Maxima

This book shows how to use Maxima. Maxima can be used to :

• both symbolic and numerical computations
• plotting 2D and 3D

Weaknesses of CAS/math software

• the limit of a discontinuous function at the point of discontinuity
• branch cuts
• zero-recognition
• keeping track of domains of validity

How to use ?

Ways to do it :

• online , without installation
• offline = after installation

How to install ?

Dependencies

• gnuplot,
• tex
  • texinfo
  • texi2html
  • texlive-latex-recommended
• GNU autotools
  • automake 1.9
  • autoconf2.13
• tk 8.4.
• lisp
  • sbcl ( sudo apt-get install sbcl)
• libgmp3-dev
• libreadline5-dev
• libncurses5-dev
• sharutils
• libxmu-dev
• libxaw7-dev
• wxWidgets library for wxMaxima
  • libwxgtk3.0-dev
  • libwxbase3.0-dev
• compiler is GNU g++ for wxWidgets library
• build-essential
• imagemagick
• ibus-gtk
• ibus-gtk3

Steps

• Download and install ready made Maxima deb packages
  • from Ubuntu repositories ( old versions )
  • from other sites
• Compile your own maxima package
• compile maxima from sources ( without creating deb packages )

Now the best is a git method

From Ubuntu repositories

You can make use of :

• Synaptic ( Graphical package manager)
• apt-get.

In second case write in command line:

sudo apt-get install maxima

This method can get old version which gets some trouble, like

 loadfile: failed to load /usr/share/maxima/5.37.2/share/draw/draw.lisp

Uninstall such Maxima version

 sudo apt-get remove maxima

From PPA

wxmaxima

From repositiories by Istvan Balhota

sudo add-apt-repository ppa:blahota/wxmaxima
sudo apt-get update
sudo apt-get install wxmaxima

From sources

Before compilation, you should have all dependent packages

CVS

cd maxima
./bootstrap    # only if you downloaded the source code from cvs
./configure --enable-clisp   # or:   --enable-sbcl
make
make check
sudo make install
make clean

This procedure does not create a deb package.

Using git

maxima/xmaxima

If you have git and SBCL then :

  git clone git://git.code.sf.net/p/maxima/code maxima
  cd maxima
  ./bootstrap
  ./configure --with-sbcl
  make

Now one can run it from inside the source directory.

cd maxima
./maxima-local

The result :

Maxima branch_5_30_base_104_ged70fb2_dirty http://maxima.sourceforge.net
using Lisp SBCL 1.0.45.0.debian
Distributed under the GNU Public License. See the file COPYING.
Dedicated to the memory of William Schelter.
The function bug_report() provides bug reporting information.
(%i1) 

Xmaxima can also be run :

cd ~/maxima
./xmaxima-local

or bash file :

#!/bin/bash

# save as a  in your home directory
# chmod +x m
# ./m

cd maxima
./xmaxima-local

wxMaxima is not present now.

Where is Maxima after standard ( as above) installation ?

  locate maxima-local

example result :

 ~/Pobrane/maxima-5.38.1/maxima-local.in
 ~/Pobrane/maxima-5.38.1/xmaxima-local.in
 ~/maxima/maxima-local
 ~/maxima/maxima-local.in
 ~/maxima/xmaxima-local
 ~/maxima/xmaxima-local.in
 ~/maxima/interfaces/xmaxima/Tkmaxima/maxima-local.tcl

Note that it is a different directory then after apt-get installation. One can have 2 ( or more ) versions of Maxima on the disk )

wxmaxima

 git clone https://github.com/wxMaxima-developers/wxmaxima

Problems

• Open a terminal
• type : sudo apt-get install gdb
• after gdb is installed type : gdb wxmaxima
• run and wait until it crashes
• and type : backtrace

Symbols

• symbols %z7 and %z9 represent any arbitrary integer value.

