Pexpect interface to Maxima#

Maxima is a free GPL’d general purpose computer algebra system whose development started in 1968 at MIT. It contains symbolic manipulation algorithms, as well as implementations of special functions, including elliptic functions and generalized hypergeometric functions. Moreover, Maxima has implementations of many functions relating to the invariant theory of the symmetric group \(S_n\). (However, the commands for group invariants, and the corresponding Maxima documentation, are in French.) For many links to Maxima documentation see http://maxima.sourceforge.net/documentation.html.

AUTHORS:

William Stein (2005-12): Initial version
David Joyner: Improved documentation
William Stein (2006-01-08): Fixed bug in parsing
William Stein (2006-02-22): comparisons (following suggestion of David Joyner)
William Stein (2006-02-24): greatly improved robustness by adding sequence numbers to IO bracketing in _eval_line
Robert Bradshaw, Nils Bruin, Jean-Pierre Flori (2010,2011): Binary library interface

This is the interface used by the maxima object:

Sage

sage: type(maxima)
<class 'sage.interfaces.maxima.Maxima'>

Python

>>> from sage.all import *
>>> type(maxima)
<class 'sage.interfaces.maxima.Maxima'>

If the string “error” (case insensitive) occurs in the output of anything from Maxima, a RuntimeError exception is raised.

EXAMPLES: We evaluate a very simple expression in Maxima.

Sage

sage: maxima('3 * 5')
15

Python

>>> from sage.all import *
>>> maxima('3 * 5')
15

We factor \(x^5 - y^5\) in Maxima in several different ways. The first way yields a Maxima object.

Sage

sage: x,y = SR.var('x,y')
sage: F = maxima.factor('x^5 - y^5')
sage: F # not tested - depends on maxima version
-((y-x)*(y^4+x*y^3+x^2*y^2+x^3*y+x^4))
sage: actual = F.sage()
sage: expected = -(y-x)*(y^4+x*y^3+x^2*y^2+x^3*y+x^4)
sage: bool(actual == expected)
True
sage: type(F)
<class 'sage.interfaces.maxima.MaximaElement'>

Python

>>> from sage.all import *
>>> x,y = SR.var('x,y')
>>> F = maxima.factor('x^5 - y^5')
>>> F # not tested - depends on maxima version
-((y-x)*(y^4+x*y^3+x^2*y^2+x^3*y+x^4))
>>> actual = F.sage()
>>> expected = -(y-x)*(y**Integer(4)+x*y**Integer(3)+x**Integer(2)*y**Integer(2)+x**Integer(3)*y+x**Integer(4))
>>> bool(actual == expected)
True
>>> type(F)
<class 'sage.interfaces.maxima.MaximaElement'>

Note that Maxima objects can also be displayed using “ASCII art”; to see a normal linear representation of any Maxima object x. Just use the print command: use str(x).

Sage

sage: print(F)
                           4      3    2  2    3      4
               - (y - x) (y  + x y  + x  y  + x  y + x )

Python

>>> from sage.all import *
>>> print(F)
                           4      3    2  2    3      4
               - (y - x) (y  + x y  + x  y  + x  y + x )

You can always use repr(x) to obtain the linear representation of an object. This can be useful for moving maxima data to other systems.

Sage

sage: F = maxima('x * y')
sage: repr(F)
'x*y'
sage: F.str()
'x*y'

Python

>>> from sage.all import *
>>> F = maxima('x * y')
>>> repr(F)
'x*y'
>>> F.str()
'x*y'

The maxima.eval command evaluates an expression in maxima and returns the result as a string not a maxima object.

Sage

sage: print(maxima.eval('factor(x^5 - 1)'))
(x-1)*(x^4+x^3+x^2+x+1)

Python

>>> from sage.all import *
>>> print(maxima.eval('factor(x^5 - 1)'))
(x-1)*(x^4+x^3+x^2+x+1)

We can create the polynomial \(f\) as a Maxima polynomial, then call the factor method on it. Notice that the notation f.factor() is consistent with how the rest of Sage works.

Sage

sage: f = maxima('x^5 + y^5')
sage: f^2
(y^5+x^5)^2
sage: f.factor()
(y+x)*(y^4-x*y^3+x^2*y^2-x^3*y+x^4)

Python

>>> from sage.all import *
>>> f = maxima('x^5 + y^5')
>>> f**Integer(2)
(y^5+x^5)^2
>>> f.factor()
(y+x)*(y^4-x*y^3+x^2*y^2-x^3*y+x^4)

Control-C interruption works well with the maxima interface, because of the excellent implementation of maxima. For example, try the following sum but with a much bigger range, and hit control-C.

