Interface to Axiom#

Todo

  • Evaluation using a file is not done. Any input line with more than a few thousand characters would hang the system, so currently it automatically raises an exception.

  • All completions of a given command.

  • Interactive help.

Axiom is a free GPL-compatible (modified BSD license) general purpose computer algebra system whose development started in 1973 at IBM. It contains symbolic manipulation algorithms, as well as implementations of special functions, including elliptic functions and generalized hypergeometric functions. Moreover, Axiom has implementations of many functions relating to the invariant theory of the symmetric group \(S_n.\) For many links to Axiom documentation see http://wiki.axiom-developer.org.

AUTHORS:

  • Bill Page (2006-10): Created this (based on Maxima interface)

    Note

    Bill Page put a huge amount of effort into the Sage Axiom interface over several days during the Sage Days 2 coding sprint. This is contribution is greatly appreciated.

  • William Stein (2006-10): misc touchup.

  • Bill Page (2007-08): Minor modifications to support axiom4sage-0.3

Note

The axiom4sage-0.3.spkg is based on an experimental version of the FriCAS fork of the Axiom project by Waldek Hebisch that uses pre-compiled cached Lisp code to build Axiom very quickly with clisp.

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

EXAMPLES: We evaluate a very simple expression in axiom.

sage: axiom('3 * 5')                     #optional - axiom
15
sage: a = axiom(3) * axiom(5); a         #optional - axiom
15

>>> from sage.all import *
>>> axiom('3 * 5')                     #optional - axiom
15
>>> a = axiom(Integer(3)) * axiom(Integer(5)); a         #optional - axiom
15

The type of a is AxiomElement, i.e., an element of the axiom interpreter.

sage: type(a)                            #optional - axiom
<class 'sage.interfaces.axiom.AxiomElement'>
sage: parent(a)                          #optional - axiom
Axiom

>>> from sage.all import *
>>> type(a)                            #optional - axiom
<class 'sage.interfaces.axiom.AxiomElement'>
>>> parent(a)                          #optional - axiom
Axiom

The underlying Axiom type of a is also available, via the type method:

sage: a.type()                           #optional - axiom
PositiveInteger

>>> from sage.all import *
>>> a.type()                           #optional - axiom
PositiveInteger

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

sage: F = axiom.factor('x^5 - y^5'); F      #optional - axiom
           4      3    2 2    3     4
- (y - x)(y  + x y  + x y  + x y + x )
sage: type(F)                               #optional - axiom
<class 'sage.interfaces.axiom.AxiomElement'>
sage: F.type()                              #optional - axiom
Factored Polynomial Integer

>>> from sage.all import *
>>> F = axiom.factor('x^5 - y^5'); F      #optional - axiom
           4      3    2 2    3     4
- (y - x)(y  + x y  + x y  + x y + x )
>>> type(F)                               #optional - axiom
<class 'sage.interfaces.axiom.AxiomElement'>
>>> F.type()                              #optional - axiom
Factored Polynomial Integer

Note that Axiom objects are normally displayed using “ASCII art”.

sage: a = axiom(2/3); a          #optional - axiom
  2
  -
  3
sage: a = axiom('x^2 + 3/7'); a      #optional - axiom
   2   3
  x  + -
       7

>>> from sage.all import *
>>> a = axiom(Integer(2)/Integer(3)); a          #optional - axiom
  2
  -
  3
>>> a = axiom('x^2 + 3/7'); a      #optional - axiom
   2   3
  x  + -
       7

The axiom.eval command evaluates an expression in axiom and returns the result as a string. This is exact as if we typed in the given line of code to axiom; the return value is what Axiom would print out.

sage: print(axiom.eval('factor(x^5 - y^5)'))   # optional - axiom
           4      3    2 2    3     4
- (y - x)(y  + x y  + x y  + x y + x )
Type: Factored Polynomial Integer

>>> from sage.all import *
>>> print(axiom.eval('factor(x^5 - y^5)'))   # optional - axiom
           4      3    2 2    3     4
- (y - x)(y  + x y  + x y  + x y + x )
Type: Factored Polynomial Integer

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

sage: f = axiom('x^5 - y^5')                  #optional - axiom
sage: f^2                                     #optional - axiom
   10     5 5    10
  y   - 2x y  + x
sage: f.factor()                              #optional - axiom
           4      3    2 2    3     4
- (y - x)(y  + x y  + x y  + x y + x )

