%%document begins here!!!!!!!! \documentclass[titlepage]{article} \usepackage{moreverb}%to get pretty listings with verbatimtab \usepackage{varioref} \usepackage{makeidx}% load up the indexing package \makeindex%tell it to make an index. \newcommand{\idea}[1]{#1}%use for putting things in concept index \newcommand{\noun}[1]{\index{#1}{\large \bf #1 }}%put all nouns in the %index \newcommand{\e}{\mathrm{e}}%%the exponential \newcommand{\dd}{\mathrm{d}}%%the differential \newcommand{\fixme}[1]{{\bf[FIXME: #1 ]}}% this should be handy... %%take care page numbering for coverpage \setcounter{page}{0} \begin{document} %puts page number on title with doc-class{article} \title{A Brief Introduction to Maxima} %Version 0.1 \author{\copyright Nels Tomlinson} \date{\today} \vspace{-1.0cm} \maketitle %\clearpage \tableofcontents \clearpage \section{Introduction} Matlab, Maple, Mathematica, and any other trademarks mentioned herein are the property of whoever owns them\dots This document is \copyright{ Nels Tomlinson, 2001}. This is a work in progress, and is NOT complete. \subsection{What is a Symbolic Math System?} A symbolic math system is a computer program intended to manipulate symbols rather than numbers. So, this lets one add $a+b$ without worrying about what values they have. One modern example of this is the TI-92 calculator, which has some symbolic math capabilities. Mathematica and Maple are two more examples. \subparagraph{What is \emph{not} a symbolic math system?} Matlab and GNU Octave are a fine pair of programs, very popular with engineers for numerical computation. They can both compute $\int_{1}^{2}\e^x \dd x$ to any reasonable number of decimal places, but by themselves\footnote{Matlab can be induced to give a symbolic answer, because it incorporates Maple, a commercial symbolic math system.}, cannot tell you that the answer is $\e^2-\e^1$. That is the answer that a symbolic math package should give by default, and this is what distinguishes a symbolic from a numeric system: it will (hopefully\dots) give the same answer that you would get by hand if you had the patience to do it right. \subsection{What is Maxima?} Maxima is a symbolic math system, written in Koyoto Common Lisp (now GNU lisp). It has been released, recently, under the GNU Public License. This means that it is libr\'e software, so one may download it and put it to work, legally. It is free, so one needn't pay anything to do this. It has been around for about 30+ years, and after this length of time, should be a fairly well--tested, mature product, with few unknown bugs. It is rather older than the two commercial symbolic math systems, Mathematica and Maple (M\&M) which have achieved commercial success . Since M\&M are made to sell, they have a number of features which can be listed on reviewer's checklists. They also have beautiful graphical user interfaces which make wonderful screen shots for reviewer's articles and for packaging. They are generally useful and have no bugs which the manufacturers publicize\footnote{Let me over--stress a point: I didn't say that they have \emph{no} bugs.}. Maxima, on the other hand, was made by academic researchers to test their theories, and developed for their own use. It has been used in hundreds of graduate--level classes in physics, computer science, and so on. It has been used in a great many large projects; the large bibliography available at {\verb* http://www.math.utah.edu:8080/pub/tex/bib/macsyma.html } should give some indication of how much research interest has involved Maxima. It is intended for use by experts, who are willing to invest some effort in learning how to use it. Thus, it is made to be powerful and easy for an expert to use, rather than superficially attractive and easy to learn. It seems to me that the authors have succeeded in three quarters of this: it is indeed fast and powerful, and is not superficially attractive\footnote{But see Section \vref{sec:gui-version}; there is now a graphical user interface which is superficially attractive! }. They have failed in one respect , though, since (to me at least) it seems no harder to learn than M\&M. The authors have generously given the public a powerful and useful tool for doing math homework on a