Abstract 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/docs.shtml/.

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 an abstract class implementing the functions shared between the Pexpect and library interfaces to Maxima.

class sage.interfaces.maxima_abstract.MaximaAbstract(name='maxima_abstract')[source]#

Bases: ExtraTabCompletion, Interface

Abstract interface to Maxima.

INPUT:

  • name – string

OUTPUT: the interface

EXAMPLES:

This class should not be instantiated directly, but through its subclasses Maxima (Pexpect interface) or MaximaLib (library interface):

sage: m = Maxima()
sage: from sage.interfaces.maxima_abstract import MaximaAbstract
sage: isinstance(m,MaximaAbstract)
True

>>> from sage.all import *
>>> m = Maxima()
>>> from sage.interfaces.maxima_abstract import MaximaAbstract
>>> isinstance(m,MaximaAbstract)
True
chdir(dir)[source]#

Change Maxima’s current working directory.

INPUT:

  • dir – string

OUTPUT: none

EXAMPLES:

sage: maxima.chdir('/')

>>> from sage.all import *
>>> maxima.chdir('/')
completions(s, verbose=True)[source]#

Return all commands that complete the command starting with the string s. This is like typing s[tab] in the Maxima interpreter.

INPUT:

  • s – string

  • verbose – boolean (default: True)

OUTPUT: array of strings

EXAMPLES:

sage: sorted(maxima.completions('gc', verbose=False))
['gcd', 'gcdex', 'gcfactor', 'gctime']

>>> from sage.all import *
>>> sorted(maxima.completions('gc', verbose=False))
['gcd', 'gcdex', 'gcfactor', 'gctime']
console()[source]#

Start the interactive Maxima console. This is a completely separate maxima session from this interface. To interact with this session, you should instead use maxima.interact().

INPUT: none

OUTPUT: none

EXAMPLES:

sage: maxima.console()             # not tested (since we can't)
Maxima 5.46.0 https://maxima.sourceforge.io
using Lisp ECL 21.2.1
Distributed under the GNU Public License. See the file COPYING.
Dedicated to the memory of William Schelter.
This is a development version of Maxima. The function bug_report()
provides bug reporting information.
(%i1)

>>> from sage.all import *
>>> maxima.console()             # not tested (since we can't)
Maxima 5.46.0 https://maxima.sourceforge.io
using Lisp ECL 21.2.1
Distributed under the GNU Public License. See the file COPYING.
Dedicated to the memory of William Schelter.
This is a development version of Maxima. The function bug_report()
provides bug reporting information.
(%i1)

sage: maxima.interact()     # not tested
  --> Switching to Maxima <--
maxima: 2+2
4
maxima:
  --> Exiting back to Sage <--

>>> from sage.all import *
>>> maxima.interact()     # not tested
  --> Switching to Maxima <--
maxima: 2+2
4
maxima:
  --> Exiting back to Sage <--
cputime(t=None)[source]#

Returns the amount of CPU time that this Maxima session has used.

INPUT:

  • t – float (default: None); If var{t} is not None, then it returns the difference between the current CPU time and var{t}.

OUTPUT: float

EXAMPLES:

sage: t = maxima.cputime()
sage: _ = maxima.de_solve('diff(y,x,2) + 3*x = y', ['x','y'], [1,1,1])
sage: maxima.cputime(t) # output random
0.568913

>>> from sage.all import *
>>> t = maxima.cputime()
>>> _ = maxima.de_solve('diff(y,x,2) + 3*x = y', ['x','y'], [Integer(1),Integer(1),Integer(1)])
>>> maxima.cputime(t) # output random
0.568913
de_solve(de, vars, ics=None)[source]#

Solves a 1st or 2nd order ordinary differential equation (ODE) in two variables, possibly with initial conditions.

INPUT:

  • de – a string representing the ODE

  • vars – a list of strings representing the two

    variables.

  • ics – a triple of numbers [a,b1,b2] representing

    y(a)=b1, y’(a)=b2

EXAMPLES:

sage: maxima.de_solve('diff(y,x,2) + 3*x = y', ['x','y'], [1,1,1])
y = 3*x-2*%e^(x-1)
sage: maxima.de_solve('diff(y,x,2) + 3*x = y', ['x','y'])
y = %k1*%e^x+%k2*%e^-x+3*x
sage: maxima.de_solve('diff(y,x) + 3*x = y', ['x','y'])
y = (%c-3*(...-x...-1)*%e^-x)*%e^x
sage: maxima.de_solve('diff(y,x) + 3*x = y', ['x','y'],[1,1])
y = -...%e^-1*(5*%e^x-3*%e*x-3*%e)...

>>> from sage.all import *
>>> maxima.de_solve('diff(y,x,2) + 3*x = y', ['x','y'], [Integer(1),Integer(1),Integer(1)])
y = 3*x-2*%e^(x-1)
>>> maxima.de_solve('diff(y,x,2) + 3*x = y', ['x','y'])
y = %k1*%e^x+%k2*%e^-x+3*x
>>> maxima.de_solve('diff(y,x) + 3*x = y', ['x','y'])
y = (%c-3*(...-x...-1)*%e^-x)*%e^x
>>> maxima.de_solve('diff(y,x) + 3*x = y', ['x','y'],[Integer(1),Integer(1)])
y = -...%e^-1*(5*%e^x-3*%e*x-3*%e)...
de_solve_laplace(de, vars, ics=None)[source]#

Solves an ordinary differential equation (ODE) using Laplace transforms.