>>> from sage.all import *
>>> f = axiom('x^5 - y^5')                  #optional - axiom
>>> f**Integer(2)                                     #optional - axiom
   10     5 5    10
  y   - 2x y  + x
>>> f.factor()                              #optional - axiom
           4      3    2 2    3     4
- (y - x)(y  + x y  + x y  + x y + x )

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

sage:  f = axiom('(x^5 - y^5)^10000')       # not tested
Interrupting Axiom...
...
<class 'exceptions.TypeError'>: Ctrl-c pressed while running Axiom

>>> from sage.all import *
>>>  f = axiom('(x^5 - y^5)^10000')       # not tested
Interrupting Axiom...
...
<class 'exceptions.TypeError'>: Ctrl-c pressed while running Axiom

sage: axiom('1/100 + 1/101')                  #optional - axiom
   201
  -----
  10100
sage: a = axiom('(1 + sqrt(2))^5'); a         #optional - axiom
     +-+
  29\|2  + 41

>>> from sage.all import *
>>> axiom('1/100 + 1/101')                  #optional - axiom
   201
  -----
  10100
>>> a = axiom('(1 + sqrt(2))^5'); a         #optional - axiom
     +-+
  29\|2  + 41
class sage.interfaces.axiom.Axiom(name='axiom', command='axiom -nox -noclef', script_subdirectory=None, logfile=None, server=None, server_tmpdir=None, init_code=[')lisp (si::readline-off)'])[source]#

Bases: PanAxiom

console()[source]#

Spawn a new Axiom command-line session.

EXAMPLES:

sage: axiom.console() #not tested
                        AXIOM Computer Algebra System
                        Version: Axiom (January 2009)
               Timestamp: Sunday January 25, 2009 at 07:08:54
-----------------------------------------------------------------------------
   Issue )copyright to view copyright notices.
   Issue )summary for a summary of useful system commands.
   Issue )quit to leave AXIOM and return to shell.
-----------------------------------------------------------------------------

>>> from sage.all import *
>>> axiom.console() #not tested
                        AXIOM Computer Algebra System
                        Version: Axiom (January 2009)
               Timestamp: Sunday January 25, 2009 at 07:08:54
-----------------------------------------------------------------------------
   Issue )copyright to view copyright notices.
   Issue )summary for a summary of useful system commands.
   Issue )quit to leave AXIOM and return to shell.
-----------------------------------------------------------------------------
sage.interfaces.axiom.AxiomElement[source]#

alias of PanAxiomElement

sage.interfaces.axiom.AxiomExpectFunction[source]#

alias of PanAxiomExpectFunction

sage.interfaces.axiom.AxiomFunctionElement[source]#

alias of PanAxiomFunctionElement

class sage.interfaces.axiom.PanAxiom(name='axiom', command='axiom -nox -noclef', script_subdirectory=None, logfile=None, server=None, server_tmpdir=None, init_code=[')lisp (si::readline-off)'])[source]#

Bases: ExtraTabCompletion, Expect

Interface to a PanAxiom interpreter.

get(var)[source]#

Get the string value of the Axiom variable var.

EXAMPLES:

sage: # optional - axiom
sage: axiom.set('xx', '2')
sage: axiom.get('xx')
'2'
sage: a = axiom('(1 + sqrt(2))^5')
sage: axiom.get(a.name())
'     +-+\r\r\n  29\\|2  + 41'

>>> from sage.all import *
>>> # optional - axiom
>>> axiom.set('xx', '2')
>>> axiom.get('xx')
'2'
>>> a = axiom('(1 + sqrt(2))^5')
>>> axiom.get(a.name())
'     +-+\r\r\n  29\\|2  + 41'
set(var, value)[source]#

Set the variable var to the given value.

EXAMPLES:

sage: axiom.set('xx', '2')    #optional - axiom
sage: axiom.get('xx')         #optional - axiom
'2'

>>> from sage.all import *
>>> axiom.set('xx', '2')    #optional - axiom
>>> axiom.get('xx')         #optional - axiom
'2'
class sage.interfaces.axiom.PanAxiomElement(parent, value, is_name=False, name=None)[source]#

Bases: ExpectElement, AxiomElement

as_type(type)[source]#

Returns self as type.