PC and for doing large--scale work on super computers. They have provided a reasonably good reference manual, which is probably all the documentation an expert needs. Unfortunately, the reference manual assumes that you are familiar with the syntax, and doesn't give examples. I'm not an expert, and since it would be ungrateful of me to whine that they didn't give me enough, I've done the only reasonable thing: as I learned how to use Maxima, I've taken notes, which I've tried to put into a readable form, which I hope will help you. \section{Obtaining and Running Maxima} Maxima may be found at \ {\verb http://www.ma.utexas.edu/maxima.html }. Both Windows and Unix versions are available. Get the most recent versions of GCL and Maxima; as I write this, that is version 5.5 beta for the Unix version. Put them into the directories \emph{from} which you wish to install them. \subsection{MSWindows Installation} Since I don't use MSWindows, I don't feel qualified to give any details, however, I understand that the windows installation is the standard windows procedure, and should not cause any problems. \subsection{Unix Installation} The Unix installation is the standard unix procedure: ./configure, make, make install. Unlike the windows version, this will custom configure and compile a version just for your machine. You will need to have write permissions for the directories in which you intend to install the packages. Thus, you may need to use the su command to become root before you type make install. If you can't do that, merely install the packages in some directory for which you have permissions. Your home directory should be fine. Use the cd command to change to the directory in which you have downloaded the tarballs and untar and unzip the files using the command: \begin{verbatim} tar -xzvf filename \end{verbatim} Cd to the GCL directory, read the readme file, and follow the directions to install. This took about 10 minutes on my little machine\footnote{233MHz Pentium II laptop, 144M ram}. Now cd to the Maxima directory, read the readme file, and again follow the directions. Maxima will take a bit longer to compile. Note that before you type ./configure, you will have to edit the configure file for Maxima. Don't panic, this is trivial. The configure file is well commented: just open it in your favorite text editor and follow the instructions. You will need to cut--and--paste in the pathname of the directory where you just built GCL. Again, that's the directory, not the pathname of any file in the directory. There is an example pathname in the configure file. If you have cd'd to the directory in which you are building Maxima, you will be able to leave the MAXDIR=`pwd' line, and comment out the other MAXDIR line. Make sure that the directories in which the Maxima and elisp files will be installed are the ones that you want, and that you have write permissions on those directories. Now you are ready. Type ./configure, make, make install. \subsection{Running Maxima}\label{sec:running-maxima} There are now two versions of Maxima: xmaxima, a GUI version based on the TCL/TK toolkit, and maxima, the traditional command--line version. Both live in the same directory, so if typing /path/maxima starts the command line version, typing /path/xmaxima will start the pretty one. In both versions of Maxima, the input and output are both ascii. This will seem rather plain to someone accustomed to seeing the beautiful on--screen output of M\&M. One types in commands, and what comes back is ascii--art: symbols are drawn, over multiple lines, using the characters available on the keyboard. An expression with sub-- and super--scripts will be written over three lines. If you are accustomed to MSWindows, this will probably seem odd. If you ever need to run Maxima on a machine which you cannot physically access (e.g., by telnetting into your work machine from home), you will realize that this is a \emph{wonderful} feature. If you simply cannot stand ascii output, try the program \idea{Symaxx}, which may be found at {\verb http://symaxx.sourceforge.net/ }. It provides a graphical front end to Maxima, and seems to work reasonably well, though it is still in its infancy. At this stage of its evolution, Symaxx