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r""" 

Library 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 

 

For this interface, Maxima is loaded into ECL which is itself loaded 

as a C library in Sage. Translations between Sage and Maxima objects 

(which are nothing but wrappers to ECL objects) is made as much as possible 

directly, but falls back to the string based conversion used by the 

classical Maxima Pexpect interface in case no new implementation has been made. 

 

This interface is the one used for calculus by Sage 

and is accessible as `maxima_calculus`:: 

 

sage: maxima_calculus 

Maxima_lib 

 

Only one instance of this interface can be instantiated, 

so the user should not try to instantiate another one, 

which is anyway set to raise an error:: 

 

sage: from sage.interfaces.maxima_lib import MaximaLib 

sage: MaximaLib() 

Traceback (most recent call last): 

... 

RuntimeError: Maxima interface in library mode can only be instantiated once 

 

Changed besselexpand to true in init_code -- automatically simplify bessel functions to trig functions when appropriate when true. Examples: 

 

For some infinite sums, a closed expression can be found. By default, "maxima" is used for that:: 

 

sage: x,n,k = var("x","n","k") 

sage: sum((-x)^n/(factorial(n)*factorial(n+3/2)),n,0,oo) 

-1/2*(2*x*cos(2*sqrt(x)) - sqrt(x)*sin(2*sqrt(x)))/(sqrt(pi)*x^2) 

 

Maxima has some flags that affect how the result gets simplified (By default, besselexpand is false in Maxima; however in 5.39 this test does not show any difference, as, apparently, another expansion path is used):: 

 

sage: maxima_calculus("besselexpand:false") 

false 

sage: x,n,k = var("x","n","k") 

sage: sum((-x)^n/(factorial(n)*factorial(n+3/2)),n,0,oo) 

-1/2*(2*x*cos(2*sqrt(x)) - sqrt(x)*sin(2*sqrt(x)))/(sqrt(pi)*x^2) 

sage: maxima_calculus("besselexpand:true") 

true 

 

""" 

 

#***************************************************************************** 

# Copyright (C) 2005 William Stein <wstein@gmail.com> 

# 

# Distributed under the terms of the GNU General Public License (GPL) 

# 

# This code is distributed in the hope that it will be useful, 

# but WITHOUT ANY WARRANTY; without even the implied warranty of 

# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU 

# General Public License for more details. 

# 

# The full text of the GPL is available at: 

# 

# http://www.gnu.org/licenses/ 

#***************************************************************************** 

from __future__ import print_function 

from __future__ import absolute_import 

from six import string_types 

 

from sage.symbolic.ring import SR 

 

from sage.libs.ecl import EclObject, ecl_eval 

 

from .maxima_abstract import (MaximaAbstract, MaximaAbstractFunction, 

MaximaAbstractElement, MaximaAbstractFunctionElement, 

MaximaAbstractElementFunction) 

from sage.docs.instancedoc import instancedoc 

 

 

## We begin here by initializing Maxima in library mode 

## i.e. loading it into ECL 

ecl_eval("(setf *load-verbose* NIL)") 

ecl_eval("(require 'maxima)") 

ecl_eval("(in-package :maxima)") 

ecl_eval("(setq $nolabels t))") 

ecl_eval("(defvar *MAXIMA-LANG-SUBDIR* NIL)") 

ecl_eval("(set-locale-subdir)") 

ecl_eval("(set-pathnames)") 

ecl_eval("(defun add-lineinfo (x) x)") 

ecl_eval('(defun principal nil (cond ($noprincipal (diverg)) ((not pcprntd) (merror "Divergent Integral"))))') 

ecl_eval("(remprop 'mfactorial 'grind)") # don't use ! for factorials (#11539) 

ecl_eval("(setf $errormsg nil)") 

 

# the following is a direct adaptation of the definition of "retrieve" 

# in the Maxima file macsys.lisp. This routine is normally responsible 

# for displaying a question and returning the answer. We change it to 

# throw an error in which the text of the question is included. We do 

# this by running exactly the same code as in the original definition 

# of "retrieve", but with *standard-output* redirected to a string. 

ecl_eval(r""" 

(defun retrieve (msg flag &aux (print? nil)) 

(declare (special msg flag print?)) 

(or (eq flag 'noprint) (setq print? t)) 

(error 

(concatenate 'string "Maxima asks: " 

(string-trim '(#\Newline) 

(with-output-to-string (*standard-output*) 

(cond ((not print?) 

(setq print? t) 

(princ *prompt-prefix*) 

(princ *prompt-suffix*) 

) 

((null msg) 

(princ *prompt-prefix*) 

(princ *prompt-suffix*) 

) 

((atom msg) 

(format t "~a~a~a" *prompt-prefix* msg *prompt-suffix*) 

) 

((eq flag t) 

(princ *prompt-prefix*) 

(mapc #'princ (cdr msg)) 

(princ *prompt-suffix*) 

) 

(t 

(princ *prompt-prefix*) 

(displa msg) 

(princ *prompt-suffix*) 

) 

)))) 

) 

) 

""") 

 

## Redirection of ECL and Maxima stdout to /dev/null 

ecl_eval(r"""(defparameter *dev-null* (make-two-way-stream 

(make-concatenated-stream) (make-broadcast-stream)))""") 

ecl_eval("(setf original-standard-output *standard-output*)") 

ecl_eval("(setf *standard-output* *dev-null*)") 

#ecl_eval("(setf *error-output* *dev-null*)") 

 

## Default options set in Maxima 

# display2d -- no ascii art output 

# keepfloat -- don't automatically convert floats to rationals 

 

init_code = ['besselexpand : true', 'display2d : false', 'domain : complex', 'keepfloat : true', 

'load(to_poly_solve)', 'load(simplify_sum)', 

'load(abs_integrate)', 'load(diag)'] 

 

 

# Turn off the prompt labels, since computing them *very 

# dramatically* slows down the maxima interpret after a while. 

# See the function makelabel in suprv1.lisp. 

# Many thanks to andrej.vodopivec@gmail.com and also 

# Robert Dodier for figuring this out! 

# See trac # 6818. 

init_code.append('nolabels : true') 

for l in init_code: 

ecl_eval("#$%s$"%l) 

## To get more debug information uncomment the next line 

## should allow to do this through a method 

#ecl_eval("(setf *standard-output* original-standard-output)") 

 

## This is the main function (ECL object) used for evaluation 

# This returns an EclObject 

maxima_eval=ecl_eval(""" 

(defun maxima-eval( form ) 

(let ((result (catch 'macsyma-quit (cons 'maxima_eval (meval form))))) 

;(princ (list "result=" result)) 

;(terpri) 

;(princ (list "$error=" $error)) 

;(terpri) 

(cond 

((and (consp result) (eq (car result) 'maxima_eval)) (cdr result)) 

((eq result 'maxima-error) 

(let ((the-jig (process-error-argl (cddr $error)))) 

(mapc #'set (car the-jig) (cadr the-jig)) 

(error (concatenate 'string 

"Error executing code in Maxima: " 

(with-output-to-string (stream) 

(apply #'mformat stream (cadr $error) 

(caddr the-jig))))) 

)) 

(t 

(let ((the-jig (process-error-argl (cddr $error)))) 

(mapc #'set (car the-jig) (cadr the-jig)) 