EXAMPLES:

sage: a = axiom(1.2); a            #optional - axiom
1.2
sage: a.as_type(axiom.DoubleFloat) #optional - axiom
1.2
sage: _.type()                     #optional - axiom
DoubleFloat

>>> from sage.all import *
>>> a = axiom(RealNumber('1.2')); a            #optional - axiom
1.2
>>> a.as_type(axiom.DoubleFloat) #optional - axiom
1.2
>>> _.type()                     #optional - axiom
DoubleFloat
comma(*args)[source]#

Returns a Axiom tuple from self and args.

EXAMPLES:

sage: # optional - axiom
sage: two = axiom(2)
sage: two.comma(3)
[2,3]
sage: two.comma(3,4)
[2,3,4]
sage: _.type()
Tuple PositiveInteger

>>> from sage.all import *
>>> # optional - axiom
>>> two = axiom(Integer(2))
>>> two.comma(Integer(3))
[2,3]
>>> two.comma(Integer(3),Integer(4))
[2,3,4]
>>> _.type()
Tuple PositiveInteger
type()[source]#

Returns the type of an AxiomElement.

EXAMPLES:

sage: axiom(x+2).type()  #optional - axiom
Polynomial Integer

>>> from sage.all import *
>>> axiom(x+Integer(2)).type()  #optional - axiom
Polynomial Integer
unparsed_input_form()[source]#

Get the linear string representation of this object, if possible (often it isn’t).

EXAMPLES:

sage: a = axiom(x^2+1); a     #optional - axiom
   2
  x  + 1
sage: a.unparsed_input_form() #optional - axiom
'x*x+1'

>>> from sage.all import *
>>> a = axiom(x**Integer(2)+Integer(1)); a     #optional - axiom
   2
  x  + 1
>>> a.unparsed_input_form() #optional - axiom
'x*x+1'
class sage.interfaces.axiom.PanAxiomExpectFunction(parent, name)[source]#

Bases: ExpectFunction

class sage.interfaces.axiom.PanAxiomFunctionElement(object, name)[source]#

Bases: FunctionElement

sage.interfaces.axiom.axiom_console()[source]#

Spawn a new Axiom command-line session.

EXAMPLES:

sage: axiom_console() #not tested
                        AXIOM Computer Algebra System
                        Version: Axiom (January 2009)
               Timestamp: Sunday January 25, 2009 at 07:08:54
-----------------------------------------------------------------------------
   Issue )copyright to view copyright notices.
   Issue )summary for a summary of useful system commands.
   Issue )quit to leave AXIOM and return to shell.
-----------------------------------------------------------------------------

>>> from sage.all import *
>>> axiom_console() #not tested
                        AXIOM Computer Algebra System
                        Version: Axiom (January 2009)
               Timestamp: Sunday January 25, 2009 at 07:08:54
-----------------------------------------------------------------------------
   Issue )copyright to view copyright notices.
   Issue )summary for a summary of useful system commands.
   Issue )quit to leave AXIOM and return to shell.
-----------------------------------------------------------------------------
sage.interfaces.axiom.is_AxiomElement(x)[source]#

Return True if x is of type AxiomElement.

EXAMPLES:

sage: from sage.interfaces.axiom import is_AxiomElement
sage: is_AxiomElement(2)
doctest:...: DeprecationWarning: the function is_AxiomElement is deprecated; use isinstance(x, sage.interfaces.abc.AxiomElement) instead
See https://github.com/sagemath/sage/issues/34804 for details.
False
sage: is_AxiomElement(axiom(2))  # optional - axiom
True

>>> from sage.all import *
>>> from sage.interfaces.axiom import is_AxiomElement
>>> is_AxiomElement(Integer(2))
doctest:...: DeprecationWarning: the function is_AxiomElement is deprecated; use isinstance(x, sage.interfaces.abc.AxiomElement) instead
See https://github.com/sagemath/sage/issues/34804 for details.
False
>>> is_AxiomElement(axiom(Integer(2)))  # optional - axiom
True
sage.interfaces.axiom.reduce_load_Axiom()[source]#

Returns the Axiom interface object defined in sage.interfaces.axiom.

EXAMPLES:

sage: from sage.interfaces.axiom import reduce_load_Axiom
sage: reduce_load_Axiom()
Axiom

>>> from sage.all import *
>>> from sage.interfaces.axiom import reduce_load_Axiom
>>> reduce_load_Axiom()
Axiom