is probably best suited for use as a super--calculator rather than a programming environment. \subsubsection{The GUI Version}\label{sec:gui-version} The X--windows version is probably the best starting point for new users who are not familiar with emacs. For MSWindows users, the GUI will be the only option. The MSWindows and Unix GUI's seem to be identical, so the information in this section should be useful for MSWindows users too. Starting xmaxima will open a window with two panes\footnote{It should go without saying that you must be running X--windows to run the GUI version on a Unix machine. Thus, you won't be able to use it via telnet. }. The upper pane contains a command--line interface to Maxima, while the lower pane contains a Maxima Primer. I recommend the Maxima Primer to your attention, since it has a number of examples. As with all GUI's, there are buttons with labels and menus, so if you want more information about using this interface, your best bet is to start Xmaxima now and explore. The actual process of getting work done in the GUI will be exactly as detailed for the command line, below. \subsubsection{Using the Command--Line Version with Emacs} The default installation of Maxima should have installed several emacs lisp files into the site--lisp subdirectory. You will need to make sure that the appropriate path is in your .emacs file, e.g., on my machine, the following command in my .emacs file does the trick: \begin{verbatim} (setq load-path (cons "/usr/lib/emacs/site-lisp" load-path)) \end{verbatim} You'll need to put the correct pathname for your machine, of course. You will also need to copy the add-defaults.el file, which came with the Maxima package, into your .emacs file. Here is what is in it: \begin{verbatim} ;;;BEGIN maxima addition (autoload 'dbl "dbl" "Make a debugger to run lisp, maxima and or gdb in" t) (autoload 'maxima-mode "maxima-mode" "Major mode for editing maxima code and interacting with debugger" t) (autoload 'gcl-mode "gcl" "Major mode for editing maxima code and interacting with debugger" t) (setq auto-mode-alist (cons '("\\.ma?[cx]\\'" . maxima-mode) auto-mode-alist)) (setq auto-mode-alist (cons '("\\.li?sp\\'" . gcl-mode) auto-mode-alist)) ;;;END maxima addition \end{verbatim} Once these preliminaries are out of the way, you are ready to begin. To run Maxima as an inferior process, use \begin{verbatim} M-x dbl \end{verbatim} which produces the following output in the new buffer: \begin{verbatim} Welcome to DBL a Debugger for Lisp, Maxima, Gdb and others. You start your program as usually would in a shell. For Lisp and Maxima the debugger commands begin with a ':', and there is completion. Typing ':' should list all the commands. In GCL these are typed when in the debugger, and in Maxima they may be typed at any time. To see the wonderful benefits of this mode, type C-h m. Note you may also use this mode to run gdb. In fact I often debug MAXIMA over GCL using gdb, thus having three debuggers at once. To run gdb and enable the automatic line display, you must supply the `--fullname' keyword as in: gdb your-file --fullname Current directory is /home/tomlinso/jerrys/ [tomlinso@localhost jerrys]% \end{verbatim} Now, at the prompt in that buffer, type maxima, and begin: \begin{verbatim} [tomlinso@localhost jerrys]% maxima GCL (GNU Common Lisp) Version(2.3.8) Fri Feb 23 18:10:05 EST 2001 Licensed under GNU Library General Public License Contains Enhancements by W. Schelter Maxima 5.5 Fri Feb 23 18:09:58 EST 2001 (with enhancements by W. Schelter). Licensed under the GNU Public License (see file COPYING) (C1) \end{verbatim} This does not seem to provide syntactic highlighting, but does seem to work well. It is a bit of a disappointment to me after using ESS, though. \subsubsection{Using the Emacs Info with Maxima}\label{sec:using-emacs-info} \fixme{Needs to be written!