{{{\left( d -f \right)}\ c} - -{f \ d} - -{5 \ {f \sp 2}}}, -{b -f}, -{a+c+d+{3 \ f}} -\right\}, -\\ -\\ -\displaystyle -\begin{array}{@{}l} -\left\{ -{{f \sp {36}} - -{{2554} \ {f \sp {30}}} - -{{399709} \ {f \sp {24}}} - -{{502276} \ {f \sp {18}}} - -{{399709} \ {f \sp {12}}} - -{{2554} \ {f \sp 6}}+1}, -\right. -\\ -\displaystyle -\left( -{{161718564} \ {f \sp {12}}} - -{161718564} -\right)\ {e \sp 2}+ -\left( --{{504205} \ {f \sp {31}}}+ -{{1287737951} \ {f \sp {25}}}+ -\right. -\\ -\displaystyle -\left. -{{201539391380} \ {f \sp {19}}}+ -{{253982817368} \ {f \sp {13}}}+ -{{201940704665} \ {f \sp 7}}+ -{{1574134601} \ f} -\right)\ e - -\\ -\displaystyle -{{2818405} \ {f \sp {32}}}+ -{{7198203911} \ {f \sp {26}}}+ -{{1126548149060} \ {f \sp {20}}}+ -\\ -\displaystyle -{{1416530563364} \ {f \sp {14}}}+ -{{1127377589345} \ {f \sp 8}}+ -{{7988820725} \ {f \sp 2}}, -\\ -\displaystyle -\left( -{{693772639560} \ {f \sp 6}} - -{693772639560} -\right)\ d - -{{462515093040} \ {f \sp 2} \ {e \sp 5}}+ -\\ -\displaystyle -{{1850060372160} \ {f \sp 3} \ {e \sp 4}} - -{{1850060372160} \ {f \sp 4} \ {e \sp 3}}+ -\left( --{{24513299931120} \ {f \sp {11}}} - -\right. -\\ -\displaystyle -\left. -{{23588269745040} \ {f \sp 5}} -\right)\ {e \sp 2}+ -\left( --{{890810428} \ {f \sp {30}}}+ -{{2275181044754} \ {f \sp {24}}}+ -\right. -\\ -\displaystyle -{{355937263869776} \ {f \sp {18}}}+ -{{413736880104344} \ {f \sp {12}}}+ -{{342849304487996} \ {f \sp 6}}+ -\\ -\displaystyle -\left. -{3704966481878} -\right)\ e - -{{4163798003} \ {f \sp {31}}}+ -{{10634395752169} \ {f \sp {25}}}+ -\\ -\displaystyle -{{1664161760192806} \ {f \sp {19}}}+ -{{2079424391370694} \ {f \sp {13}}}+ -{{1668153650635921} \ {f \sp 7}}+ -\\ -\displaystyle -{{10924274392693} \ f}, -\left( -{{12614047992} \ {f \sp 6}} - -{12614047992} -\right)\ c - -\\ -\displaystyle -{{7246825} \ {f \sp {31}}}+ -{{18508536599} \ {f \sp {25}}}+ -{{2896249516034} \ {f \sp {19}}}+ -\\ -\displaystyle -{{3581539649666} \ {f \sp {13}}}+ -{{2796477571739} \ {f \sp 7}} - -{{48094301893} \ f}, -\\ -\displaystyle -\left( -{{693772639560} \ {f \sp 6}} - -{693772639560} -\right)\ b - -{{925030186080} \ {f \sp 2} \ {e \sp 5}}+ -\\ -\displaystyle -{{2312575465200} \ {f \sp 3} \ {e \sp 4}} - -{{2312575465200} \ {f \sp 4} \ {e \sp 3}}+ -\left( --{{40007555547960} \ {f \sp {11}}} - -\right. -\\ -\displaystyle -\left. -{{35382404617560} \ {f \sp 5}} -\right)\ {e \sp 2}+ -\left( --{{3781280823} \ {f \sp {30}}}+ -{{9657492291789} \ {f \sp {24}}}+ -\right. -\\ -\displaystyle -{{1511158913397906} \ {f \sp {18}}}+ -{{1837290892286154} \ {f \sp {12}}}+ -{{1487216006594361} \ {f \sp 6}}+ -\\ -\displaystyle -\left. -{8077238712093} -\right)\ e - -{{9736390478} \ {f \sp {31}}}+ -{{24866827916734} \ {f \sp {25}}}+ -\\ -\displaystyle -{{3891495681905296} \ {f \sp {19}}}+ -{{4872556418871424} \ {f \sp {13}}}+ -{{3904047887269606} \ {f \sp 7}}+ -\\ -\displaystyle -\left. -{{27890075838538} \ f}, -{a+b+c+d+e+f} -\right\}, -\end{array} -\\ -\\ -\displaystyle -\left. -\left\{ -{{f \sp 6} -1}, -{{e \sp 2}+ -{4 \ f \ e}+ -{f \sp 2}}, -{{{\left( e -f \right)}\ d} - -{f \ e} - -{5 \ {f \sp 2}}}, -{c -f}, -{b -f}, -{a+d+e+{3 \ f}} -\right\} - -\right] -\end{array} -$$ -\returnType{Type: List RegularChain(Integer,[a,b,c,d,e,f])} - -Note that the first set of the decomposition is normalized (all -initials are integer numbers) but not the second one (normalized -triangular sets are defined in the description of the -{\tt NormalizedTriangularSetCategory} constructor). - -So apply now lexTriangular to produce normalized triangular sets. - -\spadcommand{lts := lexTriangular(lg,true)\$lextripack } -$$ -\begin{array}{@{}l} -\left[ -\begin{array}{@{}l} -\left\{ -{{f \sp 6}+1}, -{{e \sp 6} - -{3 \ f \ {e \sp 5}}+ -{3 \ {f \sp 2} \ {e \sp 4}} - -{4 \ {f \sp 3} \ {e \sp 3}}+ -{3 \ {f \sp 4} \ {e \sp 2}} - -{3 \ {f \sp 5} \ e} -1}, -\right. -\\ -\displaystyle -{{3 \ d}+ -{{f \sp 2} \ {e \sp 5}} - -{4 \ {f \sp 3} \ {e \sp 4}}+ -{4 \ {f \sp 4} \ {e \sp 3}} - -{2 \ {f \sp 5} \ {e \sp 2}} - -{2 \ e}+{2 \ f}}, -{c+f}, -\\ -\displaystyle -{{3 \ b}+ -{2 \ {f \sp 2} \ {e \sp 5}}- -{5 \ {f \sp 3} \ {e \sp 4}}+ -{5 \ {f \sp 4} \ {e \sp 3}} - -{{10} \ {f \sp 5} \ {e \sp 2}} - -{4 \ e}+{7 \ f}}, -\\ -\displaystyle -\left. -{a -{{f \sp 2} \ {e \sp 5}}+ -{3 \ {f \sp 3} \ {e \sp 4}} - -{3 \ {f \sp 4} \ {e \sp 3}}+ -{4 \ {f \sp 5} \ {e \sp 2}}+ -{3 \ e} - -{3 \ f}} -\right\}, -\end{array} -\right. -\\ -\\ -\displaystyle -{\left\{ {{f \sp 6} -1}, {e -f}, {d -f}, {{c \sp 2}+{4 \ f \ -c}+{f \sp 2}}, {b+c+{4 \ f}}, {a -f} -\right\}}, -\\ -\\ -\displaystyle -{\left\{ {{f \sp 6} -1}, {e -f}, {d -f}, {c -f}, {{b \sp 2}+{4 -\ f \ b}+{f \sp 2}}, {a+b+{4 \ f}} -\right\}}, -\\ -\\ -\displaystyle -{\left\{ {{f \sp 6} -1}, {e -f}, {{d \sp 2}+{4 \ f \ d}+{f \sp -2}}, {c+d+{4 \ f}}, {b -f}, {a -f} -\right\}}, -\\ -\\ -\displaystyle -\begin{array}{@{}l} -\left\{ -{{f \sp {36}} - -{{2554} \ {f \sp {30}}} - -{{399709} \ {f \sp {24}}} - -{{502276} \ {f \sp {18}}} - -{{399709} \ {f \sp {12}}} - -{{2554} \ {f \sp 6}}+ -1}, -\right. -\\ -\displaystyle -{{1387545279120} \ {e \sp 2}}+ -\left( -{{4321823003} \ {f \sp {31}}} - -{{11037922310209} \ {f \sp {25}}} - -\right. -\\ -\displaystyle -{{1727506390124986} \ {f \sp {19}}} - -{{2176188913464634} \ {f \sp {13}}} - -{{1732620732685741} \ {f \sp 7}} - -\\ -\displaystyle -\left. -{{13506088516033} \ f} -\right)\ e+ -{{24177661775} \ {f \sp {32}}} - -{{61749727185325} \ {f \sp {26}}} - -\\ -\displaystyle -{{9664082618092450} \ {f \sp {20}}} - -{{12152237485813570} \ {f \sp {14}}} - -{{9672870290826025} \ {f \sp 8}} - -\\ -\displaystyle -{{68544102808525} \ {f \sp 2}}, -\\ -\displaystyle -{{1387545279120} \ d}+ -\left( --{{1128983050} \ {f \sp {30}}}+ -{{2883434331830} \ {f \sp {24}}}+ -\right. -\\ -\displaystyle -{{451234998755840} \ {f \sp {18}}}+ -{{562426491685760} \ {f \sp {12}}}+ -{{447129055314890} \ {f \sp 6}} - -\\ -\displaystyle -\left. -{165557857270} -\right)\ e - -{{1816935351} \ {f \sp {31}}}+ -{{4640452214013} \ {f \sp {25}}}+ -\\ -\displaystyle -{{726247129626942} \ {f \sp {19}}}+ -{{912871801716798} \ {f \sp {13}}}+ -{{726583262666877} \ {f \sp 7}}+ -\\ -\displaystyle -{{4909358645961} \ f}, -\\ -\displaystyle -{{1387545279120} \ c}+ -{{778171189} \ {f \sp {31}}} - -{{1987468196267} \ {f \sp {25}}} - -\\ -\displaystyle -{{310993556954378} \ {f \sp {19}}} - -{{383262822316802} \ {f \sp {13}}} - -{{300335488637543} \ {f \sp 7}}+ -\\ -\displaystyle -{{5289595037041} \ f}, -\\ -\displaystyle -{{1387545279120} \ b}+ -\left( -{{1128983050} \ {f \sp {30}}} - -{{2883434331830} \ {f \sp {24}}} - -\right. -\\ -\displaystyle -{{451234998755840} \ {f \sp {18}}}- -{{562426491685760} \ {f \sp {12}}} - -{{447129055314890} \ {f \sp 6}}+ -\\ -\displaystyle -\left. -{165557857270} -\right)\ e - -{{3283058841} \ {f \sp {31}}}+ -{{8384938292463} \ {f \sp {25}}}+ -\\ -\displaystyle -{{1312252817452422} \ {f \sp {19}}}+ -{{1646579934064638} \ {f \sp {13}}}+ -{{1306372958656407} \ {f \sp 7}}+ -\\ -\displaystyle -{{4694680112151} \ f}, -\\ -\displaystyle -{{1387545279120} \ a}+ -{{1387545279120} \ e}+ -{{4321823003} \ {f \sp {31}}} - -\\ -\displaystyle -{{11037922310209} \ {f \sp {25}}} - -{{1727506390124986} \ {f \sp {19}}} - -{{2176188913464634} \ {f \sp {13}}} - -\\ -\displaystyle -\left. -{{1732620732685741} \ {f \sp 7}} - -{{13506088516033} \ f} -\right\}, -\end{array} -\\ -\\ -\displaystyle -\left. -\left\{ -{{f \sp 6} -1}, -{{e \sp 2}+{4 \ f \ e}+{f \sp 2}}, -{d+e+{4 \ f}}, -{c -f}, -{b -f}, -{a -f} -\right\} -\right] -\end{array} -$$ -\returnType{Type: List RegularChain(Integer,[a,b,c,d,e,f])} - -We check that all initials are constant. - -\spadcommand{[ [init(p) for p in (ts :: List(P))] for ts in lts] } -$$ -\begin{array}{@{}l} -\left[ -{\left[ 1, 3, 1, 3, 1, 1 \right]}, -{\left[ 1, 1, 1, 1, 1, 1 \right]}, -{\left[ 1, 1, 1, 1, 1, 1 \right]}, -{\left[ 1, 1, 1, 1, 1, 1 \right]}, -\right. -\\ -\displaystyle -\left[ {1387545279120}, {1387545279120}, {1387545279120}, -\right. -\\ -\displaystyle -\left. -{1387545279120}, {1387545279120}, 1 \right], -\\ -\displaystyle -\left. -{\left[ 1, 1, 1, 1, 1, 1\right]} -\right] -\end{array} -$$ -\returnType{Type: List List NewSparseMultivariatePolynomial(Integer,OrderedVariableList [a,b,c,d,e,f])} - -Note that each triangular set in {\bf lts} is a lexicographical -Groebner basis. Recall that a point belongs to the variety associated -with {\bf lp} if and only if it belongs to that associated with one -triangular set {\bf ts} in {\bf lts}. - -By running the \spadfunFrom{squareFreeLexTriangular}{LexTriangularPackage} -operation, we retrieve the above decomposition. - -\spadcommand{squareFreeLexTriangular(lg,true)\$lextripack } -$$ -\begin{array}{@{}l} -\left[ -\left\{ -{{f \sp 6}+1}, -{{e \sp 6} - -{3 \ f \ {e \sp 5}}+ -{3 \ {f \sp 2} \ {e \sp 4}} - -{4 \ {f \sp 3} \ {e \sp 3}}+ -{3 \ {f \sp 4} \ {e \sp 2}} - -{3 \ {f \sp 5} \ e} -1}, -\right. -\right. -\\ -\displaystyle -{{3 \ d}+ -{{f \sp 2} \ {e \sp 5}} - -{4 \ {f \sp 3} \ {e \sp 4}}+ -{4 \ {f \sp 4} \ {e \sp 3}} - -{2 \ {f \sp 5} \ {e \sp 2}} - -{2 \ e}+ -{2 \ f}}, -\\ -\displaystyle -{c+f}, -{{3 \ b}+ -{2 \ {f \sp 2} \ {e \sp 5}} - -{5 \ {f \sp 3} \ {e \sp 4}}+ -{5 \ {f \sp 4} \ {e \sp 3}} - -{{10} \ {f \sp 5} \ {e \sp 2}} - -{4 \ e}+{7 \ f}}, -\\ -\displaystyle -\left. -{a - -{{f \sp 2} \ {e \sp 5}}+ -{3 \ {f \sp 3} \ {e \sp 4}} - -{3 \ {f \sp 4} \ {e \sp 3}}+ -{4 \ {f \sp 5} \ {e \sp 2}}+ -{3 \ e} -{3 \ f}} -\right\},\hbox{\hskip 4.0cm} -\end{array} -$$ -$$ -\left\{ -{{f \sp 6} -1}, -{e -f}, -{d -f}, -{{c \sp 2}+ -{4 \ f \ c}+ -{f \sp 2}}, -{b+c+{4 \ f}}, -{a -f} -\right\},\hbox{\hskip 3.5cm} -$$ -$$ -\left\{ -{{f \sp 6} -1}, -{e -f}, -{d -f}, -{c -f}, -{{b \sp 2}+ -{4 \ f \ b}+ -{f \sp 2}}, -{a+b+{4 \ f}} -\right\},\hbox{\hskip 3.5cm} -$$ -$$ -\left\{ -{{f \sp 6} -1}, -{e -f}, -{{d \sp 2}+ -{4 \ f \ d}+ -{f \sp 2}}, -{c+d+{4 \ f}}, -{b -f}, -{a -f} -\right\},\hbox{\hskip 3.5cm} -$$ -$$ -\begin{array}{@{}l} -\left\{ -{{f \sp {36}} - -{{2554} \ {f \sp {30}}} - -{{399709} \ {f \sp {24}}} - -{{502276} \ {f \sp {18}}} - -{{399709} \ {f \sp {12}}} - -{{2554} \ {f \sp 6}}+1}, -\right. -\\ -\displaystyle -{{1387545279120} \ {e \sp 2}}+ -\left( {{4321823003} \ {f \sp {31}}} - -{{11037922310209} \ {f \sp {25}}} - -\right. -\\ -\displaystyle -{{1727506390124986} \ {f \sp {19}}} - -{{2176188913464634} \ {f \sp {13}}} - -{{1732620732685741} \ {f \sp 7}} - -\\ -\displaystyle -\left. -{{13506088516033} \ f} -\right)\ e+ -{{24177661775} \ {f \sp {32}}} - -{{61749727185325} \ {f \sp {26}}}- -\\ -\displaystyle -{{9664082618092450} \ {f \sp {20}}} - -{{12152237485813570} \ {f \sp {14}}} - -{{9672870290826025} \ {f \sp 8}} - -\\ -\displaystyle -{{68544102808525} \ {f \sp 2}}, -\\ -\displaystyle -{{1387545279120} \ d}+ -\left( -{{1128983050} \ {f \sp {30}}}+ -{{2883434331830} \ {f \sp {24}}}+ -\right. -\\ -\displaystyle -{{451234998755840} \ {f \sp {18}}}+ -{{562426491685760} \ {f \sp {12}}}+ -{{447129055314890} \ {f \sp 6}} - -\\ -\displaystyle -\left. -{165557857270} -\right)\ e - -{{1816935351} \ {f \sp {31}}}+ -{{4640452214013} \ {f \sp {25}}}+ -\\ -\displaystyle -{{726247129626942} \ {f \sp {19}}}+ -{{912871801716798} \ {f \sp {13}}}+ -{{726583262666877} \ {f \sp 7}}+ -\\ -\displaystyle -{{4909358645961} \ f}, -\\ -\displaystyle -{{1387545279120} \ c}+ -{{778171189} \ {f \sp {31}}} - -{{1987468196267} \ {f \sp {25}}} - -\\ -\displaystyle -{{310993556954378} \ {f \sp {19}}} - -{{383262822316802} \ {f \sp {13}}} - -{{300335488637543} \ {f \sp 7}}+ -\\ -\displaystyle -{{5289595037041} \ f}, -\\ -\displaystyle -{{1387545279120} \ b}+ -\left( -{{1128983050} \ {f \sp {30}}} - -{{2883434331830} \ {f \sp {24}}} - -\right. -\\ -\displaystyle -{{451234998755840} \ {f \sp {18}}} - -{{562426491685760} \ {f \sp {12}}} - -{{447129055314890} \ {f \sp 6}}+ -\\ -\displaystyle -\left. -{165557857270} -\right)\ e - -{{3283058841} \ {f \sp {31}}}+ -{{8384938292463} \ {f \sp {25}}}+ -\\ -\displaystyle -{{1312252817452422} \ {f \sp {19}}}+ -{{1646579934064638} \ {f \sp {13}}}+ -{{1306372958656407} \ {f \sp 7}}+ -\\ -\displaystyle -{{4694680112151} \ f}, -{{1387545279120} \ a}+ -{{1387545279120} \ e}+ -\\ -\displaystyle -{{4321823003} \ {f \sp {31}}} - -{{11037922310209} \ {f \sp {25}}} - -{{1727506390124986} \ {f \sp {19}}} - -\\ -\displaystyle -\left. -{{2176188913464634} \ {f \sp {13}}} - -{{1732620732685741} \ {f \sp 7}} - -{{13506088516033} \ f} -\right\},\hbox{\hskip 1.5cm} -\end{array} -$$ -$$ -\left. -\left\{ -{{f \sp 6} -1}, -{{e \sp 2}+ -{4 \ f \ e}+ -{f \sp 2}}, -{d+e+{4 \ f}}, -{c -f}, -{b -f}, -{a -f} -\right\} -\right]\hbox{\hskip 3.5cm} -$$ -\returnType{Type: List SquareFreeRegularTriangularSet(Integer,IndexedExponents OrderedVariableList [a,b,c,d,e,f],OrderedVariableList [a,b,c,d,e,f],NewSparseMultivariatePolynomial(Integer,OrderedVariableList [a,b,c,d,e,f]))} - -Thus the solutions given by {\bf lts} are pairwise different. - -We count them as follows. - -\spadcommand{reduce(+,[degree(ts) for ts in lts]) } -$$ -156 -$$ -\returnType{Type: PositiveInteger} - -We can investigate the triangular decomposition {\bf lts} by using the -{\tt ZeroDimensionalSolvePackage}. - -This requires to add an extra variable (smaller than the others) as follows. - -\spadcommand{ls2 : List Symbol := concat(ls,new()\$Symbol) } -$$ -\left[ -a, b, c, d, e, f, \%A -\right] -$$ -\returnType{Type: List Symbol} - -Then we call the package. - -\spadcommand{zdpack := ZDSOLVE(R,ls,ls2) } -$$ -ZeroDimensionalSolvePackage(Integer,[a,b,c,d,e,f],[a,b,c,d,e,f,%A]) -$$ -\returnType{Type: Domain} - -We compute a univariate representation of the variety associated with -the input system as follows. - -\spadcommand{concat [univariateSolve(ts)\$zdpack for ts in lts] } -$$ -\begin{array}{@{}l} -\left[ -\left[ -{complexRoots={{? \sp 4} -{{13} \ {? \sp 2}}+{49}}}, -\right. -\right. -\\ -\displaystyle -coordinates= -\\ -\displaystyle -\left[ -{{7 \ a}+{ \%A \sp 3} -{6 \ \%A}}, -{{{21} \ b}+{ \%A \sp 3}+ \%A}, -\right. -\\ -\displaystyle -{{{21} \ c} -{2 \ { \%A \sp 3}}+{{19} \ \%A}}, -{{7 \ d} -{ \%A \sp 3}+{6 \ \%A}}, -{{{21} \ e} -{ \%A \sp 3} - \%A}, -\\ -\displaystyle -\left. -\left. -{{{21} \ f}+{2 \ { \%A \sp 3}} -{{19} \ \%A}} -\right] -\right],\hbox{\hskip 7.5cm} -\end{array} -$$ -$$ -\begin{array}{@{}l} -\left[ -complexRoots= -{{? \sp 4}+{{11} \ {? \sp 2}}+{49}}, -\right. -\\ -\displaystyle -coordinates= -\\ -\displaystyle -\left[ -{{{35} \ a}+{3 \ { \%A \sp 3}}+{{19} \ \%A}}, -{{{35} \ b}+{ \%A \sp 3}+{{18} \ \%A}}, -{{{35} \ c} -{2 \ { \%A \sp 3}} - \%A}, -\right. -\\ -\displaystyle -\left. -\left. -{{{35} \ d} -{3 \ { \%A \sp 3}} -{{19} \ \%A}}, -{{{35} \ e} -{ \%A \sp 3} -{{18} \ \%A}}, -{{{35} \ f}+{2 \ { \%A \sp 3}}+ \%A} -\right] -\right],\hbox{\hskip 2.0cm} -\end{array} -$$ -$$ -\begin{array}{@{}l} -\left[ -complexRoots= -{? \sp 8} -{{12} \ {? \sp 7}}+{{58} \ {? \sp 6}} -{{120} \ {? \sp 5}}+ -\right. -\\ -\displaystyle -{{207} \ {? \sp 4}} - -{{360} \ {? \sp 3}}+ -{{802} \ {? \sp 2}} - -{{1332} \ ?