INPUT:

  • de – a string representing the ODE (e.g., de =

    “diff(f(x),x,2)=diff(f(x),x)+sin(x)”)

  • vars – a list of strings representing the

    variables (e.g., vars = [“x”,”f”])

  • ics – a list of numbers representing initial

    conditions, with symbols allowed which are represented by strings (eg, f(0)=1, f’(0)=2 is ics = [0,1,2])

EXAMPLES:

sage: maxima.clear('x'); maxima.clear('f')
sage: maxima.de_solve_laplace("diff(f(x),x,2) = 2*diff(f(x),x)-f(x)", ["x","f"], [0,1,2])
f(x) = x*%e^x+%e^x

>>> from sage.all import *
>>> maxima.clear('x'); maxima.clear('f')
>>> maxima.de_solve_laplace("diff(f(x),x,2) = 2*diff(f(x),x)-f(x)", ["x","f"], [Integer(0),Integer(1),Integer(2)])
f(x) = x*%e^x+%e^x

sage: maxima.clear('x'); maxima.clear('f')
sage: f = maxima.de_solve_laplace("diff(f(x),x,2) = 2*diff(f(x),x)-f(x)", ["x","f"])
sage: f
f(x) = x*%e^x*('at('diff(f(x),x,1),x = 0))-f(0)*x*%e^x+f(0)*%e^x
sage: print(f)
                                   !
                       x  d        !                  x          x
            f(x) = x %e  (-- (f(x))!     ) - f(0) x %e  + f(0) %e
                          dx       !
                                   !x = 0

>>> from sage.all import *
>>> maxima.clear('x'); maxima.clear('f')
>>> f = maxima.de_solve_laplace("diff(f(x),x,2) = 2*diff(f(x),x)-f(x)", ["x","f"])
>>> f
f(x) = x*%e^x*('at('diff(f(x),x,1),x = 0))-f(0)*x*%e^x+f(0)*%e^x
>>> print(f)
                                   !
                       x  d        !                  x          x
            f(x) = x %e  (-- (f(x))!     ) - f(0) x %e  + f(0) %e
                          dx       !
                                   !x = 0

Note

The second equation sets the values of \(f(0)\) and \(f'(0)\) in Maxima, so subsequent ODEs involving these variables will have these initial conditions automatically imposed.

demo(s)[source]#

Run Maxima’s demo for s.

INPUT:

  • s – string

OUTPUT: none

EXAMPLES:

sage: maxima.demo('cf') # not tested
read and interpret file: .../share/maxima/5.34.1/demo/cf.dem

At the '_' prompt, type ';' and <enter> to get next demonstration.
frac1:cf([1,2,3,4])
...

>>> from sage.all import *
>>> maxima.demo('cf') # not tested
read and interpret file: .../share/maxima/5.34.1/demo/cf.dem

At the '_' prompt, type ';' and <enter> to get next demonstration.
frac1:cf([1,2,3,4])
...
describe(s)[source]#

Return Maxima’s help for s.

INPUT:

  • s – string

OUTPUT:

Maxima’s help for s

EXAMPLES:

sage: maxima.help('gcd')
-- Function: gcd (<p_1>, <p_2>, <x_1>, ...)
...

>>> from sage.all import *
>>> maxima.help('gcd')
-- Function: gcd (<p_1>, <p_2>, <x_1>, ...)
...
example(s)[source]#

Return Maxima’s examples for s.

INPUT:

  • s – string

OUTPUT:

Maxima’s examples for s

EXAMPLES:

sage: maxima.example('arrays')
a[n]:=n*a[n-1]
                                a  := n a
                                 n       n - 1
a[0]:1
a[5]
                                      120
a[n]:=n
a[6]
                                       6
a[4]
                                      24
                                     done

>>> from sage.all import *
>>> maxima.example('arrays')
a[n]:=n*a[n-1]
                                a  := n a
                                 n       n - 1
a[0]:1
a[5]
                                      120
a[n]:=n
a[6]
                                       6
a[4]
                                      24
                                     done
function(args, defn, rep=None, latex=None)[source]#

Return the Maxima function with given arguments and definition.

INPUT:

  • args – a string with variable names separated by

    commas

  • defn – a string (or Maxima expression) that

    defines a function of the arguments in Maxima.

  • rep – an optional string; if given, this is how

    the function will print.

OUTPUT: Maxima function

EXAMPLES:

sage: f = maxima.function('x', 'sin(x)')
sage: f(3.2)  # abs tol 2e-16
-0.058374143427579909
sage: f = maxima.function('x,y', 'sin(x)+cos(y)')
sage: f(2, 3.5)  # abs tol 2e-16
sin(2)-0.9364566872907963
sage: f
sin(x)+cos(y)

>>> from sage.all import *
>>> f = maxima.function('x', 'sin(x)')
>>> f(RealNumber('3.2'))  # abs tol 2e-16
-0.058374143427579909
>>> f = maxima.function('x,y', 'sin(x)+cos(y)')
>>> f(Integer(2), RealNumber('3.5'))  # abs tol 2e-16
sin(2)-0.9364566872907963
>>> f
sin(x)+cos(y)

sage: g = f.integrate('z')
sage: g
(cos(y)+sin(x))*z
sage: g(1,2,3)
3*(cos(2)+sin(1))

>>> from sage.all import *
>>> g = f.integrate('z')
>>> g
(cos(y)+sin(x))*z
>>> g(Integer(1),Integer(2),Integer(3))
3*(cos(2)+sin(1))

The function definition can be a Maxima object:

sage: an_expr = maxima('sin(x)*gamma(x)')
sage: t = maxima.function('x', an_expr)
sage: t
gamma(x)*sin(x)
sage: t(2)
 sin(2)
sage: float(t(2))
0.9092974268256817
sage: loads(t.dumps())
gamma(x)*sin(x)

>>> from sage.all import *
>>> an_expr = maxima('sin(x)*gamma(x)')
>>> t = maxima.function('x', an_expr)
>>> t
gamma(x)*sin(x)
>>> t(Integer(2))
 sin(2)
>>> float(t(Integer(2)))
0.9092974268256817
>>> loads(t.dumps())
gamma(x)*sin(x)
help(s)[source]#

Return Maxima’s help for s.