(error (concatenate 'string "Maxima condition. result:" 

(princ-to-string result) "$error:" 

(with-output-to-string (stream) 

(apply #'mformat stream (cadr $error) 

(caddr the-jig))))) 

)) 

) 

) 

) 

""") 

 

## Number of instances of this interface 

maxima_lib_instances = 0 

 

## Here we define several useful ECL/Maxima objects 

# The Maxima string function can change the structure of its input 

#maxprint=EclObject("$STRING") 

maxprint=EclObject(r"""(defun mstring-for-sage (form) 

(coerce (mstring form) 'string))""").eval() 

meval=EclObject("MEVAL") 

msetq=EclObject("MSETQ") 

mlist=EclObject("MLIST") 

mequal=EclObject("MEQUAL") 

cadadr=EclObject("CADADR") 

 

max_integrate=EclObject("$INTEGRATE") 

max_sum=EclObject("$SUM") 

max_simplify_sum=EclObject("$SIMPLIFY_SUM") 

max_prod=EclObject("$PRODUCT") 

max_simplify_prod=EclObject("$SIMPLIFY_PRODUCT") 

max_ratsimp=EclObject("$RATSIMP") 

max_limit=EclObject("$LIMIT") 

max_tlimit=EclObject("$TLIMIT") 

max_plus=EclObject("$PLUS") 

max_minus=EclObject("$MINUS") 

max_use_grobner=EclObject("$USE_GROBNER") 

max_to_poly_solve=EclObject("$TO_POLY_SOLVE") 

max_at=EclObject("%AT") 

 

def stdout_to_string(s): 

r""" 

Evaluate command ``s`` and catch Maxima stdout 

(not the result of the command!) into a string. 

 

INPUT: 

 

- ``s`` - string; command to evaluate 

 

OUTPUT: string 

 

This is currently used to implement :meth:`~MaximaLibElement.display2d`. 

 

EXAMPLES:: 

 

sage: from sage.interfaces.maxima_lib import stdout_to_string 

sage: stdout_to_string('1+1') 

'' 

sage: stdout_to_string('disp(1+1)') 

'2\n\n' 

""" 

return ecl_eval(r"""(with-output-to-string (*standard-output*) 

(maxima-eval #$%s$))"""%s).python()[1:-1] 

 

def max_to_string(s): 

r""" 

Return the Maxima string corresponding to this ECL object. 

 

INPUT: 

 

- ``s`` - ECL object 

 

OUTPUT: string 

 

EXAMPLES:: 

 

sage: from sage.interfaces.maxima_lib import maxima_lib, max_to_string 

sage: ecl = maxima_lib(cos(x)).ecl() 

sage: max_to_string(ecl) 

'cos(_SAGE_VAR_x)' 

""" 

return maxprint(s).python()[1:-1] 

 

my_mread=ecl_eval(""" 

(defun my-mread (cmd) 

(caddr (mread (make-string-input-stream cmd)))) 

""") 

 

def parse_max_string(s): 

r""" 

Evaluate string in Maxima without *any* further simplification. 

 

INPUT: 

 

- ``s`` - string 

 

OUTPUT: ECL object 

 

EXAMPLES:: 

 

sage: from sage.interfaces.maxima_lib import parse_max_string 

sage: parse_max_string('1+1') 

<ECL: ((MPLUS) 1 1)> 

""" 

return my_mread('"%s;"'%s) 

 

class MaximaLib(MaximaAbstract): 

""" 

Interface to Maxima as a Library. 

 

INPUT: none 

 

OUTPUT: Maxima interface as a Library 

 

EXAMPLES:: 

 

sage: from sage.interfaces.maxima_lib import MaximaLib, maxima_lib 

sage: isinstance(maxima_lib,MaximaLib) 

True 

 

Only one such interface can be instantiated:: 

 

sage: MaximaLib() 

Traceback (most recent call last): 

... 

RuntimeError: Maxima interface in library mode can only 

be instantiated once 

""" 

def __init__(self): 

""" 

Create an instance of the Maxima interpreter. 

See ``MaximaLib`` for full documentation. 

 

TESTS:: 

 

sage: from sage.interfaces.maxima_lib import MaximaLib, maxima_lib 

sage: MaximaLib == loads(dumps(MaximaLib)) 

True 

sage: maxima_lib == loads(dumps(maxima_lib)) 

True 

 

We make sure labels are turned off (see :trac:`6816`):: 

 

sage: 'nolabels : true' in maxima_lib._MaximaLib__init_code 

True 

""" 

global maxima_lib_instances 

if maxima_lib_instances > 0: 

raise RuntimeError("Maxima interface in library mode can only be instantiated once") 

maxima_lib_instances += 1 

 

global init_code 

self.__init_code = init_code 

 

MaximaAbstract.__init__(self,"maxima_lib") 

self.__seq = 0 

 

def _coerce_from_special_method(self, x): 

r""" 

Coerce ``x`` into self trying to call a special underscore method. 

 

INPUT: 

 

- ``x`` - object to coerce into self 

 

OUTPUT: Maxima element equivalent to ``x`` 

 

EXAMPLES:: 

 

sage: from sage.interfaces.maxima_lib import maxima_lib 

sage: xmax = maxima_lib._coerce_from_special_method(x) 

sage: type(xmax) 

<class 'sage.interfaces.maxima_lib.MaximaLibElement'> 

""" 

if isinstance(x, EclObject): 

return MaximaLibElement(self,self._create(x)) 

else: 

return MaximaAbstract._coerce_from_special_method(self,x) 

 

def __reduce__(self): 

r""" 

Implement __reduce__ for ``MaximaLib``. 

 

INPUT: none 

 

OUTPUT: 

 

A couple consisting of: 

 

- the function to call for unpickling 

 

- a tuple of arguments for the function 

 

EXAMPLES:: 

 

sage: from sage.interfaces.maxima_lib import maxima_lib 

sage: maxima_lib.__reduce__() 

(<function reduce_load_MaximaLib at 0x...>, ()) 

""" 

return reduce_load_MaximaLib, tuple([]) 

 

# This outputs a string 

def _eval_line(self, line, locals=None, reformat=True, **kwds): 

r""" 

Evaluate the line in Maxima. 

 

INPUT: 

 

- ``line`` - string; text to evaluate 

 

- ``locals`` - None (ignored); this is used for compatibility with the 

Sage notebook's generic system interface. 

 

- ``reformat`` - boolean; whether to strip output or not 

 

- ``**kwds`` - All other arguments are currently ignored. 

 

OUTPUT: string representing Maxima output 

 

EXAMPLES:: 

 

sage: from sage.interfaces.maxima_lib import maxima_lib 

sage: maxima_lib._eval_line('1+1') 

'2' 

sage: maxima_lib._eval_line('1+1;') 

'2' 

sage: maxima_lib._eval_line('1+1$') 

'' 

sage: maxima_lib._eval_line('randvar : cos(x)+sin(y)$') 

'' 

sage: maxima_lib._eval_line('randvar') 

'sin(y)+cos(x)' 

""" 

result = '' 

while line: 

ind_dollar=line.find("$") 

ind_semi=line.find(";") 

if ind_dollar == -1 or (ind_semi >=0 and ind_dollar > ind_semi): 

if ind_semi == -1: 

statement = line 

line = '' 

else: 

statement = line[:ind_semi] 

line = line[ind_semi+1:] 

if statement: 

result = ((result + '\n') if result else '') + max_to_string(maxima_eval("#$%s$"%statement)) 

else: 

statement = line[:ind_dollar] 

line = line[ind_dollar+1:] 

if statement: 

maxima_eval("#$%s$" % statement) 

if not reformat: 

return result 

return ''.join([x.strip() for x in result.split()]) 

 

eval = _eval_line 

 

########################################### 

# Direct access to underlying lisp interpreter. 

########################################### 

def lisp(self, cmd): 

""" 

Send a lisp command to maxima. 