} \section{Quick Start} In order to run Maxima, type maxima or xmaxima at the shell prompt. See Section \vref{sec:running-maxima} for some details about which to choose. If you are running xmaxima, or if you have an MSWindows machine, you will start a GUI in which you can type commands, which should work as described here. I don't use MSWindows, so if you have an MSWindows machine, I really can't help you with details about the user interface which are specific to MSWindows. In order to exit Maxima, one can type \noun{quit()}; \begin{verbatimtab} (C18) quit(); [tomlinso@localhost tomlinso]% \end{verbatimtab} to end the Maxima session and return to the shell. You will sooner or later make a syntax error, and fall into a debugging level. It can happen like this: \begin{verbatimtab} (C2) describe(set-up-index); Error: Takes a symbol ((MPLUS SIMP) ((MTIMES SIMP) -1 $index) $set ...) Fast links are on: do (si::use-fast-links nil) for debugging Error signalled by CATCH. Broken at MACSYMA-TOP-LEVEL. Type :H for Help. MAXIMA>>:q (C3) \end{verbatimtab} To get back out of the debugging mode, type :q as above. Notice that the colon comes \emph{before} the q! \subsection{Getting ``Help!''} There is a manual which you should have found included with the Maxima package. I will assume that you have it available to refer to. It is a reasonably good reference, and has some introductory material in it. Some sections of it are particularly useful to beginners. In particular, the \begin{itemize} \item Introduction to Command Line \item Help \item Input and Output \item Runtime Environment \item Constants \end{itemize} sections have some value for beginners. Unfortunately, the manual has little else which is oriented towards beginners. Some sections refer to Macsyma, which was a non--libr\'e, sometimes non--free predecessor. Some sections, particularly Miscellaneous Options, are specific to DEC-10 machines. One section (Symmetries) is in French, which will limit its usefulness to Americans (we only speak English\dots). \subsubsection{Using the Manual} The manual is intended as a reference work, and so \emph{if} you know where to look, it has much of what you need. It has a complete index of functions and variables. Most sections have at least a brief introduction, and then a list of functions and variable definitions which logically fall under that heading, in alphabetic order. The functions are usually what you should look for first, as they will be what you need to actually do a job. You will also need to look at the variables, since they often control the behavior of the functions. Usually the functions will refer you to the appropriate variables, another good reason to look first at the functions. \subsubsection{Electronic Help} There is also online and electronic help available. The manual is available in html format, which is convenient (and cheaper than printing out the 230+ page manual). It is possible to read the Maxima info files using emacs. See section \vref{sec:using-emacs-info} for details. If you are using the Xwindows version (or MSWindows) there is help available in the bottom pane of the GUI. There is also some help available from the Maxima command line. The manual mentions three help functions, which seem to have similar syntax. The first is \noun{describe(command)}, e.g.:\footnote0{The stuff after this sentence, {\verb in this font}, is copied verbatim from a Maxima session, to illustrate commands and their results.}1 \begin{verbatimtab} (C63) describe(save); 0: FASSAVE :(maxima.info)Definitions for Input and Output. 1: SAVE :Definitions for Input and Output. 2: SAVEDEF :Definitions for Input and Output. 3: SAVEFACTORS :Definitions for Polynomials. Enter n, all, none, or multiple choices eg 1 3 : 0 Info from file /usr/lib/maxima-5.4/info/maxima.info: - Function: FASSAVE (ARGS) is similar to SAVE but produces a FASL file in which the sharing of subexpressions which are shared in core is preserved in the file created. Hence, expressions which have common subexpressions will consume less space when loaded back from a file created by FASSAVE rather than by SAVE. Files created with FASSAVE are reloaded using LOADFILE, just as files created with SAVE. FASSAVE returns a list of the form [,,...] where ... are the things saved. Warnings are printed out in the case of large files. FASSAVE may be used while a WRITEFILE is in progress. (D63) FALSE \end{verbatimtab} Describe seems to amplify a bit on some of the commands in the manual. If you give it all as a reply, you will need a semicolon after the all. Strangely, you don't seem to need the semicolon if