Maxima Overview:

Maxima Overview

by Burkhard Bunk 
19.3.2013 
Maxima version: 5.21.1 
wxMaxima version: 0.8.5

Documentation

/usr/share/doc/maxima-doc/html/intromax.html    introduction
/usr/share/doc/maxima-doc/html/maxima_toc.html  full help

Invocation, Help

Text mode:
      unix> maxima
      interrupt a maxima calculation with
            <CTRL>-G
      terminate maxima with
            quit();

GUI:
      unix> xmaxima
      unix> wxmaxima

Help:
      GUI ->Help
      descripe(string);
      ? string;               /* note the space after "?" */

A perfect way to restart maxima does not exist (see mailing list), use

      reset();
      kill(all);

Commands and Expressions

Terminate commands with ";" or "$" (quiet mode).

command prompts:  (%i1)  (%i2)  ..
output labels:    (%o1)  (%o2)  ..

Previous result:              %
Former output, e.g. (%o5):    %o5
Redo command, e.g. (%i5) :    ''%i5;

Operators:              + - * / ^ **  ( )  ! !!
Comparison:             = # > < >= <=           /* "#" is "not equal" */
Logical:                and or not

Text strings are written as "text".

Names

Names consist of letters, digits, % (percent) and _ (underscore). Letters are case sensitive, but predefined names are mapped to upper case. Special characters (other than "%" and "_") are allowed after declaring them alphabetic as in

      declare("$", alphabetic);

Pre-defined constants:

      %i %pi %e true false
      %gamma      Euler's constant
      %phi        (1 + sqrt(5))/2
      inf         real infinity
      minf        real (-infinity)
      infinity    complex infinity

Pre-defined functions:

      sqrt  log  exp
      sin ..  asin ..
      sinh ..  asinh ..
      gamma
      zeta
      binomial
      mod  floor
      ..

Reserved names:

      integrate   next        from        diff            
      in          at          limit       sum             
      for         and         elseif      then            
      else        do          or          if
      unless      product     while       thru            
      step                                                                     

Manage user-defined variables:

      values                  list of all user defined variables
      remvalue(var);          delete the value of var
      remvalue(all);          delete the values of all variables

Assignment

Assignment: :     (without "="!)
            ::    ??
            :=    defines a function
            ::=   defines a macro

Unassign names:
      kill(name1, name2, ..);
      kill(all);

      xmaxima:    ->File ->Restart
      wxMaxima:   ->Maxima ->Restart maxima

Lists

      lst: [el1, el2, ...];         contruct list explicitly
      lst[2];                       reference to element by index
                                          (starting from 1)

      cons(expr, alist);            prepend expr to alist
      endcons(expr, alist);         append expr to alist
      append(list1, list2, ..);     merge lists
      makelist(expr, i, i1, i2);    create list with control variable i
      makelist(expr, x, xlist);     create list with x from another list     

      length(alist);                returns #elements

      map(fct, list);               evaluate a function of one argument
      map(fct, list1, list2);       evaluate a function of two arguments

Evaluation

- automatic

Many expressions are evaluated automatically: before the are processed, substitutions and obvious simplifications are preformed.

- explicit

Sometimes, this should be prevented (e.g. when defining a differential equation), in other cases, an extra evaluation is required (e.g. after changing an item in an expression):

      'expr              don't evalute
      ''expr             do evaluate
e.g.
      a: b+c;
            c + b
      b:5$
      a;
            c + b
      ''a;
            c + 5
      'b + c;
            c + b

Evalutate variables, based on equations:

      at(expr, var=ex);
      at(expr, [var1=ex1, var2=ex2, ..]);

e.g.  at(diff(sin(x), x), x=%pi);
            -1      

Evaluate with additional settings/flags:

      ev(expr, arg1, arg2...);
      expr, arg1, arg2, ...;        same in short
            args: numer
                  float
                  bfloat
                  simpsum
                  eval
                  ...