Sage

sage: maxima('sum(1/x^2, x, 1, 10)')
1968329/1270080

Python

>>> from sage.all import *
>>> maxima('sum(1/x^2, x, 1, 10)')
1968329/1270080

Examples involving matrices#

We illustrate computing with the matrix whose \(i,j\) entry is \(i/j\), for \(i,j=1,\ldots,4\).

Sage

sage: f = maxima.eval('f[i,j] := i/j')
sage: A = maxima('genmatrix(f,4,4)'); A
matrix([1,1/2,1/3,1/4],[2,1,2/3,1/2],[3,3/2,1,3/4],[4,2,4/3,1])
sage: A.determinant()
0
sage: A.echelon()
matrix([1,1/2,1/3,1/4],[0,0,0,0],[0,0,0,0],[0,0,0,0])
sage: A.eigenvalues()
[[0,4],[3,1]]
sage: A.eigenvectors()
[[[0,4],[3,1]],[[[1,0,0,-4],[0,1,0,-2],[0,0,1,-...4/3...]],[[1,2,3,4]]]]

Python

>>> from sage.all import *
>>> f = maxima.eval('f[i,j] := i/j')
>>> A = maxima('genmatrix(f,4,4)'); A
matrix([1,1/2,1/3,1/4],[2,1,2/3,1/2],[3,3/2,1,3/4],[4,2,4/3,1])
>>> A.determinant()
0
>>> A.echelon()
matrix([1,1/2,1/3,1/4],[0,0,0,0],[0,0,0,0],[0,0,0,0])
>>> A.eigenvalues()
[[0,4],[3,1]]
>>> A.eigenvectors()
[[[0,4],[3,1]],[[[1,0,0,-4],[0,1,0,-2],[0,0,1,-...4/3...]],[[1,2,3,4]]]]

We can also compute the echelon form in Sage:

Sage

sage: B = matrix(QQ, A)
sage: B.echelon_form()
[  1 1/2 1/3 1/4]
[  0   0   0   0]
[  0   0   0   0]
[  0   0   0   0]
sage: B.charpoly('x').factor()
(x - 4) * x^3

Python

>>> from sage.all import *
>>> B = matrix(QQ, A)
>>> B.echelon_form()
[  1 1/2 1/3 1/4]
[  0   0   0   0]
[  0   0   0   0]
[  0   0   0   0]
>>> B.charpoly('x').factor()
(x - 4) * x^3

Continued Fractions#

A continued fraction \(a + 1/(b + 1/(c + \cdots))\) is represented in maxima by the list \([a, b, c, \ldots]\).

Sage

sage: maxima("cf((1 + sqrt(5))/2)")
[1,1,1,1,2]
sage: maxima("cf ((1 + sqrt(341))/2)")
[9,1,2,1,2,1,17,1,2,1,2,1,17,1,2,1,2,1,17,2]

Python

>>> from sage.all import *
>>> maxima("cf((1 + sqrt(5))/2)")
[1,1,1,1,2]
>>> maxima("cf ((1 + sqrt(341))/2)")
[9,1,2,1,2,1,17,1,2,1,2,1,17,1,2,1,2,1,17,2]

Special examples#

In this section we illustrate calculations that would be awkward to do (as far as I know) in non-symbolic computer algebra systems like MAGMA or GAP.

We compute the gcd of \(2x^{n+4} - x^{n+2}\) and \(4x^{n+1} + 3x^n\) for arbitrary \(n\).

Sage

sage: f = maxima('2*x^(n+4) - x^(n+2)')
sage: g = maxima('4*x^(n+1) + 3*x^n')
sage: f.gcd(g)
x^n

Python

>>> from sage.all import *
>>> f = maxima('2*x^(n+4) - x^(n+2)')
>>> g = maxima('4*x^(n+1) + 3*x^n')
>>> f.gcd(g)
x^n

You can plot 3d graphs (via gnuplot):

Sage

sage: maxima('plot3d(x^2-y^2, [x,-2,2], [y,-2,2], [grid,12,12])')  # not tested
[displays a 3 dimensional graph]

Python

>>> from sage.all import *
>>> maxima('plot3d(x^2-y^2, [x,-2,2], [y,-2,2], [grid,12,12])')  # not tested
[displays a 3 dimensional graph]

You can formally evaluate sums (note the nusum command):

Sage

sage: S = maxima('nusum(exp(1+2*i/n),i,1,n)')
sage: print(S)
                        2/n + 3                   2/n + 1
                      %e                        %e
               ----------------------- - -----------------------
                  1/n         1/n           1/n         1/n
               (%e    - 1) (%e    + 1)   (%e    - 1) (%e    + 1)

Python

>>> from sage.all import *
>>> S = maxima('nusum(exp(1+2*i/n),i,1,n)')
>>> print(S)
                        2/n + 3                   2/n + 1
                      %e                        %e
               ----------------------- - -----------------------
                  1/n         1/n           1/n         1/n
               (%e    - 1) (%e    + 1)   (%e    - 1) (%e    + 1)