INPUT:

  • s – string

OUTPUT:

Maxima’s help for s

EXAMPLES:

sage: maxima.help('gcd')
-- Function: gcd (<p_1>, <p_2>, <x_1>, ...)
...

>>> from sage.all import *
>>> maxima.help('gcd')
-- Function: gcd (<p_1>, <p_2>, <x_1>, ...)
...
plot2d(*args)[source]#

Plot a 2d graph using Maxima / gnuplot.

maxima.plot2d(f, ‘[var, min, max]’, options)

INPUT:

  • f – a string representing a function (such as

    f=”sin(x)”) [var, xmin, xmax]

  • options – an optional string representing plot2d

    options in gnuplot format

EXAMPLES:

sage: maxima.plot2d('sin(x)','[x,-5,5]')   # not tested
sage: opts = '[gnuplot_term, ps], [gnuplot_out_file, "sin-plot.eps"]'
sage: maxima.plot2d('sin(x)','[x,-5,5]',opts)    # not tested

>>> from sage.all import *
>>> maxima.plot2d('sin(x)','[x,-5,5]')   # not tested
>>> opts = '[gnuplot_term, ps], [gnuplot_out_file, "sin-plot.eps"]'
>>> maxima.plot2d('sin(x)','[x,-5,5]',opts)    # not tested

The eps file is saved in the current directory.

plot2d_parametric(r, var, trange, nticks=50, options=None)[source]#

Plot r = [x(t), y(t)] for t = tmin…tmax using gnuplot with options.

INPUT:

  • r – a string representing a function (such as

    r=”[x(t),y(t)]”)

  • var – a string representing the variable (such

    as var = “t”)

  • trange – [tmin, tmax] are numbers with tmintmax

  • nticks – int (default: 50)

  • options – an optional string representing plot2d

    options in gnuplot format

EXAMPLES:

sage: maxima.plot2d_parametric(["sin(t)","cos(t)"], "t",[-3.1,3.1])   # not tested

>>> from sage.all import *
>>> maxima.plot2d_parametric(["sin(t)","cos(t)"], "t",[-RealNumber('3.1'),RealNumber('3.1')])   # not tested

sage: opts = '[gnuplot_preamble, "set nokey"], [gnuplot_term, ps], [gnuplot_out_file, "circle-plot.eps"]'
sage: maxima.plot2d_parametric(["sin(t)","cos(t)"], "t", [-3.1,3.1], options=opts)   # not tested

>>> from sage.all import *
>>> opts = '[gnuplot_preamble, "set nokey"], [gnuplot_term, ps], [gnuplot_out_file, "circle-plot.eps"]'
>>> maxima.plot2d_parametric(["sin(t)","cos(t)"], "t", [-RealNumber('3.1'),RealNumber('3.1')], options=opts)   # not tested

The eps file is saved to the current working directory.

Here is another fun plot:

sage: maxima.plot2d_parametric(["sin(5*t)","cos(11*t)"], "t", [0,2*pi()], nticks=400)    # not tested

>>> from sage.all import *
>>> maxima.plot2d_parametric(["sin(5*t)","cos(11*t)"], "t", [Integer(0),Integer(2)*pi()], nticks=Integer(400))    # not tested
plot3d(*args)[source]#

Plot a 3d graph using Maxima / gnuplot.

maxima.plot3d(f, ‘[x, xmin, xmax]’, ‘[y, ymin, ymax]’, ‘[grid, nx, ny]’, options)

INPUT:

  • f – a string representing a function (such as

    f=”sin(x)”) [var, min, max]

  • args should be of the form ‘[x, xmin, xmax]’, ‘[y, ymin, ymax]’, ‘[grid, nx, ny]’, options

EXAMPLES:

sage: maxima.plot3d('1 + x^3 - y^2', '[x,-2,2]', '[y,-2,2]', '[grid,12,12]')    # not tested
sage: maxima.plot3d('sin(x)*cos(y)', '[x,-2,2]', '[y,-2,2]', '[grid,30,30]')   # not tested
sage: opts = '[gnuplot_term, ps], [gnuplot_out_file, "sin-plot.eps"]'
sage: maxima.plot3d('sin(x+y)', '[x,-5,5]', '[y,-1,1]', opts)    # not tested

>>> from sage.all import *
>>> maxima.plot3d('1 + x^3 - y^2', '[x,-2,2]', '[y,-2,2]', '[grid,12,12]')    # not tested
>>> maxima.plot3d('sin(x)*cos(y)', '[x,-2,2]', '[y,-2,2]', '[grid,30,30]')   # not tested
>>> opts = '[gnuplot_term, ps], [gnuplot_out_file, "sin-plot.eps"]'
>>> maxima.plot3d('sin(x+y)', '[x,-5,5]', '[y,-1,1]', opts)    # not tested

The eps file is saved in the current working directory.

plot3d_parametric(r, vars, urange, vrange, options=None)[source]#

Plot a 3d parametric graph with r=(x,y,z), x = x(u,v), y = y(u,v), z = z(u,v), for u = umin…umax, v = vmin…vmax using gnuplot with options.