your choice is numeric. Next is \noun{apropos(stuff)}. This doesn't seem to work, though it is mentioned in the Reference Manual in Chapter 26, Miscellaneous Options, and the function can be described with describe(apropos);. The command \noun{demo(stuff)} demonstrates certain commands. It will run a file, so use a filename as the argument. You will need to hit a semicolon followed by a carriage return to step through the examples. The command \noun{example(stuff)} will run an example file in batch mode for some commands. To find out which commands and topics you can get examples for, type examples(manual);. Actually, example(anything not in the list of things which example can provide an example of) will display the list. I recommend that you experiment with these. The examples tend to be cryptic, but do at least give a starting point for learning. \subsection{Syntax} Maxima's syntax is more--or--less standard for a programming language. That is, there is nothing {\`o}utr\'e such as is found in APL or intercal (well, not much). On the other hand, because it is ``normal'', it doesn't have the consistancy of Lisp\footnote{Yes, Lisp is rather strange if you are used to Fortran, but it \emph{is} consistent: everything is prefix notation.}. \paragraph{A quick list of some important syntactic points:} \begin{itemize} \item Everything must end with either a \noun{semicolon} (\noun{;}) or a \noun{dollar sign} (\noun{\$})! If the interpreter seems to ``hang'', you probably forgot the semicolon. Normally you will want to use a semicolon to end a line. If you don't want to see the output of your command (e.g., it might be too big), use the \$ instead. It will \idea{suppress the output}. \item The assignment operator is the colon. To assign 4 to the variable a, proceed as follows: \begin{verbatimtab} (C3) a:4; (D3) 4 (C4) \end{verbatimtab} \item To declare a function, use colon equals, like so: \begin{verbatimtab} (C4) f(x,y):=x^2+y^2; 2 2 (D4) f(x, y) := x + y (C5) \end{verbatimtab} \item To differentiate a function, use diff(f(x,y),x); \begin{verbatimtab} (C5) diff(f,x); (D5) 0 (C6) diff(f(x,y),x); (D6) 2 x (C7) \end{verbatimtab} Notice that {\verb diff(f,x);} didn't work. It seems to have tried to differentiate the variable f with respect to the variable x, rather than differentiating the function f(x,y). { \verb diff(f(x,y),x);} did what we wanted. We didn't \emph{need} to make f a function; it is also possible to differentiate expressions. \item Matrices are not arrays! An array is a data structure, while a matrix is an object on which one may commit linear algebra. One way to construct matrices is with the matrix command. \begin{verbatimtab} (C29) matrix([w,x],[y,z]); [ w x ] (D29) [ ] [ y z ] (C30) \end{verbatimtab} \end{itemize} Maxima of course has a number of built--in functions with which one can save and load data, differentiate and integrate, plot, generate Fortran or \TeX code, and so on. As mentioned above, it also has a number of global variables which one can use to affect the behavior of these functions. \subsubsection{Some Other Elements of Syntax} The \noun{square bracket}, (\noun{[]}) is used to delimit a list. The \noun{parentheses } (\ \noun{()} \ ) are used to enclose the arguments of functions. The asterisk, (\noun{*}) is used for multiplication. The \noun{caret}, (\noun{\^{}}) is used for exponentiation. The \noun{double colon}, \noun{::} is used to assign the contents of one variable to a variable which is referenced by another variable. See Section \vref{sec:expressions} for some details on this. \subsubsection{Using Earlier Results} The \idea{prompt} in Maxima is (\idea{Cnumber}), where number is a counter, telling you how many commands you have issued. When Maxima returns a result to you, it will preface it with (\idea{Dnumber}), as shown below: \begin{verbatimtab} (C2) 1+1; (D2) 2 (C3) %+3; (D3) 5 (C4) %th(2)+4; (D4) 6 (C5) D2; (D5) 2 (C6) \end{verbatimtab} There are several points to notice here. First, the (Dnumber)'s have the same number as the (Cnumber)'s which produced them. Second, you can refer to the previous result as \noun{percent}, \noun{\%}, as shown in (C3) in the example above. The command \%+3; says to take the previous result and add three to