Many of the evalutation switches (e.g. numer, simpsum) are actually flags, which are false by default, but can be enabled for subsequent use.

- numerical

Numerical evaluation is triggered by decimal numbers in expressions, but note that a dot alone doesn't make it: sqrt(2.) is not evaluated, but sqrt(2.0) is.

      float(expr);                  evaluate to floating point number
      expr, numer;                  return numerical result
      numer: true;                  numerical evaluation on (default: false)

      fpprec: digits;               precision of big floats (default: 16)
      fpprintprec: digits;          no. of printed digits
      bfloat(expr);                 evaluate to big float

example:
      fpprec: 30;
      sin(%pi);                     => 0
      sin(float(%pi));              => 1.2246063538223773E-16
      sin(bfloat(%pi));             => 1.69568553207377992879174029388B-31

Note: WxMaxima, with output format set to xml, diplays long numbers with a shortcut - compare the output of

      bfloat(%pi), fpprec:1000;

Substitution

Functions which perform substitution, with increasing level of sophistication:

      subst(..)         syntactic, symbols and complete sub-expressions only
      ratsubst(..)      similar, but employs some algebra
      at(..)            evaluation, based on equations
      ev(..)            evaluation, with equations, flags etc.

subst(..) and ratsubst(..) in more detail:

      subst(ex, var, expr);         substitute ex for var in expr
      subst(var=ex, expr);          same
      subst([var1=ex1, var2=ex2,..], expr);     multiple substitutions

      ratsubst(ex, var, expr);      substitute ex for var in expr

      subst(s, a+b, a+b+c);
            c + b + a
      ratsubst(s, a+b, a+b+c);
            s + c
      subst(1-cos(x)^2, sin(x)^2, sin(x)^4 - 5*sin(x)^2);  
               4                2
            sin (x) - 5 (1 - cos (x))
      ratsubst(1-cos(x)^2, sin(x)^2, sin(x)^4 - 5*sin(x)^2);  
               4           2
            cos (x) + 3 cos (x) - 4

Functions

Function definition, general form:

      define(f(x), expr);
      define(f(x,y), expr);         /* etc */

simple form: e.g.

      func(x) := sin(x)/x;
      radius(x,y) := sqrt(x^2 + y^2);

More complicated function definitions can be formulated with the block(..) construct.

Simplification

Polynomials

      factor(expr);           factorise polynomials (over integers only)
      expand(expr);           expand polynomials
      ratexpand(expr);        same (more efficient algorithm)

      expandwrt(expr, x, ..); expand w.r.t. specified variables
      coeff(expr, x, n);      coefficient of x^n in expr
      ratcoef(expr, x, n);    same, but simplifies expr first

      divide(pol1, pol2);     polynomial devision (with remainder)
      quotient(pol1, pol2);   quotient of polynomial devision
      remainder(pol1, pol2);  remainder of polynomial division

      realroots(pol, tol);    numerical approx. to all real roots
      realroots(pol);         tol = rootsepsilon (default: 1e-7)
      allroots(pol);          numerical approx. to all complex roots

Rational functions

      ratsimp(expr);          put on common denominator,
                                    cancel factors,
                                    expand numerator and denominator
      fullratsimp(expr);      repeated application of `ratsimp'

      factor(expr);           same as `ratsimp', but returns numerator and
                                    denominator in factored form         

      expand(expr);           expand numerator and denominator, split numerator
                                    (no common denominator)
      ratexpand(expr);        put on common denominator,
                                    cancel factors,
                                    expand numerator and denominator,
                                    split numerator
      ratdenomdivide: false;  don't split numerator (same as ratsimp?)

      facsum(expr, var, ..)   expand w.r.t. specified variables
      facsum_combine: false;  split numerator

      partfrac(expr, var);    partial fraction decomposition

examples:
      ratsimp(a/b + c/d);
            a d + b c
            ---------
               b d