INPUT:

  • x, y, z – a string representing a function (such

    as x="u2+v2", …) vars is a list or two strings representing variables (such as vars = [“u”,”v”])

  • urange – [umin, umax]

  • vrange – [vmin, vmax] are lists of numbers with

    umin umax, vmin vmax

  • options – optional string representing plot2d

    options in gnuplot format

OUTPUT: displays a plot on screen or saves to a file

EXAMPLES:

sage: maxima.plot3d_parametric(["v*sin(u)","v*cos(u)","v"], ["u","v"],[-3.2,3.2],[0,3])     # not tested
sage: opts = '[gnuplot_term, ps], [gnuplot_out_file, "sin-cos-plot.eps"]'
sage: maxima.plot3d_parametric(["v*sin(u)","v*cos(u)","v"], ["u","v"],[-3.2,3.2],[0,3],opts)      # not tested

>>> from sage.all import *
>>> maxima.plot3d_parametric(["v*sin(u)","v*cos(u)","v"], ["u","v"],[-RealNumber('3.2'),RealNumber('3.2')],[Integer(0),Integer(3)])     # not tested
>>> opts = '[gnuplot_term, ps], [gnuplot_out_file, "sin-cos-plot.eps"]'
>>> maxima.plot3d_parametric(["v*sin(u)","v*cos(u)","v"], ["u","v"],[-RealNumber('3.2'),RealNumber('3.2')],[Integer(0),Integer(3)],opts)      # not tested

The eps file is saved in the current working directory.

Here is a torus:

sage: _ = maxima.eval("expr_1: cos(y)*(10.0+6*cos(x)); expr_2: sin(y)*(10.0+6*cos(x)); expr_3: -6*sin(x);")
sage: maxima.plot3d_parametric(["expr_1","expr_2","expr_3"], ["x","y"],[0,6],[0,6])  # not tested

>>> from sage.all import *
>>> _ = maxima.eval("expr_1: cos(y)*(10.0+6*cos(x)); expr_2: sin(y)*(10.0+6*cos(x)); expr_3: -6*sin(x);")
>>> maxima.plot3d_parametric(["expr_1","expr_2","expr_3"], ["x","y"],[Integer(0),Integer(6)],[Integer(0),Integer(6)])  # not tested

Here is a Möbius strip:

sage: x = "cos(u)*(3 + v*cos(u/2))"
sage: y = "sin(u)*(3 + v*cos(u/2))"
sage: z = "v*sin(u/2)"
sage: maxima.plot3d_parametric([x,y,z],["u","v"],[-3.1,3.2],[-1/10,1/10])   # not tested

>>> from sage.all import *
>>> x = "cos(u)*(3 + v*cos(u/2))"
>>> y = "sin(u)*(3 + v*cos(u/2))"
>>> z = "v*sin(u/2)"
>>> maxima.plot3d_parametric([x,y,z],["u","v"],[-RealNumber('3.1'),RealNumber('3.2')],[-Integer(1)/Integer(10),Integer(1)/Integer(10)])   # not tested
plot_list(ptsx, ptsy, options=None)[source]#

Plots a curve determined by a sequence of points.

INPUT:

  • ptsx – [x1,…,xn], where the xi and yi are

    real,

  • ptsy – [y1,…,yn]

  • options – a string representing maxima plot2d

    options.

The points are (x1,y1), (x2,y2), etc.

This function requires maxima 5.9.2 or newer.

Note

More that 150 points can sometimes lead to the program hanging. Why?

EXAMPLES:

sage: zeta_ptsx = [(pari(1/2 + i*I/10).zeta().real()).precision(1)          # needs sage.libs.pari
....:              for i in range(70,150)]
sage: zeta_ptsy = [(pari(1/2 + i*I/10).zeta().imag()).precision(1)          # needs sage.libs.pari
....:              for i in range(70,150)]
sage: maxima.plot_list(zeta_ptsx, zeta_ptsy)        # not tested            # needs sage.libs.pari
sage: opts='[gnuplot_preamble, "set nokey"], [gnuplot_term, ps], [gnuplot_out_file, "zeta.eps"]'
sage: maxima.plot_list(zeta_ptsx, zeta_ptsy, opts)  # not tested            # needs sage.libs.pari

>>> from sage.all import *
>>> zeta_ptsx = [(pari(Integer(1)/Integer(2) + i*I/Integer(10)).zeta().real()).precision(Integer(1))          # needs sage.libs.pari
...              for i in range(Integer(70),Integer(150))]
>>> zeta_ptsy = [(pari(Integer(1)/Integer(2) + i*I/Integer(10)).zeta().imag()).precision(Integer(1))          # needs sage.libs.pari
...              for i in range(Integer(70),Integer(150))]
>>> maxima.plot_list(zeta_ptsx, zeta_ptsy)        # not tested            # needs sage.libs.pari
>>> opts='[gnuplot_preamble, "set nokey"], [gnuplot_term, ps], [gnuplot_out_file, "zeta.eps"]'
>>> maxima.plot_list(zeta_ptsx, zeta_ptsy, opts)  # not tested            # needs sage.libs.pari
plot_multilist(pts_list, options=None)[source]#

Plots a list of list of points pts_list=[pts1,pts2,…,ptsn], where each ptsi is of the form [[x1,y1],…,[xn,yn]] x’s must be integers and y’s reals options is a string representing maxima plot2d options.

INPUT:

  • pts_lst – list of points; each point must be of the form [x,y] where x is an integer and y is a real

  • var – string; representing Maxima’s plot2d options

Requires maxima 5.9.2 at least.

Note

More that 150 points can sometimes lead to the program hanging.