it. Third, you can refer to an earlier result by saying \noun{\%th(i)} for the i${}^\mathrm{th}$ previous result. On command (C4) above, I added 4 to the second previous result, which was (D2). Fourth, you can refer to the results by name. The result of the 1+1; command was (D2). In command (C5), I re--displayed it by typing its name D2. \subsection{Writing and reading files} Maxima gives several ways to save a workspace to a file. \noun{save(filename,arg1,arg2,\dots)} allows you to save the variables arg1, arg2 \dots to the file filename. \emph{filename} should be a proper Unix filename, with path. If a relative path is given (i.e., a path which does \emph{not} begin with a / ), it is taken to be relative to the present working directory. \fixme{How do we find the pwd? Relative pathnames \emph{don't} work?} Here is an example: \begin{verbatimtab} (C6) f(x,y):=x^2+y^2+2*x*y; 2 2 (D6) f(x, y) := x + y + 2 x y (C7) a:4; (C12) save("/home/tomlinso/slime.stuff",f,a); (D12) /home/tomlinso/slime.stuff \end{verbatimtab} This saves the function f and the variable a to a file in my home directory. If you do this and then take a look at the file, you will find that it is a text file, but does not look at all like the function you saw on your screen. You will get some idea about why it looks so strange in section \vref{sec:internal-details}, which details some of the internal workings of Maxima. The command \noun{loadfile(filename)} should reload a file saved by save() or \noun{fassave()}. The filename uses the same syntax here as in save. There is also the command \noun{writefile(filename)}, which will keep a log of your maxima session. If you run Maxima under emacs, this may seem redundant. The file which this command makes is \emph{not} readable by loadfile(). \fixme{Is there a way to load some ascii data, maybe a csv file?} \section{Expressions}\label{sec:expressions} \idea{Expressions} are still a bit fuzzy to me in this version of the notes, \fixme{get defuzzed!} as are functions. I won't try to define them yet. \subsection{Assignment Operators} Maxima has two assignment operators: the \noun{colon} (\noun{:}) and the \noun{double colon}, (\noun{::}). The single colon is simple to understand. A:3; sets the variable A to the value 3. \begin{verbatimtab} (C11) A:3$ (C12) A; (D12) 3 (C13) \end{verbatimtab} The double colon is a bit harder to grasp at first. Consider the following example: \begin{verbatimtab} (C24) A:3$ (C25) B:5$ (C26) C:'A; (D26) A (C27) C::B; (D27) 5 (C28) \end{verbatimtab} I assigned the value 3 to the symbol A, the value 5 to the symbol B, and then I assigned the symbol A to the symbol B. That's why (C26) says C:'A;, the apostrophe prevents Maxima from evaluating the symbol A and assigning 3 to C. Then, in (C27), I used the double colon to assign the value of B (remember, that's 5) to the value of C (remember, that's A). Here's the final result: \begin{verbatimtab} (C28) A; (D28) 5 (C29) \end{verbatimtab} A is five! The manual says ``:: assigns the value of the expression on its right to the value of the quantity on its left \dots''. Hopefully that makes more sense with an example staring at you than it did the first time I read it. This seems to be analogous to the idea of a pointer: we didn't assign the value of B to C, we assigned the value of B to what C \emph{held}, or pointed to. \section{Defining Functions} To define a \idea{function}, use function name:=function value, as follows \begin{verbatimtab} (C9) lv:=x^2+2*x*y+y^2; Improper function definition: lv -- an error. Quitting. To debug this try DEBUGMODE(TRUE);) (C10) lv(x,y):=a*x^2+2*a*b*x*y+b*y^2; 2 2 (D10) lv(x, y) := a x + 2 a b x y + b y (C11) \end{verbatimtab} Notice that we must specify what lv is a function of. The first attempt, (C9), didn't work, while (C10) specifies that lv depends on x and y, but that a and b are parameters. It appears that Maxima's function is not a function in the sense that mathematicians use the word, but rather in the computer science sense. That is, a function is a subroutine which may be called. Thus, you may find what you are looking for in Section \vref{sec:expressions}, which covers expressions. \section{Doing Calculus} \subsection{Differentiation} \subsubsection{Differential