Operators

"=" means syntactic equality. Check : is(equal(a,b))

Names

  "One quick way to check on name conflicts is to keep the html Maxima help manual up in a separate window, and have the Index panel available to type in a name you want to use. 
   The painful way to check on name conflicts is to wait for Maxima to give you a strange error message as to why what you are trying to do won’t work. If you get strange results, try changing the names of your variables and or functions. An important fact is that Maxima is case sensitive, so  you can avoid name conflicts by capitalizing the names of your user defined Maxima functions. Your Solve will not conflict with Maxima’s solve." Edwin L. Woollett

Objects

• atom
  • number
  • name
  • string
• list of atoms

Variables

• global
  • global variables category
• local

Data types

• Numbers
• Strings
• Constants
• Lists
• Arrays
• Structures

Features

Maxima recognizes certain mathematical properties of:

• functions and variables
• system. There is a list of non-mathematical, system-dependent features. See status.

which called features.

 declare (x, foo) /* gives the property foo to the function or variable x */

 declare (foo, feature) /* declares a new feature foo. */ 

 declare ([red, green, blue], feature) /* declares three new features, red, green, and blue. */

The predicate

 featurep (x, foo) 

returns true if x has the foo property, and false otherwise.

The infolist features is a list of known features. These are

   integer        noninteger      even
   odd            rational        irrational
   real           imaginary       complex
   analytic       increasing      decreasing
   oddfun         evenfun         posfun
   constant       commutative     lassociative
   rassociative   symmetric       antisymmetric
   integervalued

plus any user-defined features.

features is a list of mathematical features.

Example:

(%i1) declare (FOO, feature);
(%o1)                         done
(%i2) declare (x, FOO);
(%o2)                         done
(%i3) featurep (x, FOO);
(%o3)                         true

Declare

declare recognizes the following properties:

• additive : Tells Maxima to simplify a_i expressions by the substitution a_i(x + y + z + ...) --> a_i(x) + a_i(y) + a_i(z) + .... The substitution is carried out on the first argument only.
• alphabetic : Tells Maxima to recognize all characters in a_i (which must be a string) as alphabetic characters.
• antisymmetric, commutative, symmetric : Tells Maxima to recognize a_i as a symmetric or antisymmetric function. commutative is the same as symmetric.
• bindtest : Tells Maxima to trigger an error when a_i is evaluated unbound.
• constant : Tells Maxima to consider a_i a symbolic constant.
• even, odd : Tells Maxima to recognize a_i as an even or odd integer variable.
• evenfun, oddfun : Tells Maxima to recognize a_i as an odd or even function.
• evflag : Makes a_i known to the ev function so that a_i is bound to true during the execution of ev when a_i appears as a flag argument of ev. See evflag.
• evfun : Makes a_i known to ev so that the function named by a_i is applied when a_i appears as a flag argument of ev. See evfun.
• feature : Tells Maxima to recognize a_i as the name of a feature. Other atoms may then be declared to have the a_i property.
• increasing, decreasing : Tells Maxima to recognize a_i as an increasing or decreasing function.
• integer, noninteger : Tells Maxima to recognize a_i as an integer or noninteger variable.
• integervalued : Tells Maxima to recognize a_i as an integer-valued function.
• lassociative, rassociative : Tells Maxima to recognize a_i as a right-associative or left-associative function.
• linear : Equivalent to declaring a_i both outative and additive.
• mainvar : Tells Maxima to consider a_i a "main variable". A main variable succeeds all other constants and variables in the canonical ordering of Maxima expressions, as determined by ordergreatp.
• multiplicative : Tells Maxima to simplify a_i expressions by the substitution a_i(x * y * z * ...) --> a_i(x) * a_i(y) * a_i(z) * .... The substitution is carried out on the first argument only.
• nary : Tells Maxima to recognize a_i as an n-ary function. The nary declaration is not the same as calling the nary function. The sole effect of declare(foo, nary) is to instruct the Maxima simplifier to flatten nested expressions, for example, to simplify foo(x, foo(y, z)) to foo(x, y, z).
• nonarray : Tells Maxima to consider a_i not an array. This declaration prevents multiple evaluation of a subscripted variable name.
• nonscalar : Tells Maxima to consider a_i a nonscalar variable. The usual application is to declare a variable as a symbolic vector or matrix.
• noun : Tells Maxima to parse a_i as a noun. The effect of this is to replace instances of a_i with 'a_i or nounify(a_i), depending on the context.
• outative : Tells Maxima to simplify a_i expressions by pulling constant factors out of the first argument. When a_i has one argument, a factor is considered constant if it is a literal or declared constant. When a_i has two or more arguments, a factor is considered constant if the second argument is a symbol and the factor is free of the second argument.
• posfun : Tells Maxima to recognize a_i as a positive function.
• rational, irrational : Tells Maxima to recognize a_i as a rational or irrational real variable.
• real, imaginary, complex : Tells Maxima to recognize a_i as a real, pure imaginary, or complex variable.
• scalar : Tells Maxima to consider a_i a scalar variable.

