# Maxima

This book shows how to use Maxima.

# Directories

Start Maxima and type the command:[1]

   maxima_userdir;


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

# Numbers

## 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

↑Jump back a section

## 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))))$


# Functions

For mathematical functions use always

define()


:=


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). "[2]

# 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. " [3]

↑Jump back a section

## Array

Maxima has two array types :[4]

• 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

• 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]]

↑Jump back a section

## 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 : [5]

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
↑Jump back a section

## Matrix

"In Maxima, a matrix is implemented as a nested list (namely a list of rows of the matrix)". E.g. :[6]

(($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 : [7] 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 :[8] • 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) ↑Jump back a section ## Vectors Category: Vectors : • Vectors • eigen • express • nonscalar • nonscalarp • scalarp Category: Package vect[9] • vect • scalefactors • vect_cross • vectorpotential • vectorsimp # Algorithms ↑Jump back a section ## 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;  # Plots 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)) )\$



# 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