 

INPUT: 

 

- ``cmd`` - string 

 

OUTPUT: ECL object 

 

.. note:: 

 

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

 

EXAMPLES:: 

 

sage: from sage.interfaces.maxima_lib import maxima_lib 

sage: maxima_lib.lisp("(+ 2 17)") 

<ECL: 19> 

""" 

return ecl_eval(cmd) 

 

def set(self, var, value): 

""" 

Set the variable var to the given value. 

 

INPUT: 

 

- ``var`` - string 

 

- ``value`` - string 

 

OUTPUT: none 

 

EXAMPLES:: 

 

sage: from sage.interfaces.maxima_lib import maxima_lib 

sage: maxima_lib.set('xxxxx', '2') 

sage: maxima_lib.get('xxxxx') 

'2' 

""" 

if not isinstance(value, string_types): 

raise TypeError 

cmd = '%s : %s$'%(var, value.rstrip(';')) 

self.eval(cmd) 

 

def clear(self, var): 

""" 

Clear the variable named var. 

 

INPUT: 

 

- ``var`` - string 

 

OUTPUT: none 

 

EXAMPLES:: 

 

sage: from sage.interfaces.maxima_lib import maxima_lib 

sage: maxima_lib.set('xxxxx', '2') 

sage: maxima_lib.get('xxxxx') 

'2' 

sage: maxima_lib.clear('xxxxx') 

sage: maxima_lib.get('xxxxx') 

'xxxxx' 

""" 

try: 

self.eval('kill(%s)$'%var) 

except (TypeError, AttributeError): 

pass 

 

def get(self, var): 

""" 

Get the string value of the variable ``var``. 

 

INPUT: 

 

- ``var`` - string 

 

OUTPUT: string 

 

EXAMPLES:: 

 

sage: from sage.interfaces.maxima_lib import maxima_lib 

sage: maxima_lib.set('xxxxx', '2') 

sage: maxima_lib.get('xxxxx') 

'2' 

""" 

s = self.eval('%s;'%var) 

return s 

 

def _create(self, value, name=None): 

r""" 

Create a variable with given value and name. 

 

INPUT: 

 

- ``value`` - string or ECL object 

 

- ``name`` - string (default: None); name to use for the variable, 

an automatically generated name is used if this is none 

 

OUTPUT: 

 

- string; the name of the created variable 

 

EXAMPLES: 

 

Creation from strings:: 

 

sage: from sage.interfaces.maxima_lib import maxima_lib 

sage: maxima_lib._create('3','var3') 

'var3' 

sage: maxima_lib.get('var3') 

'3' 

sage: s = maxima_lib._create('3') 

sage: s # random output 

'sage9' 

sage: s[:4] == 'sage' 

True 

 

And from ECL object:: 

 

sage: c = maxima_lib(x+cos(19)).ecl() 

sage: maxima_lib._create(c,'m') 

'm' 

sage: maxima_lib.get('m') 

'_SAGE_VAR_x+cos(19)' 

sage: maxima_lib.clear('m') 

""" 

name = self._next_var_name() if name is None else name 

try: 

if isinstance(value,EclObject): 

maxima_eval([[msetq],cadadr("#$%s$#$"%name),value]) 

else: 

self.set(name, value) 

except RuntimeError as error: 

s = str(error) 

if "Is" in s: # Maxima asked for a condition 

self._missing_assumption(s) 

else: 

raise 

return name 

 

def _function_class(self): 

r""" 

Return the Python class of Maxima functions. 

 

INPUT: none 

 

OUTPUT: type 

 

EXAMPLES:: 

 

sage: from sage.interfaces.maxima_lib import maxima_lib 

sage: maxima_lib._function_class() 

<class 'sage.interfaces.interface.InterfaceFunction'> 

""" 

return MaximaLibFunction 

 

def _object_class(self): 

r""" 

Return the Python class of Maxima elements. 

 

INPUT: none 

 

OUTPUT: type 

 

EXAMPLES:: 

 

sage: from sage.interfaces.maxima_lib import maxima_lib 

sage: maxima_lib._object_class() 

<class 'sage.interfaces.maxima_lib.MaximaLibElement'> 

""" 

return MaximaLibElement 

 

def _function_element_class(self): 

r""" 

Return the Python class of Maxima functions of elements. 

 

INPUT: none 

 

OUTPUT: type 

 

EXAMPLES:: 

 

sage: from sage.interfaces.maxima_lib import maxima_lib 

sage: maxima_lib._function_element_class() 

<class 'sage.interfaces.interface.InterfaceFunctionElement'> 

""" 

return MaximaLibFunctionElement 

 

def _object_function_class(self): 

r""" 

Return the Python class of Maxima user-defined functions. 

 

INPUT: none 

 

OUTPUT: type 

 

EXAMPLES:: 

 

sage: from sage.interfaces.maxima_lib import maxima_lib 

sage: maxima_lib._object_function_class() 

<class 'sage.interfaces.maxima_lib.MaximaLibElementFunction'> 

""" 

return MaximaLibElementFunction 

 

## some helper functions to wrap the calculus use of the maxima interface. 

## these routines expect arguments living in the symbolic ring 

## and return something that is hopefully coercible into the symbolic 

## ring again. 

 

def sr_integral(self,*args): 

""" 

Helper function to wrap calculus use of Maxima's integration. 

 

TESTS:: 

 

sage: a,b=var('a,b') 

sage: integrate(1/(x^3 *(a+b*x)^(1/3)),x) 

Traceback (most recent call last): 

... 

ValueError: Computation failed since Maxima requested additional 

constraints; using the 'assume' command before evaluation 

*may* help (example of legal syntax is 'assume(a>0)', see 

`assume?` for more details) 

Is a positive or negative? 

sage: assume(a>0) 

sage: integrate(1/(x^3 *(a+b*x)^(1/3)),x) 

2/9*sqrt(3)*b^2*arctan(1/3*sqrt(3)*(2*(b*x + a)^(1/3) + a^(1/3))/a^(1/3))/a^(7/3) - 1/9*b^2*log((b*x + a)^(2/3) + (b*x + a)^(1/3)*a^(1/3) + a^(2/3))/a^(7/3) + 2/9*b^2*log((b*x + a)^(1/3) - a^(1/3))/a^(7/3) + 1/6*(4*(b*x + a)^(5/3)*b^2 - 7*(b*x + a)^(2/3)*a*b^2)/((b*x + a)^2*a^2 - 2*(b*x + a)*a^3 + a^4) 

sage: var('x, n') 

(x, n) 

sage: integral(x^n,x) 

Traceback (most recent call last): 

... 