Declare tells:

• evenp()
• oddp()
• integerp()
• and so on

what to return.

COmpare with:

• mode_decalre

Numbers

Check

(%i1) load(to_poly);
(%o1)  home/a/maxima/share/to_poly_solve/to_poly.lisp
(%i2) r;
(%o2) r 
(%i3) isreal_p(r);
(%o3) true     /*  When I use unspecified variables, Maxima seems to assume that they are real */ 
(%i4) z:x+y*%i;
(%o4) %i y + x
(%i5) isreal_p(z);
(%o5) isreal_p(%i y) /* maxima can't check if it is real or not */
(%i6) isreal_p(x);
(%o6) true
(%i7) isreal_p(y);
(%o7) true
(%i8) complex_number_p(z); 
(%o8) false 
(%i9) declare(z, complex);
(%o9) done
(%i10) complex_number_p(z);
(%o10)   false /* still not complex */                                                                                                              true

Random numbers

  bfloat(random(10^fpprec) / 10^fpprec); /* random bfloat with fpprec decimal digits in the range 0 to 1 */

Number types

Binary numbers

(%i1) ibase;
(%o1) 10
(%i2) obase;
(%o2) 10
(%i3) ibase:2;
(%o3) 2
(%i4) x=1001110;
(%o4) x=78

Complex numbers

Argument

Principial value of argument of complex number in turns carg produces results in the range (-pi, pi] . It can be mapped to [0, 2*pi) by adding 2*pi to the negative values

carg_t(z):=
 block(
 [t],
 t:carg(z)/(2*%pi),  /* now in turns */
 if t<0 then t:t+1, /* map from (-1/2,1/2] to [0, 1) */
 return(t)
)$

On can order list of complex points according to it's argument :

l2_ordered: sort(l2, lambda([z1,z2], is(carg(z1) < carg(z2))))$

Predicate functions

(%i1) is(0=0.0);
(%o1) false

Compare :

(%i1) a:0.0$
(%i2)is(equal(a,0));
(%o2) true

List of predicate functions ( see p at the end ) :

• abasep
• alphacharp
• alphanumericp
• atom
• bfloatp
• blockmatrixp
• cequal
• cequalignore
• cgreaterp
• cgreaterpignore
• charp
• clessp
• clesspignore
• complex_number_p from to_poly package
• constantp
• constituent
• diagmatrixp
• digitcharp
• disjointp
• elementp
• emptyp
• evenp
• featurep
• floatnump ( compare : isreal_p)
• if
• integerp
• intervalp
• is
• isreal_p from to_poly package
• lcharp
• listp
• listp
• lowercasep
• mapatom
• matrixp
• matrixp
• maybe
• member
• nonnegintegerp
• nonscalarp
• numberp
• oddp
• operatorp
• ordergreatp
• orderlessp
• picture_equalp
• picturep
• poly_depends_p
• poly_grobner_subsetp
• polynomialp
• prederror
• primep
• ratnump
• ratp
• scalarp
• sequal
• sequalignore
• setequalp
• setp
• stringp
• subsetp
• subvarp
• symbolp
• symmetricp
• taylorp
• unknown
• uppercasep
• zeroequiv
• zeromatrixp
• zn_primroot_p