EXAMPLES:

sage: xx = [i/10.0 for i in range(-10,10)]
sage: yy = [i/10.0 for i in range(-10,10)]
sage: x0 = [0 for i in range(-10,10)]
sage: y0 = [0 for i in range(-10,10)]
sage: zeta_ptsx1 = [(pari(1/2+i*I/10).zeta().real()).precision(1)           # needs sage.libs.pari
....:               for i in range(10)]
sage: zeta_ptsy1 = [(pari(1/2+i*I/10).zeta().imag()).precision(1)           # needs sage.libs.pari
....:               for i in range(10)]
sage: maxima.plot_multilist([[zeta_ptsx1,zeta_ptsy1], [xx,y0], [x0,yy]])    # not tested
sage: zeta_ptsx1 = [(pari(1/2+i*I/10).zeta().real()).precision(1)           # needs sage.libs.pari
....:               for i in range(10,150)]
sage: zeta_ptsy1 = [(pari(1/2+i*I/10).zeta().imag()).precision(1)           # needs sage.libs.pari
....:               for i in range(10,150)]
sage: maxima.plot_multilist([[zeta_ptsx1,zeta_ptsy1], [xx,y0], [x0,yy]])    # not tested
sage: opts='[gnuplot_preamble, "set nokey"]'
sage: maxima.plot_multilist([[zeta_ptsx1,zeta_ptsy1], [xx,y0], [x0,yy]],    # not tested
....:                       opts)

>>> from sage.all import *
>>> xx = [i/RealNumber('10.0') for i in range(-Integer(10),Integer(10))]
>>> yy = [i/RealNumber('10.0') for i in range(-Integer(10),Integer(10))]
>>> x0 = [Integer(0) for i in range(-Integer(10),Integer(10))]
>>> y0 = [Integer(0) for i in range(-Integer(10),Integer(10))]
>>> zeta_ptsx1 = [(pari(Integer(1)/Integer(2)+i*I/Integer(10)).zeta().real()).precision(Integer(1))           # needs sage.libs.pari
...               for i in range(Integer(10))]
>>> zeta_ptsy1 = [(pari(Integer(1)/Integer(2)+i*I/Integer(10)).zeta().imag()).precision(Integer(1))           # needs sage.libs.pari
...               for i in range(Integer(10))]
>>> maxima.plot_multilist([[zeta_ptsx1,zeta_ptsy1], [xx,y0], [x0,yy]])    # not tested
>>> zeta_ptsx1 = [(pari(Integer(1)/Integer(2)+i*I/Integer(10)).zeta().real()).precision(Integer(1))           # needs sage.libs.pari
...               for i in range(Integer(10),Integer(150))]
>>> zeta_ptsy1 = [(pari(Integer(1)/Integer(2)+i*I/Integer(10)).zeta().imag()).precision(Integer(1))           # needs sage.libs.pari
...               for i in range(Integer(10),Integer(150))]
>>> maxima.plot_multilist([[zeta_ptsx1,zeta_ptsy1], [xx,y0], [x0,yy]])    # not tested
>>> opts='[gnuplot_preamble, "set nokey"]'
>>> maxima.plot_multilist([[zeta_ptsx1,zeta_ptsy1], [xx,y0], [x0,yy]],    # not tested
...                       opts)
solve_linear(eqns, vars)[source]#

Wraps maxima’s linsolve.

INPUT:

  • eqns – a list of m strings; each representing a linear question in m = n variables

  • vars – a list of n strings; each representing a variable

EXAMPLES:

sage: eqns = ["x + z = y","2*a*x - y = 2*a^2","y - 2*z = 2"]
sage: vars = ["x","y","z"]
sage: maxima.solve_linear(eqns, vars)
[x = a+1,y = 2*a,z = a-1]

>>> from sage.all import *
>>> eqns = ["x + z = y","2*a*x - y = 2*a^2","y - 2*z = 2"]
>>> vars = ["x","y","z"]
>>> maxima.solve_linear(eqns, vars)
[x = a+1,y = 2*a,z = a-1]
unit_quadratic_integer(n)[source]#

Finds a unit of the ring of integers of the quadratic number field \(\QQ(\sqrt{n})\), \(n>1\), using the qunit maxima command.

INPUT:

  • n – an integer

EXAMPLES:

sage: u = maxima.unit_quadratic_integer(101); u
a + 10
sage: u.parent()
Number Field in a with defining polynomial x^2 - 101 with a = 10.04987562112089?
sage: u = maxima.unit_quadratic_integer(13)
sage: u
5*a + 18
sage: u.parent()
Number Field in a with defining polynomial x^2 - 13 with a = 3.605551275463990?

>>> from sage.all import *
>>> u = maxima.unit_quadratic_integer(Integer(101)); u
a + 10
>>> u.parent()
Number Field in a with defining polynomial x^2 - 101 with a = 10.04987562112089?
>>> u = maxima.unit_quadratic_integer(Integer(13))
>>> u
5*a + 18
>>> u.parent()
Number Field in a with defining polynomial x^2 - 13 with a = 3.605551275463990?
version()[source]#

Return the version of Maxima that Sage includes.

INPUT: none

OUTPUT: none

EXAMPLES:

sage: maxima.version()  # random
'5.41.0'

>>> from sage.all import *
>>> maxima.version()  # random
'5.41.0'
class sage.interfaces.maxima_abstract.MaximaAbstractElement(parent, value, is_name=False, name=None)[source]#

Bases: ExtraTabCompletion, InterfaceElement

Element of Maxima through an abstract interface.