ValueError: Computation failed since Maxima requested additional 

constraints; using the 'assume' command before evaluation 

*may* help (example of legal syntax is 'assume(n>0)', 

see `assume?` for more details) 

Is n equal to -1? 

sage: assume(n+1>0) 

sage: integral(x^n,x) 

x^(n + 1)/(n + 1) 

sage: forget() 

sage: assumptions() # Check the assumptions really were forgotten 

[] 

 

Make sure the abs_integrate package is being used, 

:trac:`11483`. The following are examples from the Maxima 

abs_integrate documentation:: 

 

sage: integrate(abs(x), x) 

1/2*x*abs(x) 

 

:: 

 

sage: integrate(sgn(x) - sgn(1-x), x) 

abs(x - 1) + abs(x) 

 

This is a known bug in Sage symbolic limits code, see 

:trac:`17892` and https://sourceforge.net/p/maxima/bugs/3237/ :: 

 

sage: integrate(1 / (1 + abs(x-5)), x, -5, 6) # not tested -- known bug 

log(11) + log(2) 

 

:: 

 

sage: integrate(1/(1 + abs(x)), x) 

1/2*(log(x + 1) + log(-x + 1))*sgn(x) + 1/2*log(x + 1) - 1/2*log(-x + 1) 

 

:: 

 

sage: integrate(cos(x + abs(x)), x) 

-1/2*x*sgn(x) + 1/4*(sgn(x) + 1)*sin(2*x) + 1/2*x 

 

The last example relies on the following simplification:: 

 

sage: maxima("realpart(signum(x))") 

signum(x) 

 

An example from sage-support thread e641001f8b8d1129:: 

 

sage: f = e^(-x^2/2)/sqrt(2*pi) * sgn(x-1) 

sage: integrate(f, x, -Infinity, Infinity) 

-erf(1/2*sqrt(2)) 

 

From :trac:`8624`:: 

 

sage: integral(abs(cos(x))*sin(x),(x,pi/2,pi)) 

1/2 

 

:: 

 

sage: integrate(sqrt(x + sqrt(x)), x).canonicalize_radical() 

1/12*((8*x - 3)*x^(1/4) + 2*x^(3/4))*sqrt(sqrt(x) + 1) + 1/8*log(sqrt(sqrt(x) + 1) + x^(1/4)) - 1/8*log(sqrt(sqrt(x) + 1) - x^(1/4)) 

 

And :trac:`11594`:: 

 

sage: integrate(abs(x^2 - 1), x, -2, 2) 

4 

 

This definite integral returned zero (incorrectly) in at least 

Maxima 5.23. The correct answer is now given (:trac:`11591`):: 

 

sage: f = (x^2)*exp(x) / (1+exp(x))^2 

sage: integrate(f, (x, -infinity, infinity)) 

1/3*pi^2 

 

Sometimes one needs different simplification settings, such as 

``radexpand``, to compute an integral (see :trac:`10955`):: 

 

sage: f = sqrt(x + 1/x^2) 

sage: maxima = sage.calculus.calculus.maxima 

sage: maxima('radexpand') 

true 

sage: integrate(f, x) 

integrate(sqrt(x + 1/x^2), x) 

sage: maxima('radexpand: all') 

all 

sage: g = integrate(f, x); g 

2/3*sqrt(x^3 + 1) - 1/3*log(sqrt(x^3 + 1) + 1) + 1/3*log(sqrt(x^3 + 1) - 1) 

sage: (f - g.diff(x)).canonicalize_radical() 

0 

sage: maxima('radexpand: true') 

true 

 

The following integral was computed incorrectly in versions of 

Maxima before 5.27 (see :trac:`12947`):: 

 

sage: a = integrate(x*cos(x^3),(x,0,1/2)).n() 

sage: a.real() 

0.124756040961038 

sage: a.imag().abs() < 3e-17 

True 

 

""" 

try: 

return max_to_sr(maxima_eval(([max_integrate],[sr_to_max(SR(a)) for a in args]))) 

except RuntimeError as error: 

s = str(error) 

if "Divergent" in s or "divergent" in s: 

# in pexpect interface, one looks for this - e.g. integrate(1/x^3,x,-1,3) gives a principal value 

# if "divergent" in s or 'Principal Value' in s: 

raise ValueError("Integral is divergent.") 

elif "Is" in s: # Maxima asked for a condition 

self._missing_assumption(s) 

else: 

raise 

 

def sr_sum(self,*args): 

""" 

Helper function to wrap calculus use of Maxima's summation. 

 

TESTS: 

 

Check that :trac:`16224` is fixed:: 

 

sage: k = var('k') 

sage: sum(x^(2*k)/factorial(2*k), k, 0, oo).canonicalize_radical() 

cosh(x) 

 

:: 

 

sage: x, y, k, n = var('x, y, k, n') 

sage: sum(binomial(n,k) * x^k * y^(n-k), k, 0, n) 

(x + y)^n 

sage: q, a = var('q, a') 

sage: sum(a*q^k, k, 0, oo) 

Traceback (most recent call last): 

... 

ValueError: Computation failed since Maxima requested additional 

constraints; using the 'assume' command before evaluation *may* help 

(example of legal syntax is 'assume(abs(q)-1>0)', see `assume?` 

for more details) 

Is abs(q)-1 positive, negative or zero? 

sage: assume(q > 1) 

sage: sum(a*q^k, k, 0, oo) 

Traceback (most recent call last): 

... 

ValueError: Sum is divergent. 

sage: forget() 

sage: assume(abs(q) < 1) 

sage: sum(a*q^k, k, 0, oo) 

-a/(q - 1) 

sage: forget() 

sage: assumptions() # check the assumptions were really forgotten 

[] 

 

Taking the sum of all natural numbers informs us that the sum 

is divergent. Maxima (before 5.29.1) used to ask questions 

about `m`, leading to a different error (see :trac:`11990`):: 

 

sage: m = var('m') 

sage: sum(m, m, 0, infinity) 

Traceback (most recent call last): 

... 

ValueError: Sum is divergent. 

 

An error with an infinite sum in Maxima (before 5.30.0, 

see :trac:`13712`):: 

 

sage: n = var('n') 

sage: sum(1/((2*n-1)^2*(2*n+1)^2*(2*n+3)^2), n, 0, oo) 

3/256*pi^2 

 

Maxima correctly detects division by zero in a symbolic sum 

(see :trac:`11894`):: 

 

sage: sum(1/(m^4 + 2*m^3 + 3*m^2 + 2*m)^2, m, 0, infinity) 

Traceback (most recent call last): 

... 

RuntimeError: ECL says: Error executing code in Maxima: Zero to negative power computed. 

 

Similar situation for :trac:`12410`:: 

 

sage: x = var('x') 

sage: sum(1/x*(-1)^x, x, 0, oo) 

Traceback (most recent call last): 

... 

RuntimeError: ECL says: Error executing code in Maxima: Zero to negative power computed. 

 

""" 

try: 

return max_to_sr(maxima_eval([[max_ratsimp],[[max_simplify_sum],([max_sum],[sr_to_max(SR(a)) for a in args])]])); 

except RuntimeError as error: 

s = str(error) 

if "divergent" in s: 

# in pexpect interface, one looks for this; 

# could not find an example where 'Pole encountered' occurred, though 

# if "divergent" in s or 'Pole encountered' in s: 

raise ValueError("Sum is divergent.") 

elif "Is" in s: # Maxima asked for a condition 

self._missing_assumption(s) 

else: 

raise 

 

def sr_prod(self,*args): 

""" 

Helper function to wrap calculus use of Maxima's product. 