See also :

• declare
  • property:
    • rational, irrational, real, imaginary, complex,
    • even, odd,
    • decreasing, increasing
    • evenfun, oddfun

Number Theory

There are functions and operators useful with integer numbers

Elementary number theory

In Maxima there are some elementary functions like the factorial n! and the double factorial n!! defined as ${\frac {n!}{k!}}$ where $k$ is the greatest integer less than or equal to $n/2$

Divisibility

Some of the most important functions for integer numbers have to do with divisibility:

gcd, ifactor, mod, divisors... 

all of them well documented in help. you can view it with the '?' command.

Function ifactors takes positive integer and returns a list of pairs : prime factor and its exponent. For example :

a:ifactors(8);
[[2,3]]

It means that : $8=2^{3}\,$

Other Functions

Continuus fractions :

(%i6) cfdisrep([1,1,1,1]);
(%o6) 1+1/(1+1/(1+1/1))
(%i7) float(%), numer;
(%o7) 1.666666666666667

Functions

Definition

For mathematical functions use always

define() 

instead of

:= 

There is only one small difference between them:

"The function definition operator. f(x_1, ..., x_n) := expr defines a function named f with arguments x_1, …, x_n and function body expr. := never evaluates the function body (unless explicitly evaluated by quote-quote )."

"Defines a function named f with arguments x_1, …, x_n and function body expr. define always evaluates its second argument (unless explicitly quoted). "

return

"return doesn't mean the same thing in Maxima as it does in other programming languages. It means only to return a value from a block; that's not enough for what you need here." Robert Dodier

If you wish to make a return from some expression inside the function then you must use block and return.

   block ([], expr1, ..., if (a > 10) then return(a), ..., exprn)

is itself an expression, and so could take the place of the right hand side of a function definition. Here it may happen that the return happens earlier than the last expression.

Local variables

  block ([x], expr1, ..., exprn)

The first [] in the block, may contain a list of variables and variable assignments, such as [a: 3, b, c: []], which would cause the three variables a,b,and c to not refer to their global values, but rather have these special values for as long as the code executes inside the block, or inside functions called from inside the block. This is called dynamic binding, since the variables last from the start of the block to the time it exits. Once you return from the block, or throw out of it, the old values (if any) of the variables will be restored. It is certainly a good idea to protect your variables in this way. Note that the assignments in the block variables, are done in parallel. This means, that if you had used c: a in the above, the value of c would have been the value of a at the time you just entered the block, but before a was bound. Thus doing something like:

   block ([a: a], expr1, ... a: a+3, ..., exprn)

will protect the external value of a from being altered, but would let you access what that value was. Thus the right hand side of the assignments, is evaluated in the entering context, before any binding occurs. Using just block ([x], ...) would cause the x to have itself as value, just as if it would have if you entered a fresh Maxima session.

See also description by Darren Irvine

compile

• "The modedeclare statement is important for compiled code
• Compiling can greatly speed numerical code; for mostly symbolic code, the speed up is not great.
• There are some differences between compiled and uncompiled code -- be careful." Barton Willis

mode_declare

"mode_declare tells the compiler that the compiled code needs not to be generic enough to be able to deal with all kind of objects. If your function only takes floats, but no symbolic arguments the compiled code might run much faster." Gunter Königsmann

Debugging

Directories

Start Maxima and type the command:

   maxima_userdir;

this will tell you the directory that is being used as your user directory.