EXAMPLES:

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

sage: from sage.interfaces.maxima_lib import maxima_lib
sage: xp = maxima(x)
sage: type(xp)
<class 'sage.interfaces.maxima.MaximaElement'>
sage: xl = maxima_lib(x)
sage: type(xl)
<class 'sage.interfaces.maxima_lib.MaximaLibElement'>

>>> from sage.all import *
>>> from sage.interfaces.maxima_lib import maxima_lib
>>> xp = maxima(x)
>>> type(xp)
<class 'sage.interfaces.maxima.MaximaElement'>
>>> xl = maxima_lib(x)
>>> type(xl)
<class 'sage.interfaces.maxima_lib.MaximaLibElement'>
comma(args)[source]#

Form the expression that would be written ‘self, args’ in Maxima.

INPUT:

  • args – string

OUTPUT: Maxima object

EXAMPLES:

sage: maxima('sqrt(2) + I').comma('numer')
I+1.41421356237309...
sage: maxima('sqrt(2) + I*a').comma('a=5')
5*I+sqrt(2)

>>> from sage.all import *
>>> maxima('sqrt(2) + I').comma('numer')
I+1.41421356237309...
>>> maxima('sqrt(2) + I*a').comma('a=5')
5*I+sqrt(2)
derivative(var='x', n=1)[source]#

Return the n-th derivative of self.

INPUT:

  • var – variable (default: ‘x’)

  • n – integer (default: 1)

OUTPUT: n-th derivative of self with respect to the variable var

EXAMPLES:

sage: f = maxima('x^2')
sage: f.diff()
2*x
sage: f.diff('x')
2*x
sage: f.diff('x', 2)
2
sage: maxima('sin(x^2)').diff('x',4)
16*x^4*sin(x^2)-12*sin(x^2)-48*x^2*cos(x^2)

>>> from sage.all import *
>>> f = maxima('x^2')
>>> f.diff()
2*x
>>> f.diff('x')
2*x
>>> f.diff('x', Integer(2))
2
>>> maxima('sin(x^2)').diff('x',Integer(4))
16*x^4*sin(x^2)-12*sin(x^2)-48*x^2*cos(x^2)

sage: f = maxima('x^2 + 17*y^2')
sage: f.diff('x')
34*y*'diff(y,x,1)+2*x
sage: f.diff('y')
34*y

>>> from sage.all import *
>>> f = maxima('x^2 + 17*y^2')
>>> f.diff('x')
34*y*'diff(y,x,1)+2*x
>>> f.diff('y')
34*y
diff(var='x', n=1)[source]#

Return the n-th derivative of self.

INPUT:

  • var – variable (default: ‘x’)

  • n – integer (default: 1)

OUTPUT: n-th derivative of self with respect to the variable var

EXAMPLES:

sage: f = maxima('x^2')
sage: f.diff()
2*x
sage: f.diff('x')
2*x
sage: f.diff('x', 2)
2
sage: maxima('sin(x^2)').diff('x',4)
16*x^4*sin(x^2)-12*sin(x^2)-48*x^2*cos(x^2)

>>> from sage.all import *
>>> f = maxima('x^2')
>>> f.diff()
2*x
>>> f.diff('x')
2*x
>>> f.diff('x', Integer(2))
2
>>> maxima('sin(x^2)').diff('x',Integer(4))
16*x^4*sin(x^2)-12*sin(x^2)-48*x^2*cos(x^2)

sage: f = maxima('x^2 + 17*y^2')
sage: f.diff('x')
34*y*'diff(y,x,1)+2*x
sage: f.diff('y')
34*y

>>> from sage.all import *
>>> f = maxima('x^2 + 17*y^2')
>>> f.diff('x')
34*y*'diff(y,x,1)+2*x
>>> f.diff('y')
34*y
dot(other)[source]#

Implements the notation self . other.

INPUT:

  • other – matrix; argument to dot.

OUTPUT: Maxima matrix

EXAMPLES:

sage: A = maxima('matrix ([a1],[a2])')
sage: B = maxima('matrix ([b1, b2])')
sage: A.dot(B)
matrix([a1*b1,a1*b2],[a2*b1,a2*b2])

>>> from sage.all import *
>>> A = maxima('matrix ([a1],[a2])')
>>> B = maxima('matrix ([b1, b2])')
>>> A.dot(B)
matrix([a1*b1,a1*b2],[a2*b1,a2*b2])
imag()[source]#

Return the imaginary part of this Maxima element.

INPUT: none

OUTPUT: Maxima real

EXAMPLES:

sage: maxima('2 + (2/3)*%i').imag()
2/3

>>> from sage.all import *
>>> maxima('2 + (2/3)*%i').imag()
2/3
integral(var='x', min=None, max=None)[source]#

Return the integral of self with respect to the variable x.

INPUT:

  • var – variable

  • min – default: None

  • max – default: None

OUTPUT:

  • the definite integral if xmin is not None

  • an indefinite integral otherwise

EXAMPLES:

sage: maxima('x^2+1').integral()
x^3/3+x
sage: maxima('x^2+ 1 + y^2').integral('y')
y^3/3+x^2*y+y
sage: maxima('x / (x^2+1)').integral()
log(x^2+1)/2
sage: maxima('1/(x^2+1)').integral()
atan(x)
sage: maxima('1/(x^2+1)').integral('x', 0, infinity)
%pi/2
sage: maxima('x/(x^2+1)').integral('x', -1, 1)
0

>>> from sage.all import *
>>> maxima('x^2+1').integral()
x^3/3+x
>>> maxima('x^2+ 1 + y^2').integral('y')
y^3/3+x^2*y+y
>>> maxima('x / (x^2+1)').integral()
log(x^2+1)/2
>>> maxima('1/(x^2+1)').integral()
atan(x)
>>> maxima('1/(x^2+1)').integral('x', Integer(0), infinity)
%pi/2
>>> maxima('x/(x^2+1)').integral('x', -Integer(1), Integer(1))
0

sage: f = maxima('exp(x^2)').integral('x',0,1)
sage: f.sage()
-1/2*I*sqrt(pi)*erf(I)
sage: f.numer()
1.46265174590718...