 

TESTS:: 

 

sage: from sage.calculus.calculus import symbolic_product 

sage: _ = var('n') 

sage: symbolic_product(x,x,1,n) 

factorial(n) 

sage: symbolic_product(2*x,x,1,n) 

2^n*factorial(n) 

 

""" 

try: 

return max_to_sr(maxima_eval([[max_ratsimp],[[max_simplify_prod],([max_prod],[sr_to_max(SR(a)) for a in args])]])); 

except RuntimeError as error: 

s = str(error) 

if "divergent" in s: 

raise ValueError("Product is divergent.") 

elif "Is" in s: # Maxima asked for a condition 

self._missing_assumption(s) 

else: 

raise 

 

 

def sr_limit(self, expr, v, a, dir=None): 

""" 

Helper function to wrap calculus use of Maxima's limits. 

 

TESTS:: 

 

sage: f = (1+1/x)^x 

sage: limit(f,x = oo) 

e 

sage: limit(f,x = 5) 

7776/3125 

sage: limit(f,x = 1.2) 

2.06961575467... 

sage: var('a') 

a 

sage: limit(x^a,x=0) 

Traceback (most recent call last): 

... 

ValueError: Computation failed since Maxima requested additional 

constraints; using the 'assume' command before evaluation 

*may* help (example of legal syntax is 'assume(a>0)', see `assume?` 

for more details) 

Is a positive, negative or zero? 

sage: assume(a>0) 

sage: limit(x^a,x=0) 

Traceback (most recent call last): 

... 

ValueError: Computation failed ... 

Is a an integer? 

sage: assume(a,'integer') 

sage: assume(a,'even') # Yes, Maxima will ask this too 

sage: limit(x^a,x=0) 

0 

sage: forget() 

sage: assumptions() # check the assumptions were really forgotten 

[] 

 

The second limit below was computed incorrectly prior to 

Maxima 5.24 (:trac:`10868`):: 

 

sage: f(n) = 2 + 1/factorial(n) 

sage: limit(f(n), n=infinity) 

2 

sage: limit(1/f(n), n=infinity) 

1/2 

 

The limit below was computed incorrectly prior to Maxima 5.30 

(see :trac:`13526`):: 

 

sage: n = var('n') 

sage: l = (3^n + (-2)^n) / (3^(n+1) + (-2)^(n+1)) 

sage: l.limit(n=oo) 

1/3 

 

The following limit computation used to incorrectly return 0 

or infinity, depending on the domain (see :trac:`15033`):: 

 

sage: m = sage.calculus.calculus.maxima 

sage: _ = m.eval('domain: real') # much faster than 'domain: complex' 

sage: limit(gamma(x + 1/2)/(sqrt(x)*gamma(x)), x=infinity) 

1 

sage: _ = m.eval('domain: complex') 

 

""" 

try: 

L = [sr_to_max(SR(aa)) for aa in [expr, v, a]] 

if dir == "plus": 

L.append(max_plus) 

elif dir == "minus": 

L.append(max_minus) 

return max_to_sr(maxima_eval(([max_limit], L))) 

except RuntimeError as error: 

s = str(error) 

if "Is" in s: # Maxima asked for a condition 

self._missing_assumption(s) 

else: 

raise 

 

def sr_tlimit(self, expr, v, a, dir=None): 

""" 

Helper function to wrap calculus use of Maxima's Taylor series limits. 

 

TESTS:: 

 

sage: f = (1+1/x)^x 

sage: limit(f, x = I, taylor=True) 

(-I + 1)^I 

""" 

L = [sr_to_max(SR(aa)) for aa in [expr, v, a]] 

if dir == "plus": 

L.append(max_plus) 

elif dir == "minus": 

L.append(max_minus) 

return max_to_sr(maxima_eval(([max_tlimit], L))) 

 

def _missing_assumption(self,errstr): 

""" 

Helper function for unified handling of failed computation because an 

assumption was missing. 

 

EXAMPLES:: 

 

sage: from sage.interfaces.maxima_lib import maxima_lib 

sage: maxima_lib._missing_assumption('Is xyz a thing?') 

Traceback (most recent call last): 

... 

ValueError: Computation failed ... 

Is xyz a thing? 

""" 

j = errstr.find('Is ') 

errstr = errstr[j:] 

jj = 2 

if errstr[3] == ' ': 

jj = 3 

k = errstr.find(' ',jj+1) 

 

outstr = "Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume("\ 

+ errstr[jj+1:k] +">0)', see `assume?` for more details)\n" + errstr 

outstr = outstr.replace('_SAGE_VAR_','') 

raise ValueError(outstr) 

 

def is_MaximaLibElement(x): 

r""" 

Returns True if x is of type MaximaLibElement. 

 

EXAMPLES:: 

 

sage: from sage.interfaces.maxima_lib import maxima_lib, is_MaximaLibElement 

sage: m = maxima_lib(1) 

sage: is_MaximaLibElement(m) 

True 

sage: is_MaximaLibElement(1) 

False 

""" 

return isinstance(x, MaximaLibElement) 

 

 

@instancedoc 

class MaximaLibElement(MaximaAbstractElement): 

r""" 

Element of Maxima through library interface. 

 

EXAMPLES: 

 

Elements of this class should not be created directly. 

The targeted parent should be used instead:: 

 

sage: from sage.interfaces.maxima_lib import maxima_lib 

sage: maxima_lib(4) 

4 

sage: maxima_lib(log(x)) 

log(_SAGE_VAR_x) 

""" 

 

def ecl(self): 

r""" 

Return the underlying ECL object of this MaximaLib object. 

 

INPUT: none 

 

OUTPUT: ECL object 

 

EXAMPLES:: 

 

sage: from sage.interfaces.maxima_lib import maxima_lib 

sage: maxima_lib(x+cos(19)).ecl() 

<ECL: ((MPLUS SIMP) ((%COS SIMP) 19) |$_SAGE_VAR_x|)> 

""" 

try: 

return self._ecl 

except AttributeError: 

self._ecl=maxima_eval("#$%s$"%self._name) 

return self._ecl 

 

def to_poly_solve(self,vars,options=""): 

r""" 

Use Maxima's to_poly_solver package. 

 

INPUT: 

 

- ``vars`` - symbolic expressions 

 

- ``options`` - string (default="") 

 

OUTPUT: Maxima object 

 

EXAMPLES: 

 

The zXXX below are names for arbitrary integers and 

subject to change:: 

 

sage: from sage.interfaces.maxima_lib import maxima_lib 

sage: sol = maxima_lib(sin(x) == 0).to_poly_solve(x) 

sage: sol.sage() 

[[x == pi*z...]] 

""" 

if options.find("use_grobner=true") != -1: 

cmd=EclObject([[max_to_poly_solve], self.ecl(), sr_to_max(vars), 

[[mequal],max_use_grobner,True]]) 

else: 

cmd=EclObject([[max_to_poly_solve], self.ecl(), sr_to_max(vars)]) 

return self.parent()(maxima_eval(cmd)) 

 

def display2d(self, onscreen=True): 

r""" 

Return the 2d representation of this Maxima object. 