infolists

• is a system variable
• Default value: []
• is a list of the names of all of the information lists in Maxima. These are
  • labels : All bound %i, %o, and %t labels.
  • values : All bound atoms which are user variables, not Maxima options or switches, created by : or :: or functional binding.
  • functions : All user-defined functions, created by := or define.
  • arrays : All declared and undeclared arrays, created by :, ::, or :=.
  • macros : All user-defined macro functions, created by ::=.
  • myoptions : All options ever reset by the user (whether or not they are later reset to their default values).
  • rules : All user-defined pattern matching and simplification rules, created by tellsimp, tellsimpafter, defmatch, or defrule.
  • aliases : All atoms which have a user-defined alias, created by the alias, ordergreat, orderless functions or by declaring the atom as a noun with declare.
  • dependencies : All atoms which have functional dependencies, created by the depends, dependencies, or gradef functions.
  • gradefs : All functions which have user-defined derivatives, created by the gradef function.
  • props : All atoms which have any property other than those mentioned above, such as properties established by atvalue or matchdeclare, etc., as well as properties established in the declare function.
  • let_rule_packages : All user-defined let rule packages plus the special package default_let_rule_package. (default_let_rule_package is the name of the rule package used when one is not explicitly set by the user.)

reset

reset : resets many (global) system variables

Use:

 reset()

example output :

 (%o1) [_, __, load_pathname, labels, %, linenum, ratprint, tr-unique, 
                                                     lispdisp, multiplicities]

trace

In Maxima :

:lisp (setf *debugger-hook* nil)

manual debuging

print partial reults using :

• print
• playback

values

values:

• is a system variable
• is a list ... which contains the names of currently assigned scalar values which have been set by:
  • the user interactively
  • by packages which have been loaded"
• initial value is []

dependencies

dependencies :

• is a system variable
• is the list of atoms which have functional dependencies, assigned by:
  • depends
  • the function dependencies
  • gradef

status

 status (feature);

example output:

(%o34) [sb-bsd-sockets-addrinfo, cl, mk-defsystem, cltl2, 64-bit, 
64-bit-registers, alien-callbacks, ansi-cl, ash-right-vops, 
c-stack-is-control-stack, common-lisp, compare-and-swap-vops, 
complex-float-vops, cycle-counter, elf, float-eql-vops, 
fp-and-pc-standard-save, gencgc, ieee-floating-point, inline-constants, 
integer-eql-vop, interleaved-raw-slots, largefile, linkage-table, linux, 
little-endian, memory-barrier-vops, multiply-high-vops, os-provides-blksize-t, 
os-provides-dladdr, os-provides-dlopen, os-provides-getprotoby-r, 
os-provides-poll, os-provides-putwc, os-provides-suseconds-t, 
package-local-nicknames, precise-arg-count-error, raw-instance-init-vops, 
sb-after-xc-core, sb-core-compression, sb-doc, sb-eval, sb-futex, sb-ldb, 
sb-package-locks, sb-simd-pack, sb-source-locations, sb-test, sb-thread, 
sb-unicode, sb-xref-for-internals, sbcl, stack-allocatable-closures, 
stack-allocatable-fixed-objects, stack-allocatable-lists, 
stack-allocatable-vectors, stack-grows-downward-not-upward, symbol-info-vops, 
unix, unwind-to-frame-and-call-vop, x86-64]

Data structures

"Maxima's current array/matrix semantics are a mess, I must say, with at least four different kinds of object (hasharrays, explicit lists, explicit matrices, Lisp arrays) supporting subscripting with varying semantics. "

"Maxima's concept of arrays, lists, and matrices is pretty confused, since various ideas have accreted in the many years of the project... Yes, this is a mess. Sorry about that. These are all interesting ideas, but there is no unifying framework. "

Array

Maxima has two array types :

• Undeclared arrays ( implemented as Lisp hash tables )
• Declared arrays ( implemented as Lisp arrays )
  • created by the array function
  • created by the make_array function.

Category: Arrays :

• array
• arrayapply
• arrayinfo
• arraymake
• arrays - global variable
• fillarray
• listarray
• make_array
• rearray
• remarray
• subvar
• use_fast_arrays

See also :

• array functions defined by := and define.

Undeclared array

Assignment creates an undeclared array.