      (x-1)/(x+1)^2 - 1/(x-1);
             x - 1       1
            -------- - -----
                   2   x - 1
            (x + 1)
      ratexpand(%);
                    4 x
            - ---------------
               3    2
              x  + x  - x - 1
      factor(%);
                    4 x
            - ----------------
                             2
              (x - 1) (x + 1)


      r: (u+v)^2*u/((u^2-v^2)*v);
                           2
                  u (v + u)
                  -----------
                      2    2
                  v (u  - v )

      ratsimp(r);
                    u v + u
                  - --------
                     2
                    v  - u v

      factor(r);
                    u (v + u)
                  - ---------
                    v (v - u)

      ratexpand(r);
                        2
                       u         u
                  - -------- - -----
                     2         v - u
                    v  - u v

      s: a*b/(c*d+c*e) + f*b/(c*d+c*e);
                     b f         a b
                  --------- + ---------
                  c e + c d   c e + c d
      factor(s);
                  b (f + a)
                  ---------
                  c (e + d)
      ratsimp(s);
                  b f + a b
                  ---------
                  c e + c d

Notes: 
the following appear to be equivalent:

      ratsimp(expr);          and   ratexpand(expr), ratdenomdivide: false;
      ratsimp(expr), factor;  and   factor(expr);

but factor(expr) does not understand algebraic!

Summary: simplify rational functions with ratsimp(expr), possibly combined with factor and/or algebraic. Use ratexpand(expr), possibly with algebraic, if you prefer to split the numerator.

Roots

      rootscontract(expr);                products of roots -> roots of products
      ratsimp(expr), algebraic;           rationalise denominators

      radcan(expr);                       canonical form, involving roots, logs,
      radcan(expr), algebraic;                   and exponentials

examples:

      ex: 1/(sqrt(a)+sqrt(b));
                    1
            -----------------
            sqrt(b) + sqrt(a)
      ratsimp(ex), algebraic;
            sqrt(b) - sqrt(a)
            -----------------
                  b - a

      sqrt(x^2);
            abs(x)
      sqrt(x^2), radexpand:all;
            x
      

In some cases, sqrtdenest can disentangle nested square roots:

      load(sqdnst);
      sqrtdenest(expr);
e.g.
      sqrt(sqrt(7)+ 4);
            sqrt(sqrt(7) + 4)
      sqrtdenest(%);
            sqrt(7)      1
            ------- + -------
            sqrt(2)   sqrt(2)
      factor(%);
            sqrt(7) + 1
            -----------
              sqrt(2)

Logarithms

      logexpand:all;                enables automatic expansion of products

      logcontract(expr);            contracts sums of logs to logs of products
                                          and _integer_ multiples of logs to
                                          logs of powers

      radcan(expr);                 canonical form, involving roots, logs,
      radcan(expr), algebraic;             and exponentials

examples:

      log(a^b);
            log(a) b

      log(a*b), logexpand:all;
            log(b) + log(a)

      logcontract(2*log(a) + 3*log(b));
                 2  3
            log(a  b )

Trigonometric functions

      trigsimp(expr);         use sin(x)^2 + cos(x)^2 = 1 etc
      trigexpand(expr);       use addition theorems etc
      trigreduce(expr);       powers -> multiple arguments
                              products -> sums
      trigrat(expr);          simplify rational expressions of trigonometric
                                    functions as well as linear arguments
                                    involving %pi/n
      halfangles:true;        replace half angles by roots

      exponentialize(expr);   trig/hyperb -> exponentials
      demoivre(expr);         complex exponentials -> trig (not hyperb)

      logarc(expr);           arc trig/hyperb -> logarithms

trigexpand is a flag as well (and an evflag), but the other trigX aren't!

trigsimp(..) in combination with roots is tricky:

      trigsimp(sqrt(sinh(x)^2 + 1));
            cosh(x)
      trigsimp(sqrt(cosh(x)^2 - 1));
            sqrt(cosh(x) - 1) sqrt(cosh(x) + 1)

instead of the expected abs(sinh(x)). It does not work for sin() and cos(x) either. Is this caused by abs(..)? 
?HOW TO

There is no command to convert real exponentials to hyperbolic functions - use ratsubst(..) instead.