 

INPUT: 

 

- ``onscreen`` - boolean (default: True); whether to print or return 

 

OUTPUT: 

 

The representation is printed if onscreen is set to True 

and returned as a string otherwise. 

 

EXAMPLES:: 

 

sage: from sage.interfaces.maxima_lib import maxima_lib 

sage: F = maxima_lib('x^5 - y^5').factor() 

sage: F.display2d() 

4 3 2 2 3 4 

- (y - x) (y + x y + x y + x y + x ) 

""" 

self._check_valid() 

P = self.parent() 

P._eval_line('display2d : true$') 

s = stdout_to_string('disp(%s)'%self.name()) 

#s = P._eval_line('disp(%s)$'%self.name()) 

P._eval_line('display2d : false$') 

s = s.strip('\r\n') 

 

# if ever want to dedent, see 

# http://mail.python.org/pipermail/python-list/2006-December/420033.html 

if onscreen: 

print(s) 

else: 

return s 

 

 

MaximaLibFunctionElement = MaximaAbstractFunctionElement 

MaximaLibFunction = MaximaAbstractFunction 

 

@instancedoc 

class MaximaLibElementFunction(MaximaLibElement, MaximaAbstractElementFunction): 

pass 

 

 

# The (unique) instance 

maxima_lib = MaximaLib() 

maxima = maxima_lib 

 

 

def reduce_load_MaximaLib(): 

r""" 

Unpickle the (unique) Maxima library interface. 

 

EXAMPLES:: 

 

sage: from sage.interfaces.maxima_lib import reduce_load_MaximaLib 

sage: reduce_load_MaximaLib() 

Maxima_lib 

""" 

return maxima_lib 

 

 

############################################# 

# Smart translations between SR and Maxima 

############################################# 

 

import sage.rings.real_double 

import sage.symbolic.expression 

import sage.symbolic.integration.integral 

from sage.symbolic.operators import FDerivativeOperator, add_vararg, mul_vararg 

 

car=EclObject("car") 

cdr=EclObject("cdr") 

caar=EclObject("caar") 

cadr=EclObject("cadr") 

cddr=EclObject("cddr") 

caddr=EclObject("caddr") 

caaadr=EclObject("caaadr") 

cadadr=EclObject("cadadr") 

meval=EclObject("meval") 

NIL=EclObject("NIL") 

lisp_length=EclObject("length") 

 

## Dictionaries for standard operators 

sage_op_dict = { 

sage.functions.other.abs : "MABS", 

add_vararg : "MPLUS", 

sage.symbolic.expression.operator.truediv : "MQUOTIENT", 

sage.symbolic.expression.operator.eq : "MEQUAL", 

sage.symbolic.expression.operator.ge : "MGEQP", 

sage.symbolic.expression.operator.gt : "MGREATERP", 

sage.symbolic.expression.operator.le : "MLEQP", 

sage.symbolic.expression.operator.lt : "MLESSP", 

mul_vararg : "MTIMES", 

sage.symbolic.expression.operator.ne : "MNOTEQUAL", 

sage.symbolic.expression.operator.neg : "MMINUS", 

sage.symbolic.expression.operator.pow : "MEXPT", 

sage.symbolic.expression.operator.or_ : "MOR", 

sage.symbolic.expression.operator.and_ : "MAND", 

sage.functions.log.ln : "%LOG", 

sage.functions.log.log : "%LOG", 

sage.functions.log.lambert_w : "%LAMBERT_W", 

sage.functions.other.factorial : "MFACTORIAL", 

sage.functions.error.erf : "%ERF", 

sage.functions.gamma.gamma_inc : "%GAMMA_INCOMPLETE", 

} 

#we compile the dictionary 

sage_op_dict = dict([(k,EclObject(sage_op_dict[k])) for k in sage_op_dict]) 

max_op_dict = dict([(sage_op_dict[k],k) for k in sage_op_dict]) 

 

 

## Here we correct the dictionaries for some simple operators 

 

def sage_rat(x,y): 

r""" 

Return quotient x/y. 

 

INPUT: 

 

- ``x`` - integer 

 

- ``y`` - integer 

 

OUTPUT: rational 

 

EXAMPLES:: 

 

sage: from sage.interfaces.maxima_lib import sage_rat 

sage: sage_rat(1,7) 

1/7 

""" 

return x/y 

 

mplus=EclObject("MPLUS") 

mtimes=EclObject("MTIMES") 

rat=EclObject("RAT") 

max_op_dict[mplus]=add_vararg 

max_op_dict[mtimes]=mul_vararg 

max_op_dict[rat]=sage_rat 

 

 

## Here we build dictionaries for operators needing special conversions. 

ratdisrep = EclObject("ratdisrep") 

mrat = EclObject("MRAT") 

mqapply = EclObject("MQAPPLY") 

max_li = EclObject("$LI") 

max_psi = EclObject("$PSI") 

max_hyper = EclObject("$%F") 

max_array = EclObject("ARRAY") 

mdiff = EclObject("%DERIVATIVE") 

max_lambert_w = sage_op_dict[sage.functions.log.lambert_w] 

max_harmo = EclObject("$GEN_HARMONIC_NUMBER") 

 

def mrat_to_sage(expr): 

r""" 

Convert a Maxima MRAT expression to Sage SR. 

 

INPUT: 

 

- ``expr`` - ECL object; a Maxima MRAT expression 

 

OUTPUT: symbolic expression 

 

Maxima has an optimised representation for multivariate 

rational expressions. The easiest way to translate those 

to SR is by first asking Maxima to give the generic representation 

of the object. That is what RATDISREP does in Maxima. 

 

EXAMPLES:: 

 

sage: from sage.interfaces.maxima_lib import maxima_lib, mrat_to_sage 

sage: var('x y z') 

(x, y, z) 

sage: c = maxima_lib((x+y^2+z^9)/x^6+z^8/y).rat() 

sage: c 

(_SAGE_VAR_y*_SAGE_VAR_z^9+_SAGE_VAR_x^6*_SAGE_VAR_z^8+_SAGE_VAR_y^3+_SAGE_VAR_x*_SAGE_VAR_y)/(_SAGE_VAR_x^6*_SAGE_VAR_y) 

sage: c.ecl() 

<ECL: ((MRAT SIMP (|$_SAGE_VAR_x| |$_SAGE_VAR_y| |$_SAGE_VAR_z|) 

...> 

sage: mrat_to_sage(c.ecl()) 

(x^6*z^8 + y*z^9 + y^3 + x*y)/(x^6*y) 

""" 

return max_to_sr(meval(EclObject([[ratdisrep],expr]))) 

 

def mqapply_to_sage(expr): 

r""" 

Special conversion rule for MQAPPLY expressions. 

 

INPUT: 

 

- ``expr`` - ECL object; a Maxima MQAPPLY expression 

 

OUTPUT: symbolic expression 

 

MQAPPLY is used for function as li[x](y) and psi[x](y). 