(%i1) c[99] : 789;
(%o1)                          789
(%i2) c[99];
(%o2)                          789
(%i3) c;
(%o3)                           c
(%i4) arrayinfo (c);
(%o4)                   [hashed, 1, [99]]
(%i5) listarray (c);
(%o5)                         [789]

"If the user assigns to a subscripted variable before declaring the corresponding array, an undeclared array is created. Undeclared arrays, otherwise known as hashed arrays (because hash coding is done on the subscripts), are more general than declared arrays. The user does not declare their maximum size, and they grow dynamically by hashing as more elements are assigned values. The subscripts of undeclared arrays need not even be numbers. However, unless an array is rather sparse, it is probably more efficient to declare it when possible than to leave it undeclared. " ( from Maxima CAS doc)

"created if one does :

b[x+1]:y^2 

(and b is not already an array, a list, or a matrix - if it were one of these an error would be caused since x+1 would not be a valid subscript for an art-q array, a list or a matrix).

Its indices (also known as keys) may be any object. It only takes one key at a time (b[x+1,u]:y would ignore the u). Referencing is done by b[x+1] ==> y^2. Of course the key may be a list, e.g.

 b[[x+1,u]]:y 

would be valid. "( from Maxima CAS doc)

Declared array

• declare array ( give a name and allocate memmory)
• fill array
• process array

array function

Function: array (name, type, dim_1, …, dim_n)

Creates an n-dimensional array. Here :

• type can be fixnum for integers of limited size or flonum for floating-point numbers
• n may be less than or equal to 5. The subscripts for the i'th dimension are the integers running from 0 to dim_i.

(%i5) array(A,2,2);
(%o5)                                  A
(%i8) arrayinfo(A);
(%o8)                      [declared, 2, [2,2]]

Because subscripts of array A elements are goes from 0 to dim so array A has :

( dim_1 + 1) * (dim_2 + 1) = 3*3 = 9 

elements.

The array function can be used to transform an undeclared array into a declared array.

arraymake function

Function: arraymake (A, [i_1, …, i_n])

The result is an unevaluated array reference.

(%i12) arraymake(A,[3,3,3]);
(%o12)                    array_make(3, 3, 3)
                                           3, 3, 3
(%i13) arrayinfo(A);
       arrayinfo: A is not an array.

Function: make_array

Function: make_array (type, dim_1, …, dim_n)

Creates and returns a Lisp array.

Type may be :

• any
• flonum
• fixnum
• hashed
• functional.

There are n indices, and the i'th index runs from 0 to dim_i - 1.

The advantage of make_array over array is that the return value doesn't have a name, and once a pointer to it goes away, it will also go away. For example, if y: make_array (...) then y points to an object which takes up space, but after y: false, y no longer points to that object, so the object can be garbage collected.

Examples:

(%i9) A : array_make(3,3,3);
(%o9)                         array_make(3, 3, 3)
(%i10) arrayinfo(A);
(%o10)                     [declared, 3, [3, 3, 3]]

List

Assignment to an element of a list.

(%i1) b : [1, 2, 3];
(%o1)                       [1, 2, 3]
(%i2) b[3] : 456;
(%o2)                          456
(%i3) b;
(%o3)                      [1, 2, 456]

(%i1) mylist : [[a,b],[c,d]];
(%o1) [[a, b], [c, d]]

Intresting examples by Burkhard Bunk :

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

Category: Lists :

• [
• ]
• append
• assoc
• cons
• copylist
• create_list
• delete
• eighth
• endcons
• fifth
• first
• flatten
• fourth
• fullsetify
• join
• last
• length
• listarith
• listp
• lmax
• lmin
• lreduce
• makelist
• member
• ninth
• permut
• permutations
• pop
• push
• random_permutation
• rest
• reverse
• rreduce
• second
• setify
• seventh
• sixth
• some
• sort
• sublist
• sublist_indices
• tenth
• third
• tree_reduce
• xreduce

product

/* example by Burton Willis */
(%i28) P1(s) := rectform(product(s[i],i, 1, length(s)))$

(%i29) P2(s) := xreduce(lambda([a,b], rectform(a*b)),s)$

(%i30) s : makelist(random(1.0)+%i*random(1.0),10^4)$

(%i31) P1(s);
Evaluation took 32.3080 seconds (32.3860 elapsed) using 17944.301 MB.
(%o31) 0.0