Examples:

      sin(x/2), halfangles;
            sqrt(1 - cos(x))
            ----------------
                sqrt(2)

?HOW TO:

      sin(x) + cos(x) = sqrt(2) * sin(x + %pi/4)
      try exponentialize(...)

?HOW TO

      ex1: cos(x) + cos(y);
      ex2: 2 * cos((x+y)/2) * cos((x-y)/2);
ex2 -> ex1:
      trigreduce(ex2), ratsimp;
or    trigrat(ex2);
ex1 -> ex2 ??

logarc examples:

      asinh(x), logarc;
                      2
            log(sqrt(x  + 1) + x)
      
      acosh(x), logarc;
                  sqrt(x + 1)   sqrt(x - 1)
            2 log(----------- + -----------)
                    sqrt(2)       sqrt(2)
      %, logcontract, expand, rootscontract;
                      2
            log(sqrt(x  - 1) + x)

Number Theory

Compute the prime factorisation of a number:

      factor(n)               basic method
      ifactors(n)             more efficient algorithm
      ifactor_verbose: true   show details

Limit

      limit(f(x), x, a);
      limit(f(x), x, a, dir);       direction dir = plus, minus

Differentiation

      diff(expr, x);
      diff(expr, x, n);       /* n-th derivative */
      diff(expr, x, y);       /* mixed partial derivative */

Convert the derivative to a function with define(..):

      f(x) := sin(x);               /* works */
      diff(f(x), x);
            cos(x)                  /* ok */
      g(x) := diff(f(x), x);        /* doesn't work */
      define(g(x), diff(f(x), x));
            g(x) := cos(x)          /* works */

Compute the derivative at a specific value with at(..):

      at(diff((x-a)^2, x, 2), x=a);
            2

Integration

      integrate(f(x), x);           indefinite integral
      integrate(f(x), x, a, b);     definite integral
      defint(f(x), x, a, b);        same
      ldefint(f(x), x, a, b);       same, but taking limits at the boundaries

Examples:

      assume(a>0)$
      declare(a, noninteger)$
      facts(a);
            [a > 0, kind(a, noninteger)]
      integrate(x^a * exp(-x), x, 0, inf);
            gamma(a + 1)
      kill(all)$
      facts();
            []

      integrate(1/(a - cos(x)), x, 0, %pi);

      Is  (a - 1) (a + 1)  positive, negative, or zero? pos;
                2
      Is  sqrt(a  - 1) - a  positive or negative? neg;
                2
      Is  sqrt(a  - 1) - a + 1  positive, negative, or zero? pos;
          !      2         !
      Is  !sqrt(a  - 1) + a! - 1  positive, negative, or zero? pos;

                          2
              2 %pi sqrt(a  - 1) - 2 %pi a
            - ----------------------------
                         2         2
              2 (a sqrt(a  - 1) - a  + 1)

      ratsimp(%), algebraic;

                %pi
            ------------
                  2
            sqrt(a  - 1)

Summation

Sums are defined with sum(..) and evaluated (symbolically) with simpsum:

      sum(expr, n, n1, n2);
      ev(sum(...), simpsum);        sum and simplify
      sum(...), simpsum;            same in short
      simpsum: true;                enable summations

Example:

      sum(k^2, k, 1, n);
                                     n
                                    ====
                                    \      2
                                     >    k
                                    /
                                    ====
                                    k = 1
      %, simpsum;
                                   3      2
                                2 n  + 3 n  + n
                                ---------------
                                       6

The same is achieved with

      sum(k^2, k, 1, n), simpsum;
or
      simpsum: true;
      sum(k^2, k, 1, n);