>>> from sage.all import *
>>> f = maxima('exp(x^2)').integral('x',Integer(0),Integer(1))
>>> f.sage()
-1/2*I*sqrt(pi)*erf(I)
>>> f.numer()
1.46265174590718...
integrate(var='x', min=None, max=None)[source]#

Return the integral of self with respect to the variable x.

INPUT:

  • var – variable

  • min – default: None

  • max – default: None

OUTPUT:

  • the definite integral if xmin is not None

  • an indefinite integral otherwise

EXAMPLES:

sage: maxima('x^2+1').integral()
x^3/3+x
sage: maxima('x^2+ 1 + y^2').integral('y')
y^3/3+x^2*y+y
sage: maxima('x / (x^2+1)').integral()
log(x^2+1)/2
sage: maxima('1/(x^2+1)').integral()
atan(x)
sage: maxima('1/(x^2+1)').integral('x', 0, infinity)
%pi/2
sage: maxima('x/(x^2+1)').integral('x', -1, 1)
0

>>> from sage.all import *
>>> maxima('x^2+1').integral()
x^3/3+x
>>> maxima('x^2+ 1 + y^2').integral('y')
y^3/3+x^2*y+y
>>> maxima('x / (x^2+1)').integral()
log(x^2+1)/2
>>> maxima('1/(x^2+1)').integral()
atan(x)
>>> maxima('1/(x^2+1)').integral('x', Integer(0), infinity)
%pi/2
>>> maxima('x/(x^2+1)').integral('x', -Integer(1), Integer(1))
0

sage: f = maxima('exp(x^2)').integral('x',0,1)
sage: f.sage()
-1/2*I*sqrt(pi)*erf(I)
sage: f.numer()
1.46265174590718...

>>> from sage.all import *
>>> f = maxima('exp(x^2)').integral('x',Integer(0),Integer(1))
>>> f.sage()
-1/2*I*sqrt(pi)*erf(I)
>>> f.numer()
1.46265174590718...
nintegral(var='x', a=0, b=1, desired_relative_error='1e-8', maximum_num_subintervals=200)[source]#

Return a numerical approximation to the integral of self from a to b.

INPUT:

  • var – variable to integrate with respect to

  • a – lower endpoint of integration

  • b – upper endpoint of integration

  • desired_relative_error – (default: ‘1e-8’) the

    desired relative error

  • maximum_num_subintervals – (default: 200)

    maxima number of subintervals

OUTPUT:

  • approximation to the integral

  • estimated absolute error of the

    approximation

  • the number of integrand evaluations

  • an error code:

    • 0 – no problems were encountered

    • 1 – too many subintervals were done

    • 2 – excessive roundoff error

    • 3 – extremely bad integrand behavior

    • 4 – failed to converge

    • 5 – integral is probably divergent or slowly convergent

    • 6 – the input is invalid

EXAMPLES:

sage: maxima('exp(-sqrt(x))').nintegral('x',0,1)
(0.5284822353142306, 4.163...e-11, 231, 0)

>>> from sage.all import *
>>> maxima('exp(-sqrt(x))').nintegral('x',Integer(0),Integer(1))
(0.5284822353142306, 4.163...e-11, 231, 0)

Note that GP also does numerical integration, and can do so to very high precision very quickly:

sage: gp('intnum(x=0,1,exp(-sqrt(x)))')
0.5284822353142307136179049194             # 32-bit
0.52848223531423071361790491935415653022   # 64-bit
sage: _ = gp.set_precision(80)
sage: gp('intnum(x=0,1,exp(-sqrt(x)))')
0.52848223531423071361790491935415653021675547587292866196865279321015401702040079

>>> from sage.all import *
>>> gp('intnum(x=0,1,exp(-sqrt(x)))')
0.5284822353142307136179049194             # 32-bit
0.52848223531423071361790491935415653022   # 64-bit
>>> _ = gp.set_precision(Integer(80))
>>> gp('intnum(x=0,1,exp(-sqrt(x)))')
0.52848223531423071361790491935415653021675547587292866196865279321015401702040079
numer()[source]#

Return numerical approximation to self as a Maxima object.

INPUT: none

OUTPUT: Maxima object

EXAMPLES:

sage: a = maxima('sqrt(2)').numer(); a
1.41421356237309...
sage: type(a)
<class 'sage.interfaces.maxima.MaximaElement'>

>>> from sage.all import *
>>> a = maxima('sqrt(2)').numer(); a
1.41421356237309...
>>> type(a)
<class 'sage.interfaces.maxima.MaximaElement'>
partial_fraction_decomposition(var='x')[source]#

Return the partial fraction decomposition of self with respect to the variable var.

INPUT:

  • var – string

OUTPUT: Maxima object

EXAMPLES:

sage: f = maxima('1/((1+x)*(x-1))')
sage: f.partial_fraction_decomposition('x')
1/(2*(x-1))-1/(2*(x+1))
sage: print(f.partial_fraction_decomposition('x'))
                     1           1
                 --------- - ---------
                 2 (x - 1)   2 (x + 1)

>>> from sage.all import *
>>> f = maxima('1/((1+x)*(x-1))')
>>> f.partial_fraction_decomposition('x')
1/(2*(x-1))-1/(2*(x+1))
>>> print(f.partial_fraction_decomposition('x'))
                     1           1
                 --------- - ---------
                 2 (x - 1)   2 (x + 1)
real()[source]#

Return the real part of this Maxima element.