}+{1369}, -\\ -\displaystyle -coordinates= -\\ -\displaystyle -\left[ -{{43054532} \ a}+{{33782} \ { \%A \sp 7}} - -{{546673} \ { \%A \sp 6}}+ -{{3127348} \ { \%A \sp 5}} -{{6927123} \ { \%A \sp 4}}+ -\right. -\\ -\displaystyle -{{4365212} \ { \%A \sp 3}} - -{{25086957} \ { \%A \sp 2}}+ -{{39582814} \ \%A} -{107313172}, -\\ -\displaystyle -{{43054532} \ b} -{{33782} \ { \%A \sp 7}}+ -{{546673} \ { \%A \sp 6}} - -{{3127348} \ { \%A \sp 5}}+ -\\ -\displaystyle -{{6927123} \ { \%A \sp 4}} - -{{4365212} \ { \%A \sp 3}}+ -{{25086957} \ { \%A \sp 2}} - -\\ -\displaystyle -{{39582814} \ \%A}+{107313172}, -\\ -\displaystyle -{{21527266} \ c} -{{22306} \ { \%A \sp 7}}+ -{{263139} \ { \%A \sp 6}} - -{{1166076} \ { \%A \sp 5}}+{{1821805} \ { \%A \sp 4}} - -\\ -\displaystyle -{{2892788} \ { \%A \sp 3}}+ -{{10322663} \ { \%A \sp 2}} - -{{9026596} \ \%A}+{12950740}, -\\ -\displaystyle -{{43054532} \ d}+ -{{22306} \ { \%A \sp 7}} - -{{263139} \ { \%A \sp 6}}+ -\\ -\displaystyle -{{1166076} \ { \%A \sp 5}} - -{{1821805} \ { \%A \sp 4}}+ -{{2892788} \ { \%A \sp 3}} - -\\ -\displaystyle -{{10322663} \ { \%A \sp 2}}+ -{{30553862} \ \%A} -{12950740}, -\\ -\displaystyle -{{43054532} \ e} - -{{22306} \ { \%A \sp 7}}+ -{{263139} \ { \%A \sp 6}} - -\\ -\displaystyle -{{1166076} \ { \%A \sp 5}}+ -{{1821805} \ { \%A \sp 4}} - -{{2892788} \ { \%A \sp 3}}+ -\\ -\displaystyle -{{10322663} \ { \%A \sp 2}} - -{{30553862} \ \%A}+{12950740}, -\\ -\displaystyle -{{21527266} \ f}+ -{{22306} \ { \%A \sp 7}} - -{{263139} \ { \%A \sp 6}}+ -\\ -\displaystyle -{{1166076} \ { \%A \sp 5}} - -{{1821805} \ { \%A \sp 4}}+ -{{2892788} \ { \%A \sp 3}} - -\\ -\displaystyle -\left. -\left. -{{10322663} \ { \%A \sp 2}}+ -{{9026596} \ \%A} -{12950740} -\right] -\right],\hbox{\hskip 5.0cm} -\end{array} -$$ -$$ -\begin{array}{@{}l} -\left[ -complexRoots= -{? \sp 8}+ -{{12} \ {? \sp 7}}+ -{{58} \ {? \sp 6}}+ -{{120} \ {? \sp 5}}+ -\right. -\\ -\displaystyle -{{207} \ {? \sp 4}}+ -{{360} \ {? \sp 3}}+ -{{802} \ {? \sp 2}}+ -{{1332} \ ?}+{1369}, -\\ -\displaystyle -coordinates= -\\ -\displaystyle -\left[ -{{43054532} \ a}+ -{{33782} \ { \%A \sp 7}}+ -{{546673} \ { \%A \sp 6}}+ -{{3127348} \ { \%A \sp 5}}+ -\right. -\\ -\displaystyle -{{6927123} \ { \%A \sp 4}}+ -{{4365212} \ {