Series expansion

      powerseries(expr, var, point);      symbolic, possibly infinite
      taylor(expr, var, point, order);    truncated at given order

      niceindices(expr);                  rewrite symbolic sums

Equations

Equations (single or systems) are solved by solve.

      solve(eqn, var);
      solve([eqn1, eqn2, ..], [var1, var2, ..]);

It returns a list of solutions resp. solution vectors. 
Examples:

      sol: solve(x^2 + p*x + q, x);
                   sqrt(p  - 4 q) + p      sqrt(p  - 4 q) - p
            [x = - ------------------, x = ------------------]
                           2                       2
      x1: x, sol[1];
              sqrt(p  - 4 q) + p
            - ------------------
                      2
      x2: x, sol[2];
              sqrt(p  - 4 q) - p
            - ------------------
                      2

      eqn1: x + y = 4;
            y + x = 4
      eqn2: x - y = 2;
            x - y = 2
      sol: solve([eqn1, eqn2], [x,y]);
            [[x = 3, y = 1]]
      x, sol;
            3
      y, sol;
            1
Check:
      map(is, ev([eqn1, eqn2], sol));
            [true, true]

Maybe ev(..) needs more flags (e.g. ratexpand).

Check multiple solutions (equation is formulated as f(x) = 0):

      f(x):= x^2 + 2*b*x + c;
                     2
            f(x) := x  + 2 b x + c
      sol: solve(f(x), x);
                         2                     2
            [x = - sqrt(b  - c) - b, x = sqrt(b  - c) - b]
      map(f, map(rhs, sol)), expand;
            [0, 0]

?HOW TO:

      solve(sin(x) + cos(x) = 1/2, x);

      eqn: sin(x) + cos(x) = 1/2;
      solve(eqn, x);                no success

      eqnx: exponentialize(eqn);
      sol: solve(eqnx, x);          solution in terms of complex logs

      ratsimp(sol);
      %, numer;

      rectform(sol);                imaginary parts are obsolete
      ratsimp(%);                   nicer formula
      %, numer;

?HOW TO:

      solve(s + sqrt(1-s^2) = 1/2, s);

      solveradcan: true;            doesn't help here
                                    (solve calls radcan)

      eq1: s + r = 1/2;             aux variable r = sqrt(1 - s^2) > 0
      eq2: r^2 = 1 - s^2;
      solve([eq1, eq2], [s, r]);
                    sqrt(7) - 1      sqrt(7) + 1
            [[s = - -----------, r = -----------], 
                         4                4
                  sqrt(7) + 1        sqrt(7) - 1
             [s = -----------, r = - -----------]]
                       4                  4

Solution is s = (1 - sqrt(7))/4 .

?HOW TO:

      solve([sqrt(x) + y = 0, sqrt(x) = 1], [x, y]);
            []
      eliminate([sqrt(x) + y = 0, sqrt(x) = 1], [x]);
            [1]                     ??? BUG ???
      eliminate([sqrt(x) + y = 0, sqrt(x) = 1], [y]);
            [ sqrt(x) - 1 ]         ok

Note that eliminate() uses resultant(), which is supposed to work with polynomials. 
Eliminate variables from a set of (polynomial?) equations:

      eliminate([eqn1, eqn2, ..], [var1, var2, ..]);

Note: there is a new solver package ("topoly" or "to_poly"), see appendix.

Numerical solution

      find_root(expr, x, a, b)

For polynomials: see realroots(pol), allroots(pol).