(%i32) P2(s);
Evaluation took 0.0620 seconds (0.0620 elapsed) using 8.343 MB.
(%o32) 0.0

Matrix

"In Maxima, a matrix is implemented as a nested list (namely a list of rows of the matrix)". E.g. :

(($MATRIX) ((MLIST) 1 2 3) ((MLIST) 4 5 6))

Construction of matrices from nested lists :

(%i3) M:matrix([1,2,3],[4,5,6],[0,-1,-2]);
                                [ 1   2    3  ]
                                [             ]
(%o3)                           [ 4   5    6  ]
                                [             ]
                                [ 0  - 1  - 2 ]

Intresting examples by Burkhard Bunk :

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);

Category: Matrices :

• addcol
• addrow
• adjoint
• augcoefmatrix
• cauchy_matrix
• charpoly
• coefmatrix
• col
• columnvector
• covect
• copymatrix
• determinant
• detout
• diag
• diagmatrix
• doallmxops
• domxexpt
• domxmxops
• domxnctimes
• doscmxops
• doscmxplus
• echelon
• eigen
• ematrix
• entermatrix
• genmatrix
• ident
• invert
• list_matrix_entries
• lmxchar
• matrix
• matrix_element_add
• matrix_element_mult
• matrix_element_transpose
• matrixmap
• matrixp
• mattrace
• minor
• ncharpoly
• newdet
• nonscalar
• nonscalarp
• permanent
• rank
• ratmx
• row
• scalarmatrixp
• scalarp
• setelmx
• sparse
• submatrix
• tracematrix
• transpose
• triangularize
• zeromatrix

Function: matrix

Function: matrix (row_1, …, row_n)

Returns a rectangular matrix which has the rows row_1, …, row_n. Each row is a list of expressions. All rows must be the same length.

. operator

"The dot operator, for matrix (non-commutative) multiplication. When "." is used in this way, spaces should be left on both sides of it, e.g.

A . B

This distinguishes it plainly from a decimal point in a floating point number.

The operator . represents noncommutative multiplication and scalar product. When the operands are 1-column or 1-row matrices a and b, the expression a.b is equivalent to sum (a[i]*b[i], i, 1, length(a)). If a and b are not complex, this is the scalar product, also called the inner product or dot product, of a and b. The scalar product is defined as conjugate(a).b when a and b are complex; innerproduct in the eigen package provides the complex scalar product.

When the operands are more general matrices, the product is the matrix product a and b. The number of rows of b must equal the number of columns of a, and the result has number of rows equal to the number of rows of a and number of columns equal to the number of columns of b.

To distinguish . as an arithmetic operator from the decimal point in a floating point number, it may be necessary to leave spaces on either side. For example, 5.e3 is 5000.0 but 5 . e3 is 5 times e3.

There are several flags which govern the simplification of expressions involving ., namely dot0nscsimp, dot0simp, dot1simp, dotassoc, dotconstrules, dotdistrib, dotexptsimp, dotident, and dotscrules." ( from Maxima CAS doc)

Vectors

Category: Vectors :

• Vectors
• eigen
• express
• nonscalar
• nonscalarp
• scalarp

Category: Package vect

• vect
• scalefactors
• vect_cross
• vectorpotential
• vectorsimp

Numerical methods

 "If you are doing purely numerical computation and are concerned about speed, use a compiled numerical programming language. Maxima is intended for use if you have symbolic mathematical symbols, too,
 and while it works for numbers, most components of the system are on the lookout for non-numeric inputs, and this checking takes time. 
 It is possible to speed up certain kinds of numeric computations in Maxima by using compile() and mode_declare()  together. " RJF

Newton method

Functions

• newton for equation with function of one variable
• mnewton is an implementation of Newton's method for solving nonlinear equations in one or more variables.