INPUT: none

OUTPUT: Maxima real

EXAMPLES:

sage: maxima('2 + (2/3)*%i').real()
2

>>> from sage.all import *
>>> maxima('2 + (2/3)*%i').real()
2
str()[source]#

Return string representation of this Maxima object.

INPUT: none

OUTPUT: string

EXAMPLES:

sage: maxima('sqrt(2) + 1/3').str()
'sqrt(2)+1/3'

>>> from sage.all import *
>>> maxima('sqrt(2) + 1/3').str()
'sqrt(2)+1/3'
subst(val)[source]#

Substitute a value or several values into this Maxima object.

INPUT:

  • val – string representing substitution(s) to perform

OUTPUT: Maxima object

EXAMPLES:

sage: maxima('a^2 + 3*a + b').subst('b=2')
a^2+3*a+2
sage: maxima('a^2 + 3*a + b').subst('a=17')
b+340
sage: maxima('a^2 + 3*a + b').subst('a=17, b=2')
342

>>> from sage.all import *
>>> maxima('a^2 + 3*a + b').subst('b=2')
a^2+3*a+2
>>> maxima('a^2 + 3*a + b').subst('a=17')
b+340
>>> maxima('a^2 + 3*a + b').subst('a=17, b=2')
342
class sage.interfaces.maxima_abstract.MaximaAbstractElementFunction(parent, name, defn, args, latex)[source]#

Bases: MaximaAbstractElement

Create a Maxima function with the parent parent, name name, definition defn, arguments args and latex representation latex.

INPUT:

  • parent – an instance of a concrete Maxima interface

  • name – string

  • defn – string

  • args – string; comma separated names of arguments

  • latex – string

OUTPUT: Maxima function

EXAMPLES:

sage: maxima.function('x,y','e^cos(x)')
e^cos(x)

>>> from sage.all import *
>>> maxima.function('x,y','e^cos(x)')
e^cos(x)
arguments(split=True)[source]#

Returns the arguments of this Maxima function.

INPUT:

  • split – boolean; if True return a tuple of strings, otherwise return a string of comma-separated arguments

OUTPUT:

  • a string if split is False

  • a list of strings if split is True

EXAMPLES:

sage: f = maxima.function('x,y','sin(x+y)')
sage: f.arguments()
['x', 'y']
sage: f.arguments(split=False)
'x,y'
sage: f = maxima.function('', 'sin(x)')
sage: f.arguments()
[]

>>> from sage.all import *
>>> f = maxima.function('x,y','sin(x+y)')
>>> f.arguments()
['x', 'y']
>>> f.arguments(split=False)
'x,y'
>>> f = maxima.function('', 'sin(x)')
>>> f.arguments()
[]
definition()[source]#

Returns the definition of this Maxima function as a string.

INPUT: none

OUTPUT: string

EXAMPLES:

sage: f = maxima.function('x,y','sin(x+y)')
sage: f.definition()
'sin(x+y)'

>>> from sage.all import *
>>> f = maxima.function('x,y','sin(x+y)')
>>> f.definition()
'sin(x+y)'
integral(var)[source]#

Returns the integral of self with respect to the variable var.

INPUT:

  • var – a variable

OUTPUT: Maxima function

Note that integrate is an alias of integral.

EXAMPLES:

sage: x,y = var('x,y')
sage: f = maxima.function('x','sin(x)')
sage: f.integral(x)
-cos(x)
sage: f.integral(y)
sin(x)*y

>>> from sage.all import *
>>> x,y = var('x,y')
>>> f = maxima.function('x','sin(x)')
>>> f.integral(x)
-cos(x)
>>> f.integral(y)
sin(x)*y
integrate(var)[source]#

Returns the integral of self with respect to the variable var.

INPUT:

  • var – a variable

OUTPUT: Maxima function

Note that integrate is an alias of integral.

EXAMPLES:

sage: x,y = var('x,y')
sage: f = maxima.function('x','sin(x)')
sage: f.integral(x)
-cos(x)
sage: f.integral(y)
sin(x)*y

>>> from sage.all import *
>>> x,y = var('x,y')
>>> f = maxima.function('x','sin(x)')
>>> f.integral(x)
-cos(x)
>>> f.integral(y)
sin(x)*y
sage.interfaces.maxima_abstract.maxima_console()[source]#

Spawn a new Maxima command-line session.

EXAMPLES:

sage: from sage.interfaces.maxima_abstract import maxima_console
sage: maxima_console()                    # not tested
Maxima 5.46.0 https://maxima.sourceforge.io
...

>>> from sage.all import *
>>> from sage.interfaces.maxima_abstract import maxima_console
>>> maxima_console()                    # not tested
Maxima 5.46.0 https://maxima.sourceforge.io
...
sage.interfaces.maxima_abstract.maxima_version()[source]#

Return Maxima version.

Currently this calls a new copy of Maxima.

EXAMPLES:

sage: from sage.interfaces.maxima_abstract import maxima_version
sage: maxima_version()  # random
'5.41.0'

>>> from sage.all import *
>>> from sage.interfaces.maxima_abstract import maxima_version
>>> maxima_version()  # random
'5.41.0'
sage.interfaces.maxima_abstract.reduce_load_MaximaAbstract_function(parent, defn, args, latex)[source]#

Unpickle a Maxima function.

EXAMPLES:

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

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