We formally compute the limit as \(n\to\infty\) of \(2S/n\) as follows:

Sage

sage: T = S*maxima('2/n')
sage: T.tlimit('n','inf')
%e^3-%e

Python

>>> from sage.all import *
>>> T = S*maxima('2/n')
>>> T.tlimit('n','inf')
%e^3-%e

Miscellaneous#

Obtaining digits of \(\pi\):

Sage

sage: maxima.eval('fpprec : 100')
'100'
sage: maxima(pi).bfloat()
3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117068b0

Python

>>> from sage.all import *
>>> maxima.eval('fpprec : 100')
'100'
>>> maxima(pi).bfloat()
3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117068b0

Defining functions in maxima:

Sage

sage: maxima.eval('fun[a] := a^2')
'fun[a]:=a^2'
sage: maxima('fun[10]')
100

Python

>>> from sage.all import *
>>> maxima.eval('fun[a] := a^2')
'fun[a]:=a^2'
>>> maxima('fun[10]')
100

Interactivity#

Unfortunately maxima doesn’t seem to have a non-interactive mode, which is needed for the Sage interface. If any Sage call leads to maxima interactively answering questions, then the questions can’t be answered and the maxima session may hang. See the discussion at http://www.ma.utexas.edu/pipermail/maxima/2005/011061.html for some ideas about how to fix this problem. An example that illustrates this problem is maxima.eval('integrate (exp(a*x), x, 0, inf)').

Latex Output#

To TeX a maxima object do this:

Sage

sage: latex(maxima('sin(u) + sinh(v^2)'))
\sinh v^2+\sin u

Python

>>> from sage.all import *
>>> latex(maxima('sin(u) + sinh(v^2)'))
\sinh v^2+\sin u

Here’s another example:

Sage

sage: g = maxima('exp(3*%i*x)/(6*%i) + exp(%i*x)/(2*%i) + c')
sage: latex(g)
-...{{i\,e^{3\,i\,x}}\over{6}}...-{{i\,e^{i\,x}}\over{2}}+c

Python

>>> from sage.all import *
>>> g = maxima('exp(3*%i*x)/(6*%i) + exp(%i*x)/(2*%i) + c')
>>> latex(g)
-...{{i\,e^{3\,i\,x}}\over{6}}...-{{i\,e^{i\,x}}\over{2}}+c

Long Input#

The MAXIMA interface reads in even very long input (using files) in a robust manner, as long as you are creating a new object.

Note

Using maxima.eval for long input is much less robust, and is not recommended.

Sage

sage: t = '"%s"'%10^10000   # ten thousand character string.
sage: a = maxima(t)

Python

>>> from sage.all import *
>>> t = '"%s"'%Integer(10)**Integer(10000)   # ten thousand character string.
>>> a = maxima(t)

class sage.interfaces.maxima.Maxima(script_subdirectory=None, logfile=None, server=None, init_code=None)[source]#

Bases: MaximaAbstract, Expect

Interface to the Maxima interpreter.

EXAMPLES:

Sage

sage: m = Maxima()
sage: m == maxima
False

Python

>>> from sage.all import *
>>> m = Maxima()
>>> m == maxima
False

clear(var)[source]#

Clear the variable named var.

EXAMPLES:

Sage

sage: maxima.set('xxxxx', '2')
sage: maxima.get('xxxxx')
'2'
sage: maxima.clear('xxxxx')
sage: maxima.get('xxxxx')
'xxxxx'

Python

>>> from sage.all import *
>>> maxima.set('xxxxx', '2')
>>> maxima.get('xxxxx')
'2'
>>> maxima.clear('xxxxx')
>>> maxima.get('xxxxx')
'xxxxx'

get(var)[source]#

Get the string value of the variable var.

EXAMPLES:

Sage

sage: maxima.set('xxxxx', '2')
sage: maxima.get('xxxxx')
'2'

Python

>>> from sage.all import *
>>> maxima.set('xxxxx', '2')
>>> maxima.get('xxxxx')
'2'

lisp(cmd)[source]#

Send a lisp command to Maxima.

Note

The output of this command is very raw - not pretty.

EXAMPLES:

Sage

sage: maxima.lisp("(+ 2 17)")   # random formatted output
 :lisp (+ 2 17)
19
(

Python

>>> from sage.all import *
>>> maxima.lisp("(+ 2 17)")   # random formatted output
 :lisp (+ 2 17)
19
(

set(var, value)[source]#

Set the variable var to the given value.

