r"""
Interface to Singular
AUTHORS:
- David Joyner and William Stein (2005): first version
- Martin Albrecht (2006-03-05): code so singular.[tab] and x =
singular(...), x.[tab] includes all singular commands.
- Martin Albrecht (2006-03-06): This patch adds the equality symbol to
singular. Also fix a problem in which " " as prompt means comparison
will break all further communication with Singular.
- Martin Albrecht (2006-03-13): added current_ring() and
current_ring_name()
- William Stein (2006-04-10): Fixed problems with ideal constructor
- Martin Albrecht (2006-05-18): added sage_poly.
- Simon King (2010-11-23): Reduce the overhead caused by waiting for
the Singular prompt by doing garbage collection differently.
- Simon King (2011-06-06): Make conversion from Singular to Sage more flexible.
Introduction
------------
This interface is extremely flexible, since it's exactly like
typing into the Singular interpreter, and anything that works there
should work here.
The Singular interface will only work if Singular is installed on
your computer; this should be the case, since Singular is included
with Sage. The interface offers three pieces of functionality:
#. ``singular_console()`` - A function that dumps you
into an interactive command-line Singular session.
#. ``singular(expr, type='def')`` - Creation of a
Singular object. This provides a Pythonic interface to Singular.
For example, if ``f=singular(10)``, then
``f.factorize()`` returns the factorization of
`10` computed using Singular.
#. ``singular.eval(expr)`` - Evaluation of arbitrary
Singular expressions, with the result returned as a string.
Of course, there are polynomial rings and ideals in Sage as well
(often based on a C-library interface to Singular). One can convert
an object in the Singular interpreter interface to Sage by the
method ``sage()``.
Tutorial
--------
EXAMPLES: First we illustrate multivariate polynomial
factorization::
sage: R1 = singular.ring(0, '(x,y)', 'dp')
sage: R1
// characteristic : 0
// number of vars : 2
// block 1 : ordering dp
// : names x y
// block 2 : ordering C
sage: f = singular('9x16 - 18x13y2 - 9x12y3 + 9x10y4 - 18x11y2 + 36x8y4 + 18x7y5 - 18x5y6 + 9x6y4 - 18x3y6 - 9x2y7 + 9y8')
sage: f
9*x^16-18*x^13*y^2-9*x^12*y^3+9*x^10*y^4-18*x^11*y^2+36*x^8*y^4+18*x^7*y^5-18*x^5*y^6+9*x^6*y^4-18*x^3*y^6-9*x^2*y^7+9*y^8
sage: f.parent()
Singular
::
sage: F = f.factorize(); F
[1]:
_[1]=9
_[2]=x^6-2*x^3*y^2-x^2*y^3+y^4
_[3]=-x^5+y^2
[2]:
1,1,2
::
sage: F[1]
9,
x^6-2*x^3*y^2-x^2*y^3+y^4,
-x^5+y^2
sage: F[1][2]
x^6-2*x^3*y^2-x^2*y^3+y^4
We can convert `f` and each exponent back to Sage objects
as well.
::
sage: g = f.sage(); g
9*x^16 - 18*x^13*y^2 - 9*x^12*y^3 + 9*x^10*y^4 - 18*x^11*y^2 + 36*x^8*y^4 + 18*x^7*y^5 - 18*x^5*y^6 + 9*x^6*y^4 - 18*x^3*y^6 - 9*x^2*y^7 + 9*y^8
sage: F[1][2].sage()
x^6 - 2*x^3*y^2 - x^2*y^3 + y^4
sage: g.parent()
Multivariate Polynomial Ring in x, y over Rational Field
This example illustrates polynomial GCD's::
sage: R2 = singular.ring(0, '(x,y,z)', 'lp')
sage: a = singular.new('3x2*(x+y)')
sage: b = singular.new('9x*(y2-x2)')
sage: g = a.gcd(b)
sage: g
x^2+x*y
This example illustrates computation of a Groebner basis::
sage: R3 = singular.ring(0, '(a,b,c,d)', 'lp')
sage: I = singular.ideal(['a + b + c + d', 'a*b + a*d + b*c + c*d', 'a*b*c + a*b*d + a*c*d + b*c*d', 'a*b*c*d - 1'])
sage: I2 = I.groebner()
sage: I2
c^2*d^6-c^2*d^2-d^4+1,
c^3*d^2+c^2*d^3-c-d,
b*d^4-b+d^5-d,
b*c-b*d^5+c^2*d^4+c*d-d^6-d^2,
b^2+2*b*d+d^2,
a+b+c+d
The following example is the same as the one in the Singular - Gap
interface documentation::
sage: R = singular.ring(0, '(x0,x1,x2)', 'lp')
sage: I1 = singular.ideal(['x0*x1*x2 -x0^2*x2', 'x0^2*x1*x2-x0*x1^2*x2-x0*x1*x2^2', 'x0*x1-x0*x2-x1*x2'])
sage: I2 = I1.groebner()
sage: I2
x1^2*x2^2,
x0*x2^3-x1^2*x2^2+x1*x2^3,
x0*x1-x0*x2-x1*x2,
x0^2*x2-x0*x2^2-x1*x2^2
sage: I2.sage()
Ideal (x1^2*x2^2, x0*x2^3 - x1^2*x2^2 + x1*x2^3, x0*x1 - x0*x2 - x1*x2, x0^2*x2 - x0*x2^2 - x1*x2^2) of Multivariate Polynomial Ring in x0, x1, x2 over Rational Field
This example illustrates moving a polynomial from one ring to
another. It also illustrates calling a method of an object with an
argument.