Ordinary Differential Equations

ode2

Try to solve general ODEs of first or second order with ode2:

      ode2(eqn, y, x);

E.g. eqn of 2nd order:
      eqn: 'diff(x,t,2) + r*'diff(x,t)^2 = 0;
general solution, with constants of integration %k1, %k2:
      sol: ode2(eqn, x, t);
                      log(r t + %k1 r)
                  x = ---------------- + %k2
                             r
impose initial conditions:
      ic2(sol, t=0, x=x0, 'diff(x,t)=v0), logcontract, ratexpand;
                           log(r t v0 + 1)
                  x = x0 + ---------------
                                  r
convert solution equation to a function:
      define(x(t), rhs(%));

[TO DO: for the same deqn with exponent 2 -> 3, ic2() fails to solve for the initial conditions. This looks like a problem with solve...] 
Boundary conditions are imposed with bc2(..). 
For an equation of first order, the initial conditions are specified with ic1(..).

desolve

Solve a linear ODE with desolve (using Laplace transformation):

      eqn: 'diff(f(x), x) = 2*f(x);       /* linear ODE of 1st order */
      sol: desolve(eqn, f(x));
                          2 x
            f(x) = f(0) %e      

[Note the different format of derivatives in the equation.] 
Initial values at x = 0 can be imposed with atvalue before calling desolve:

      eqn: 'diff(f(x), x) = 2*f(x);
      atvalue(f(x), x=0, k);              /* initial value at x=0 */
      sol: desolve(eqn, f(x));            /* solution as an equation */
                       2 x
            f(x) = k %e
      define(f(x), rhs(%));               /* solution function */
                        2 x
            f(x) := k %e

An example of second order:

      eqn: 'diff(f(t), t, 2) + r*'diff(f(t), t) + f(t) = sin(omega*t);
      atvalue(f(t), t=0, 1);
      atvalue('diff(f(t), t), t=0, 0);

      desolve(eqn, f(t));     /* omega nonzero, -2 < r < 2 */

      define(f(t), rhs(%));
      plot2d(ev(f(t), omega=1.1, r=0.1), [t, 0, 100]);

A linear system is solved with desolve as follows:

      eqn1: 'diff(f(x), x) = c*f(x) - g(x);
      eqn2: 'diff(g(x), x) = c*g(x) + f(x);
      atvalue(f(x), x=0, 1);
      atvalue(g(x), x=0, 0);

      sol: desolve([eqn1, eqn2], [f(x), g(x)]);

      define(f(x), rhs(sol[1]));
      define(g(x), rhs(sol[2]));
      c: 0.1;
      plot2d([parametric, f(t), g(t)], [t, 0, 10], [nticks, 100]);

desolve requires that the inverse Laplace transform (ilt) is applied to a rational function with a denominator of first or second order.

Vectors and Matrices

      A: matrix([a, b, c], [d, e, f], [g, h, i]);     /* (3x3) matrix */
      u: matrix([x, y, z]);                           /* row vector */
      v: transpose(matrix([r, s, t]));                /* column vector */

Reference to elements etc:

      u[1,2];                 /* second element of u */
      v[2,1];                 /* second element of v */
      A[2,3];   or  A[2][3];  /* (2,3) element */
      A[2];                         /* second row of A */
      transpose(transpose(A)[2]);   /* second column of A */

Element by element operations:

      A + B;      A - B;
      A * B;      A / B;
      A ^ s;      s ^ A;

Matrix operations:

      A . B;                  /* matrix multiplication */
      A ^^ s;                 /* matrix exponentiation (including inverse) */
      transpose(A);
      determinant(A);
      invert(A);

Properties

      assume(pred);           predicate `pred' e.g. a > 0, equal(b,3), notequal(c,0)
      is(expr);               check whether expr is true, based on assumptions
      forget(pred);           remove `pred' from assume database

      features                list of mathematical properties
      declare(var, prop);
      remove(var, prop);

      facts(item);            list properties associated with item
      facts();                list all properties

      domain:real             default
      domain:complex

Examples:

      facts(a);               list of properties involving a
      forget(facts());        remove all properties (but not the features?)
      forget(facts(a));       remove all properties (not features?) involving a

kill(all) clears the facts database (among other things).