 

EXAMPLES:: 

 

sage: from sage.interfaces.maxima_lib import maxima_lib, mqapply_to_sage 

sage: c = maxima_lib('li[2](3)') 

sage: c.ecl() 

<ECL: ((MQAPPLY SIMP) (($LI SIMP ARRAY) 2) 3)> 

sage: mqapply_to_sage(c.ecl()) 

dilog(3) 

""" 

if caaadr(expr) == max_li: 

return sage.functions.log.polylog(max_to_sr(cadadr(expr)), 

max_to_sr(caddr(expr))) 

if caaadr(expr) == max_psi: 

return sage.functions.gamma.psi(max_to_sr(cadadr(expr)), 

max_to_sr(caddr(expr))) 

if caaadr(expr) == max_hyper: 

return sage.functions.hypergeometric.hypergeometric(mlist_to_sage(car(cdr(cdr(expr)))), 

mlist_to_sage(car(cdr(cdr(cdr(expr))))), 

max_to_sr(car(cdr(cdr(cdr(cdr(expr))))))) 

else: 

op=max_to_sr(cadr(expr)) 

max_args=cddr(expr) 

args=[max_to_sr(a) for a in max_args] 

return op(*args) 

 

def mdiff_to_sage(expr): 

r""" 

Special conversion rule for %DERIVATIVE expressions. 

 

INPUT: 

 

- ``expr`` - ECL object; a Maxima %DERIVATIVE expression 

 

OUTPUT: symbolic expression 

 

EXAMPLES:: 

 

sage: from sage.interfaces.maxima_lib import maxima_lib, mdiff_to_sage 

sage: f = maxima_lib('f(x)').diff('x',4) 

sage: f.ecl() 

<ECL: ((%DERIVATIVE SIMP) (($F SIMP) $X) $X 4)> 

sage: mdiff_to_sage(f.ecl()) 

diff(f(x), x, x, x, x) 

""" 

return max_to_sr(expr.cadr()).diff(*[max_to_sr(e) for e in expr.cddr()]) 

 

def mlist_to_sage(expr): 

r""" 

Special conversion rule for MLIST expressions. 

 

INPUT: 

 

- ``expr`` - ECL object; a Maxima MLIST expression (i.e., a list) 

 

OUTPUT: a Python list of converted expressions. 

 

EXAMPLES:: 

 

sage: from sage.interfaces.maxima_lib import maxima_lib, mlist_to_sage 

sage: L=maxima_lib("[1,2,3]") 

sage: L.ecl() 

<ECL: ((MLIST SIMP) 1 2 3)> 

sage: mlist_to_sage(L.ecl()) 

[1, 2, 3] 

""" 

return [max_to_sr(x) for x in expr.cdr()] 

 

def max_at_to_sage(expr): 

r""" 

Special conversion rule for AT expressions. 

 

INPUT: 

 

- ``expr`` - ECL object; a Maxima AT expression 

 

OUTPUT: symbolic expression 

 

EXAMPLES:: 

 

sage: from sage.interfaces.maxima_lib import maxima_lib, max_at_to_sage 

sage: a=maxima_lib("'at(f(x,y,z),[x=1,y=2,z=3])") 

sage: a 

'at(f(x,y,z),[x=1,y=2,z=3]) 

sage: max_at_to_sage(a.ecl()) 

f(1, 2, 3) 

sage: a=maxima_lib("'at(f(x,y,z),x=1)") 

sage: a 

'at(f(x,y,z),x=1) 

sage: max_at_to_sage(a.ecl()) 

f(1, y, z) 

""" 

arg=max_to_sr(expr.cadr()) 

subsarg=caddr(expr) 

if caar(subsarg)==mlist: 

subsvalues=dict( (v.lhs(),v.rhs()) for v in max_to_sr(subsarg)) 

else: 

v=max_to_sr(subsarg) 

subsvalues=dict([(v.lhs(),v.rhs())]) 

return SR(arg).subs(subsvalues) 

 

def dummy_integrate(expr): 

r""" 

We would like to simply tie Maxima's integrate to 

sage.calculus.calculus.dummy_integrate, but we're being 

imported there so to avoid circularity we define it here. 

 

INPUT: 

 

- ``expr`` - ECL object; a Maxima %INTEGRATE expression 

 

OUTPUT: symbolic expression 

 

EXAMPLES:: 

 

sage: from sage.interfaces.maxima_lib import maxima_lib, dummy_integrate 

sage: f = maxima_lib('f(x)').integrate('x') 

sage: f.ecl() 

<ECL: ((%INTEGRATE SIMP) (($F SIMP) $X) $X)> 

sage: dummy_integrate(f.ecl()) 

integrate(f(x), x) 

 

:: 

 

sage: f = maxima_lib('f(x)').integrate('x',0,10) 

sage: f.ecl() 

<ECL: ((%INTEGRATE SIMP) (($F SIMP) $X) $X 0 10)> 

sage: dummy_integrate(f.ecl()) 

integrate(f(x), x, 0, 10) 

""" 

args=[max_to_sr(a) for a in cdr(expr)] 

if len(args) == 4 : 

return sage.symbolic.integration.integral.definite_integral(*args, 

hold=True) 

else: 

return sage.symbolic.integration.integral.indefinite_integral(*args, 

hold=True) 

 

def max_harmonic_to_sage(expr): 

""" 

EXAMPLES:: 

 

sage: from sage.interfaces.maxima_lib import maxima_lib, max_to_sr 

sage: c=maxima_lib(harmonic_number(x,2)) 

sage: c.ecl() 

<ECL: (($GEN_HARMONIC_NUMBER SIMP) 2 |$_SAGE_VAR_x|)> 

sage: max_to_sr(c.ecl()) 

harmonic_number(x, 2) 

""" 

return sage.functions.log.harmonic_number(max_to_sr(caddr(expr)), 

max_to_sr(cadr(expr))) 

 

## The dictionaries 

special_max_to_sage={ 

mrat : mrat_to_sage, 

mqapply : mqapply_to_sage, 

mdiff : mdiff_to_sage, 

EclObject("%INTEGRATE") : dummy_integrate, 

max_at : max_at_to_sage, 

mlist : mlist_to_sage, 

max_harmo : max_harmonic_to_sage 

} 

 

special_sage_to_max={ 

sage.functions.log.polylog : lambda N,X : [[mqapply],[[max_li, max_array],N],X], 

sage.functions.gamma.psi1 : lambda X : [[mqapply],[[max_psi, max_array],0],X], 

sage.functions.gamma.psi2 : lambda N,X : [[mqapply],[[max_psi, max_array],N],X], 

sage.functions.log.lambert_w : lambda N,X : [[max_lambert_w], X] if N==EclObject(0) else [[mqapply],[[max_lambert_w, max_array],N],X], 

sage.functions.log.harmonic_number : lambda N,X : [[max_harmo],X,N], 

sage.functions.hypergeometric.hypergeometric : lambda A, B, X : [[mqapply],[[max_hyper, max_array],lisp_length(A.cdr()),lisp_length(B.cdr())],A,B,X] 

} 

 

 

## Dictionaries for symbols 

sage_sym_dict={} 

max_sym_dict={} 

 

 

## Generic conversion functions 

 

max_i=EclObject("$%I") 

def pyobject_to_max(obj): 

r""" 

Convert a (simple) Python object into a Maxima object. 

 

INPUT: 

 

- ``expr`` - Python object 

 

OUTPUT: ECL object 

 

.. note:: 

 

This uses functions defined in sage.libs.ecl. 