newton

load :

load("newton");
(%o1)                                                                                                   /home/a/maxima/share/numeric/newton.mac
(%i2) load("newton1");
(%o2)                                                                                                   /home/a/maxima/share/numeric/newton1.mac

Code :

/* 
 
Maxima CAS code 
from /maxima/share/numeric/newton1.mac 
input : 
 exp  =  function of one variable, x
 var = variable 
 x0 = initial value of variable
 eps = 

The search begins with x = x_0 and proceeds until abs(expr) < eps (with expr evaluated at the current value of x). 

output : xn = an approximate solution of expr = 0 by Newton's method
*/ 


newton(exp,var,x0,eps):=
block(
  [xn,s,numer],
   numer:true,
   s:diff(exp,var),
   xn:x0,
   
   loop, 
    if abs(subst(xn,var,exp))<eps 
         then return(xn),
    xn:xn-subst(xn,var,exp)/subst(xn,var,s),
   go(loop) 
)$

mnewton

Boundaries of Components of Mandelbrot set by Newton method. Image and Maxima CAS code

One can use Newton method for solving systems of multiple nonlinear functions. It is implemented in mnewton function. See directory :

/Maxima..../share/contrib/mnewton.mac

which uses linsolve_by_lu defined in :

 /share/linearalgebra/lu.lisp

See this image for more code.

(%i1) load("mnewton")$

(%i2) mnewton([x1+3*log(x1)-x2^2, 2*x1^2-x1*x2-5*x1+1],
              [x1, x2], [5, 5]);
(%o2) [[x1 = 3.756834008012769, x2 = 2.779849592817897]]
(%i3) mnewton([2*a^a-5],[a],[1]);
(%o3)             [[a = 1.70927556786144]]
(%i4) mnewton([2*3^u-v/u-5, u+2^v-4], [u, v], [2, 2]);
(%o4) [[u = 1.066618389595407, v = 1.552564766841786]]

Algorithms

Stack ( LIFO)

Stack implementation using list:

/* create stack */
stack:[1];
/* push on stack */
stack:endcons(2,stack);
stack:endcons(3,stack);
block
(
  loop,
  stack:delete(last(stack),stack), /* pop from stack */
  disp(stack), /* display */
  if is(not emptyp(stack)) then go(loop)
);
stack;

Vectors made with Maxima CAS

Plots

Complex Functions

• Visualizing Complex Functions with Conformal Mapping ( themaximalist )

Topologist's sine curve

plot2d (sin(1/x), [x, 0, 1])$

Topologist's comb :

Topologist's comb

load(draw); /* by Mario Rodríguez Riotorto  http://riotorto.users.sourceforge.net/gnuplot/index.html  */

draw2d(
 title     = "Topologist\\47s comb", /* Syntax for postscript enhanced option; character 47 = '  */
 xrange        = [0.0,1.2],
 yrange        = [0,1.2],
 file_name = "comb2",
 terminal      = svg,
 points_joined = impulses, /* vertical segments are drawn from points to the x-axis  */
 color         = "blue",
 points(makelist(1/x,x,1,1000),makelist(1,y,1,1000)) )$

Output, warnings

• ratprint : false; /* remove "rat :replaced " */
• using $ instead of ; at the end of the line ( not inside function where , ends the line )
• return from the function
• "warning: encountered undefined variable ... in translation." This is warning from compile when function uses global variable without using in in tha arguments list. You have to include the variables in the function's list

Example programs

• Iteration of complex numbers, stack and drawing a list of points

Julia set using IIM

• Iteration of complex numbers, comparing complex numbers, finding period of cycle

critical orbit with 3-period cycle

• Drawing Julia set and critical orbit. Computing period

Drawing Julia set and critical orbit. Computing period

• drawing curves in 2D plane

Mandelbrot lemniscates-image and source

• drawing points

Drawing 2D points. Image and source code

help

Commands

• dependencies
• infolists
• describe

See also

• Images with Maxima CAS source code
• syntax highlight for Gedit
• questions tagged Maxima on stackoverflow

References

Chaotic dynamics with Maxima A. Morante and J. A. Vallejo