Graphics

      plot2d(expr, range);                /* one curve */
      plot2d([expr1, expr2], range);      /* two curves */
      plot2d([parametric, expr1, expr2], range);      /* parametric */
examples:
      plot2d(sin(x), [x, 0, 10]);
      plot2d(tan(x), [x, 0, 10], [y, -2, 2]);   /* truncate vertically */
      plot2d([8*sin(x), exp(x)], [x, -2, 2]);
      plot2d([parametric, t*cos(t), t*sin(t)], [t, 0, 10], [nticks, 100]);

      plot2d(sin(x), [x, 0, 10],  [gnuplot_term, ps],
            [gnuplot_out_file, "filename"]);    /* write PS file */

      plot3d(expr, range1, range2);       /* 3D mesh plot */
example:
      plot3d(sin(x)^2 * sin(y)^2, [x, -2, 2], [y, -2, 2]);

It appears that in a parametric plot, the parameter has to be named "t".

I/O

Console interaction:

      print("text");
      print(expr1, expr2, ..);
      disp(expr1, expr2, ..);
      display(expr1, expr2, ..);

      z: read("what is z?");              /* terminate reply with ; */

Read/write data (in matrix or list form) from/to a file (space-separated numbers):

      load("numericalio");

      read_matrix("filename");
      write_data(matrix, "filename");

      read_nested_list("filename");
      write_data(list, "filename");

File search and display:

      file_search("filename");      check for the existence of a file, using
                                    file_search_maxima etc as search paths
      file_search("filename",["path/"]);  use specified path
                                          (relative or absolute path)
      file_search_maxima            search path list, for load etc
      file_search_usage             search path list, e.g. for printfile
      printfile("filename");        display contents of file

e.g. prepend a directory as follows:

      file_search_maxima: cons("/home/me/work/", file_search_maxima);

Notes: 
o terminate path elements with / 
o It appears that the current working directory (where maxima was started) is always searched first.

Running scripts:

      batch("filename");            run maxima commands from file
      batchload("filename");        same, in quiet mode
      load("filename");             run maxima and lisp code from file

Command display and session log

Disable 2D display, use 1-line output:

      display2d:false

Convert expression to TeX format:

      tex(expr);

Save command output and input to file:

      with_stdout("filename", commands);  writes output of commands to file
            file_output_append: true;     switch to append mode

      stringout("filename", expr1, ..);   write expressions in a form
                                                suitable for maxima input

Session transcript:

      writefile("filename");        session transcript in console output format
      appendfile("filename");       same, but append to file
      closefile();                  terminate session transcript

Control Structures

      for var: first step incr thru limit do body
      for var: first step incr while cond do body
      for var: first step incr unless cond do body

      if cond then body
      if cond then body1 else body2

      return(expr);         /* abnormal termination of the "for" loop */

step 1 can be omitted. 
body is a single command or a comma separated list of commands. 
for returns done upon normal termination. 
if returns the last result of the executed commands or false. 
The following example computes sqrt(10) to floating point precision:

      x: 1.;
      for n: 1 thru 10 do
            (x0: x, x: .5*(x + 10./x), if x = x0 then return(x));

In this example, the control variable is obsolete and can be omitted:

      x: 1.;
      do (x0: x, x: .5*(x + 10./x), if x = x0 then return(x));

Scripts

      batch("filename");            run Maxima script 
      /* .. */                      comments in script

      unix> maxima -b filename      run in terminal (from the command line)

Startup scripts

      $HOME/.maxima/maximarc

      maxima_userdir                directory for startup file `maxima-init.mac'
                                    (Unix default: $HOME/.maxima)      

Checking the installation

      build_info();
      run_testsuite();
      run_testsuite(true);          show bugs only

wxMaxima

      ->Edit ->Configure
         ->Options
            Language: (select)
         ->Style
            [x] Use greek font

displays variables starting with % as (small or capital) greek letters, e.g. %omega and %Omega. 
Beware: %pi and %phi are predefined.