INPUT:

var – string
value – string

EXAMPLES:

Sage

sage: maxima.set('xxxxx', '2')
sage: maxima.get('xxxxx')
'2'

Python

>>> from sage.all import *
>>> maxima.set('xxxxx', '2')
>>> maxima.get('xxxxx')
'2'

set_seed(seed=None)[source]#

http://maxima.sourceforge.net/docs/manual/maxima_10.html make_random_state (n) returns a new random state object created from an integer seed value equal to n modulo 2^32. n may be negative.

EXAMPLES:

Sage

sage: m = Maxima()
sage: m.set_seed(1)
1
sage: [m.random(100) for i in range(5)]
[45, 39, 24, 68, 63]

Python

>>> from sage.all import *
>>> m = Maxima()
>>> m.set_seed(Integer(1))
1
>>> [m.random(Integer(100)) for i in range(Integer(5))]
[45, 39, 24, 68, 63]

class sage.interfaces.maxima.MaximaElement(parent, value, is_name=False, name=None)[source]#

Bases: MaximaAbstractElement, ExpectElement

Element of Maxima through Pexpect interface.

EXAMPLES:

Elements of this class should not be created directly. The targeted parent should be used instead:

Sage

sage: maxima(3)
3
sage: maxima(cos(x)+e^234)
cos(_SAGE_VAR_x)+%e^234

Python

>>> from sage.all import *
>>> maxima(Integer(3))
3
>>> maxima(cos(x)+e**Integer(234))
cos(_SAGE_VAR_x)+%e^234

display2d(onscreen=True)[source]#

Return the 2d string representation of this Maxima object.

EXAMPLES:

Sage

sage: F = maxima('x^5 - y^5').factor()
sage: F.display2d()
                       4      3    2  2    3      4
           - (y - x) (y  + x y  + x  y  + x  y + x )

Python

>>> from sage.all import *
>>> F = maxima('x^5 - y^5').factor()
>>> F.display2d()
                       4      3    2  2    3      4
           - (y - x) (y  + x y  + x  y  + x  y + x )

class sage.interfaces.maxima.MaximaElementFunction(parent, name, defn, args, latex)[source]#

Bases: MaximaElement, MaximaAbstractElementFunction

Maxima user-defined functions.

EXAMPLES:

Elements of this class should not be created directly. The method function of the targeted parent should be used instead:

Sage

sage: maxima.function('x,y','h(x)*y')
h(x)*y

Python

>>> from sage.all import *
>>> maxima.function('x,y','h(x)*y')
h(x)*y

sage.interfaces.maxima.is_MaximaElement(x)[source]#

Return True if x is of type MaximaElement.

EXAMPLES:

Sage

sage: from sage.interfaces.maxima import is_MaximaElement
sage: is_MaximaElement(1)
doctest:...: DeprecationWarning: the function is_MaximaElement is deprecated; use isinstance(x, sage.interfaces.abc.MaximaElement) instead
See https://github.com/sagemath/sage/issues/34804 for details.
False
sage: m = maxima(1)
sage: is_MaximaElement(m)
True

Python

>>> from sage.all import *
>>> from sage.interfaces.maxima import is_MaximaElement
>>> is_MaximaElement(Integer(1))
doctest:...: DeprecationWarning: the function is_MaximaElement is deprecated; use isinstance(x, sage.interfaces.abc.MaximaElement) instead
See https://github.com/sagemath/sage/issues/34804 for details.
False
>>> m = maxima(Integer(1))
>>> is_MaximaElement(m)
True

sage.interfaces.maxima.reduce_load_Maxima()[source]#

Unpickle a Maxima Pexpect interface.

EXAMPLES:

Sage

sage: from sage.interfaces.maxima import reduce_load_Maxima
sage: reduce_load_Maxima()
Maxima

Python

>>> from sage.all import *
>>> from sage.interfaces.maxima import reduce_load_Maxima
>>> reduce_load_Maxima()
Maxima

sage.interfaces.maxima.reduce_load_Maxima_function(parent, defn, args, latex)[source]#

Unpickle a Maxima function.

EXAMPLES:

Sage

sage: from sage.interfaces.maxima import reduce_load_Maxima_function
sage: f = maxima.function('x,y','sin(x+y)')
sage: _,args = f.__reduce__()
sage: g = reduce_load_Maxima_function(*args)
sage: g == f
True

Python

>>> from sage.all import *
>>> from sage.interfaces.maxima import reduce_load_Maxima_function
>>> f = maxima.function('x,y','sin(x+y)')
>>> _,args = f.__reduce__()
>>> g = reduce_load_Maxima_function(*args)
>>> g == f
True

Pexpect interface to Maxima#

Tutorial#

Examples involving matrices#

Laplace Transforms#

Continued Fractions#

Special examples#

Miscellaneous#

Interactivity#

Latex Output#

Long Input#