Equations} \subsection{Integration} \section{Working with Matrix Algebra} \subsection{Building and Manipulating Matrices} Matrices are not arrays! An array is a data structure, while a matrix is an objecton on which one may commit linear algebra. One way to construct matrices is with the \noun{entermatrix(m,n)} command, which allows one to enter a matrix with m rows and n columns, with prompting. The manual gives a good example of this, which I won't repeat. Another way to construct matrices is with the \noun{matrix} command. \begin{verbatimtab} (C29) matrix([w,x],[y,z]); [ w x ] (D29) [ ] [ y z ] (C30) row(matrix([w,x],[y,z]),1); (D30) [ w x ] (C31) row(matrix([w,x],[y,z]),2); (D31) [ y z ] \end{verbatimtab} The matrix command takes as its arguments a series of lists. Each list makes up one row of the matrix. The \noun{row} command takes a matrix and a number as arguments, and returns the row specified by the number. There is also a \noun{submatrix} command, which takes a number, a matrix, and a number as arguments. The first number specifies the row to cast out, the second number specifies the column to cast out, and the function returns the submatrix which is lect after casting out these rows and columns. \begin{verbatimtab} (C7) m:matrix([a,b,c],[d,e,f],[g,h,i]); [ A b c ] [ ] (D7) [ d e f ] [ ] [ g h i ] (C8) submatrix(2,m,2); [ A c ] (D8) [ ] [ g i ] (C9) submatrix(1,m,3); [ d e ] (D9) [ ] [ g h ] (C10) \end{verbatimtab} In this example, we first defined the matrix m, then cast out the second row and column on line (C8). Finally in line (C9), we cast out the first row and the last column. The \noun{diagmatrix(n,x)} command produces a diagonal matrix of size n, with all the diagonal elements x. Plainly, x=1 will produce an n by n identity matrix. So will \noun{ident(n)}. \begin{verbatimtab} (C11) diagmatrix(4,3); [ 3 0 0 0 ] [ ] [ 0 3 0 0 ] (D11) [ ] [ 0 0 3 0 ] [ ] [ 0 0 0 3 ] (C12) \end{verbatimtab} A third way to build a matrix is with the \noun{genmatrix(array,i2,j2,i1,j1)} command. This fills a matrix from an array. The manual provides examples. \subsection{Doing Linear Algebra} \paragraph{Matrix Multiplication} To multiply two matrices together, the \noun{dot operator}, or \noun{.} is used. To make certain that the interpreter does not mistake it for a decimal point, it is important that at least one space should be left on each side of it. \begin{verbatimtab} (C26) A:matrix([b,c],[d,e]); [ b c ] (D26) [ ] [ d e ] (C27) f:matrix([g,h],[i,j]); [ g h ] (D27) [ ] [ i j ] (C28) A . f; [ c i + b g c j + b h ] (D28) [ ] [ e i + d g e j + d h ] (C29) \end{verbatimtab} And a quadratic form: \begin{verbatimtab} (C29) matrix([1,2]) . A . transpose(matrix([1,2])); (D29) 2 (2 e + d) + 2 c + b (C30) \end{verbatimtab} The \noun{transpose()} function, of course, gives the transpose of its matrix argument. To square a matrix, use the dot operator, as in A . A, or use the exponential version: \begin{verbatimtab} (C7) a^^2; [ 2 ] [ c d + b c e + b c ] (D7) [ ] [ 2 ] [ d e + b d e + c d ] (C8) a . a; [ 2 ] [ c d + b c e + b c ] (D8) [ ] [ 2 ] [ d e + b d e + c d ] (C9) \end{verbatimtab} The \noun{invert()} function also has a self--explanatory name. Notice that the \noun{doallmxops} variable, if true, causes the determinant to be moved inside the inverse. When false, it causes the determinant to be left outside the matrix. The Reference Manual also mentions the variable \noun{detout} in this regard, but it seems to have no effect. \begin{verbatimtab} (C42) doallmxops:true; (D42) TRUE (C43) invert(a); [ e c ] [ --------- - --------- ] [ b e - c d b e - c d ] (D43) [ ] [ d b ] [ - --------- --------- ] [ b e - c d b e - c d ] (C44) doallmxops:false; (D44) FALSE (C45) invert(A); [ e - c ] [ ] [ - d b ] (D45) ------------ b e - c d (C46) \end{verbatimtab} \paragraph{Row Reduction} The \noun{echelon(matrix)} command returns the echelon form of the matrix argument. \section{Getting Numbers} \subsection{Evaluating: the ev() function} \section{Plotting} \section{Internal Details} \label{sec:internal-details} \subsection{Lists} \subsection{Maxima and Lisp} \section{\emph{Making} it Work: What to do When Maxima is Stubborn} \subsection{Pickapart} \clearpage \printindex \end{document}