Math display modes:

      set_display('xml);      nice format (default)
      set_display('ascii);    multi-line mode, like maxima with display2d:true
      set_display('none);     one-line mode, like maxima with display2d:false

May be set via ->Maxima ->Change 2d display. 
In xml mode, long numbers are displayed with a shortcut.

wxMaxima-0.8.x notes:

In 0.8.0/0.8.1

• the input line was removed,
• commands are terminated with <SHIFT>-<ENTER>, a la Mathematica.

The latter is made configurable in 0.8.2:

  [x] Enter evaluates cells

In wxMaxima-0.8.5, %alpha etc are represented as greek letters by default (no need to configure).

Documentation

Tutorials

Antonio Cangiano, A 10 minute tutorial for solving Math problems with Maxima 
Boris Gaertner, The Computer Algebra Program Maxima - a Tutorial 
Moses Glasner, The Maxima Primer 
Richard H. Rand, Introduction to MACSYMA 
wxMaxima tutorials  [in *.wxm.zip format]

Advanced

Maxima Beginner's FAQ [with many tips for advanced usage, in fact] 
Paulo Ney de Souza et al., The Maxima Book (19-Sept-2004) 
Moses Glasner, A Maxima Guide for Calculus Students 
Robert Dodier, Minimal Maxima 
Edwin L. Woollett, Maxima by Example 
P. Lutus, Symbolic Mathematics Using Maxima 
Maxima, Maple, and Mathematica: the Summary 
Gilberto E. Urroz, Maxima Book

Plotting

Examples of the Maxima Gnuplot interface 
A Maxima-Gnuplot interface

Manuals

Maxima Manual  [5.28.0] 
Michael Clarkson, DOE-Maxima Reference Manual  [Maxima manual v5.9 reformatted]

Help

Maxima mailing list and archives 
Maxima user interface tips

In German

Johann Weilharter, Mathematik mit Computerunterstützung: CAS Maxima  [Sammlung von Unterrichtsmaterial] 
Walter Wegscheider, Maxima 5.xx unter der Oberfl?che wxMaxima 
Robert Gl?ckner, Einführung in Maxima 
H.-G. Gr?be, Skript zu "Einführung in das symbolische Rechnen"  [not using Maxima]

More links

see maxima.sourceforge.net

Appendices

Experimental solver to_poly_solve()

by Barton Willis, 2006ff

provided e.g. with Maxima-5.15.0 + 5.17.1: 
files: 
topoly.lisp (engine) 
topoly_solver.mac (wrapper).

They also work with Maxima-5.10.

      load(topoly_solver);
            /home/b/maxima/5.15.0/topoly_solver.mac
      to_poly_solve(s + sqrt(1-s^2) = 1/2, s);
                    sqrt(7) - 1
            [[s = - -----------]]
                         4
      to_poly_solve([sqrt(x) + y = 0, sqrt(x) = 1], [x, y]);
            [[x = 1, y = - 1]]
      to_poly_solve(sin(x) + cos(x) = 1/2, x);
            Nonalgebraic argument given to 'topoly'

In Maxima-5.18.1, there is a new version (which does not work with Maxima-5.10?) - it solves the last example as well:

      load(to_poly_solver);
      to_poly_solve(sin(x) + cos(x) = 1/2, x);
            %union{TO DO}

In Maxima-5.21.1, the to_poly_solver package is auto-loaded as needed.

More special functions

There is work in progress to implement the Exponential Integral and the Incomplete Gamma Function, see mailing list.

GCL

GCL (GNU Common Lisp) prompt and commands:

      MAXIMA>>(help)          show GCL help
      MAXIMA>>(run)           restart Maxima session
      MAXIMA>>(bye)           exit
      MAXIMA>>(by)            exit

TO DO

%union(..) 
wxMaxima cells

源地址： http://www-com.physik.hu-berlin.de/~bunk/kurs/maxima/maxima.html