 

EXAMPLES:: 

 

sage: from sage.interfaces.maxima_lib import pyobject_to_max 

sage: pyobject_to_max(4) 

<ECL: 4> 

sage: pyobject_to_max('z') 

<ECL: Z> 

sage: var('x') 

x 

sage: pyobject_to_max(x) 

Traceback (most recent call last): 

... 

TypeError: Unimplemented type for python_to_ecl 

""" 

if isinstance(obj,sage.rings.rational.Rational): 

return EclObject(obj) if (obj.denom().is_one()) else EclObject([[rat], obj.numer(),obj.denom()]) 

elif isinstance(obj,sage.rings.number_field.number_field_element_quadratic.NumberFieldElement_quadratic) and obj.parent().defining_polynomial().list() == [1,0,1]: 

re, im = obj.list() 

return EclObject([[mplus], pyobject_to_max(re), [[mtimes], pyobject_to_max(im), max_i]]) 

return EclObject(obj) 

 

# This goes from SR to EclObject 

def sr_to_max(expr): 

r""" 

Convert a symbolic expression into a Maxima object. 

 

INPUT: 

 

- ``expr`` - symbolic expression 

 

OUTPUT: ECL object 

 

EXAMPLES:: 

 

sage: from sage.interfaces.maxima_lib import sr_to_max 

sage: var('x') 

x 

sage: sr_to_max(x) 

<ECL: $X> 

sage: sr_to_max(cos(x)) 

<ECL: ((%COS) $X)> 

sage: f = function('f')(x) 

sage: sr_to_max(f.diff()) 

<ECL: ((%DERIVATIVE) (($F) $X) $X 1)> 

 

TESTS: 

 

We should be able to convert derivatives evaluated at a point, 

:trac:`12796`:: 

 

sage: from sage.interfaces.maxima_lib import sr_to_max, max_to_sr 

sage: f = function('f') 

sage: f_prime = f(x).diff(x) 

sage: max_to_sr(sr_to_max(f_prime(x = 1))) 

D[0](f)(1) 

 

""" 

global sage_op_dict, max_op_dict 

global sage_sym_dict, max_sym_dict 

if isinstance(expr,list) or isinstance(expr,tuple): 

return EclObject(([mlist],[sr_to_max(e) for e in expr])) 

op = expr.operator() 

if op: 

# Stolen from sage.symbolic.expression_conversion 

# Should be defined in a function and then put in special_sage_to_max 

# For that, we should change the API of the functions there 

# (we need to have access to op, not only to expr.operands() 

if isinstance(op, FDerivativeOperator): 

from sage.symbolic.ring import is_SymbolicVariable 

args = expr.operands() 

if (not all(is_SymbolicVariable(v) for v in args) or 

len(args) != len(set(args))): 

# An evaluated derivative of the form f'(1) is not a 

# symbolic variable, yet we would like to treat it 

# like one. So, we replace the argument `1` with a 

# temporary variable e.g. `t0` and then evaluate the 

# derivative f'(t0) symbolically at t0=1. See trac 

# #12796. 

temp_args = [SR.var("t%s"%i) for i in range(len(args))] 

f = sr_to_max(op.function()(*temp_args)) 

params = op.parameter_set() 

deriv_max = [[mdiff],f] 

for i in set(params): 

deriv_max.extend([sr_to_max(temp_args[i]), EclObject(params.count(i))]) 

at_eval=sr_to_max([temp_args[i]==args[i] for i in range(len(args))]) 

return EclObject([[max_at],deriv_max,at_eval]) 

 

f = sr_to_max(op.function()(*args)) 

params = op.parameter_set() 

deriv_max = [] 

[deriv_max.extend([sr_to_max(args[i]), EclObject(params.count(i))]) for i in set(params)] 

l = [[mdiff],f] 

l.extend(deriv_max) 

return EclObject(l) 

elif (op in special_sage_to_max): 

return EclObject(special_sage_to_max[op](*[sr_to_max(o) for o in expr.operands()])) 

elif op == tuple: 

return EclObject( ([mlist],list(sr_to_max(op) for op in expr.operands())) ) 

elif not (op in sage_op_dict): 

# Maxima does some simplifications automatically by default 

# so calling maxima(expr) can change the structure of expr 

#op_max=caar(maxima(expr).ecl()) 

# This should be safe if we treated all special operators above 

#furthermore, this should already use any _maxima_ methods on op, so use any 

#conversion methods that are registered in pynac. 

op_max=maxima(op).ecl() 

if op_max in max_op_dict: 

raise RuntimeError("Encountered operator mismatch in sr-to-maxima translation") 

sage_op_dict[op]=op_max 

max_op_dict[op_max]=op 

return EclObject(([sage_op_dict[op]], 

[sr_to_max(o) for o in expr.operands()])) 

elif expr.is_symbol() or expr._is_registered_constant_(): 

if not expr in sage_sym_dict: 

sym_max=maxima(expr).ecl() 

sage_sym_dict[expr]=sym_max 

max_sym_dict[sym_max]=expr 

return sage_sym_dict[expr] 

else: 

try: 

return pyobject_to_max(expr.pyobject()) 

except TypeError: 

return maxima(expr).ecl() 

 

# This goes from EclObject to SR 

from sage.libs.pynac.pynac import symbol_table 

max_to_pynac_table = symbol_table['maxima'] 

 

 

def max_to_sr(expr): 

r""" 

Convert a Maxima object into a symbolic expression. 

 

INPUT: 

 

- ``expr`` - ECL object 

 

OUTPUT: symbolic expression 

 

EXAMPLES:: 

 

sage: from sage.interfaces.maxima_lib import maxima_lib, max_to_sr 

sage: f = maxima_lib('f(x)') 

sage: f.ecl() 

<ECL: (($F SIMP) $X)> 

sage: max_to_sr(f.ecl()) 

f(x) 

 

TESTS:: 

 

sage: from sage.interfaces.maxima_lib import sr_to_max, max_to_sr 

sage: f = function('f')(x).diff() 

sage: bool(max_to_sr(sr_to_max(f)) == f) 

True 

""" 

if expr.consp(): 

op_max=caar(expr) 

if op_max in special_max_to_sage: 

return special_max_to_sage[op_max](expr) 

if not(op_max in max_op_dict): 

op_max_str=maxprint(op_max).python()[1:-1] 

if op_max_str in max_to_pynac_table: 

op = max_to_pynac_table[op_max_str] 

else: 

# This could be unsafe if the conversion to SR 

# changes the structure of expr 

sage_expr=SR(maxima(expr)) 

op=sage_expr.operator() 

if op in sage_op_dict: 

raise RuntimeError("Encountered operator mismatch in maxima-to-sr translation") 

max_op_dict[op_max]=op 

sage_op_dict[op]=op_max 

else: 

op=max_op_dict[op_max] 

max_args=cdr(expr) 

args=[max_to_sr(a) for a in max_args] 

return op(*args) 

elif expr.symbolp(): 

if not(expr in max_sym_dict): 

sage_symbol=SR(maxima(expr)) 

sage_sym_dict[sage_symbol]=expr 

max_sym_dict[expr]=sage_symbol 

return max_sym_dict[expr] 

else: 

e=expr.python() 

if isinstance(e,float): 

return sage.rings.real_double.RealDoubleElement(e) 

return e