::
sage: R = singular.ring(0, '(x,y,z)', 'dp')
sage: f = singular('x3+y3+(x-y)*x2y2+z2')
sage: f
x^3*y^2-x^2*y^3+x^3+y^3+z^2
sage: R1 = singular.ring(0, '(x,y,z)', 'ds')
sage: f = R.fetch(f)
sage: f
z^2+x^3+y^3+x^3*y^2-x^2*y^3
We can calculate the Milnor number of `f`::
sage: _=singular.LIB('sing.lib') # assign to _ to suppress printing
sage: f.milnor()
4
The Jacobian applied twice yields the Hessian matrix of
`f`, with which we can compute.
::
sage: H = f.jacob().jacob()
sage: H
6*x+6*x*y^2-2*y^3,6*x^2*y-6*x*y^2, 0,
6*x^2*y-6*x*y^2, 6*y+2*x^3-6*x^2*y,0,
0, 0, 2
sage: H.sage()
[6*x + 6*x*y^2 - 2*y^3 6*x^2*y - 6*x*y^2 0]
[ 6*x^2*y - 6*x*y^2 6*y + 2*x^3 - 6*x^2*y 0]
[ 0 0 2]
sage: H.det() # This is a polynomial in Singular
72*x*y+24*x^4-72*x^3*y+72*x*y^3-24*y^4-48*x^4*y^2+64*x^3*y^3-48*x^2*y^4
sage: H.det().sage() # This is the corresponding polynomial in Sage
72*x*y + 24*x^4 - 72*x^3*y + 72*x*y^3 - 24*y^4 - 48*x^4*y^2 + 64*x^3*y^3 - 48*x^2*y^4
The 1x1 and 2x2 minors::
sage: H.minor(1)
2,
6*y+2*x^3-6*x^2*y,
6*x^2*y-6*x*y^2,
6*x^2*y-6*x*y^2,
6*x+6*x*y^2-2*y^3
sage: H.minor(2)
12*y+4*x^3-12*x^2*y,
12*x^2*y-12*x*y^2,
12*x^2*y-12*x*y^2,
12*x+12*x*y^2-4*y^3,
-36*x*y-12*x^4+36*x^3*y-36*x*y^3+12*y^4+24*x^4*y^2-32*x^3*y^3+24*x^2*y^4
::
sage: _=singular.eval('option(redSB)')
sage: H.minor(1).groebner()
1
Computing the Genus
-------------------
We compute the projective genus of ideals that define curves over
`\QQ`. It is *very important* to load the
``normal.lib`` library before calling the
``genus`` command, or you'll get an error message.
EXAMPLE::
sage: singular.lib('normal.lib')
sage: R = singular.ring(0,'(x,y)','dp')
sage: i2 = singular.ideal('y9 - x2*(x-1)^9 + x')
sage: i2.genus()
40
Note that the genus can be much smaller than the degree::
sage: i = singular.ideal('y9 - x2*(x-1)^9')
sage: i.genus()
0
An Important Concept
--------------------
AUTHORS:
- Neal Harris
The following illustrates an important concept: how Sage interacts
with the data being used and returned by Singular. Let's compute a
Groebner basis for some ideal, using Singular through Sage.
::
sage: singular.lib('poly.lib')
sage: singular.ring(32003, '(a,b,c,d,e,f)', 'lp')
// characteristic : 32003
// number of vars : 6
// block 1 : ordering lp
// : names a b c d e f
// block 2 : ordering C
sage: I = singular.ideal('cyclic(6)')
sage: g = singular('groebner(I)')
Traceback (most recent call last):
...
TypeError: Singular error:
...
We restart everything and try again, but correctly.
::
sage: singular.quit()
sage: singular.lib('poly.lib'); R = singular.ring(32003, '(a,b,c,d,e,f)', 'lp')
sage: I = singular.ideal('cyclic(6)')
sage: I.groebner()
f^48-2554*f^42-15674*f^36+12326*f^30-12326*f^18+15674*f^12+2554*f^6-1,
...
It's important to understand why the first attempt at computing a
basis failed. The line where we gave singular the input
'groebner(I)' was useless because Singular has no idea what 'I' is!
Although 'I' is an object that we computed with calls to Singular
functions, it actually lives in Sage. As a consequence, the name
'I' means nothing to Singular. When we called
``I.groebner()``, Sage was able to call the groebner
function on'I' in Singular, since 'I' actually means something to
Sage.
Long Input
----------
The Singular interface reads in even very long input (using files)
in a robust manner, as long as you are creating a new object.
::
sage: t = '"%s"'%10^15000 # 15 thousand character string (note that normal Singular input must be at most 10000)
sage: a = singular.eval(t)
sage: a = singular(t)
TESTS:
We test an automatic coercion::
sage: a = 3*singular('2'); a
6
sage: type(a)
<class 'sage.interfaces.singular.SingularElement'>
sage: a = singular('2')*3; a
6
sage: type(a)
<class 'sage.interfaces.singular.SingularElement'>
Create a ring over GF(9) to check that ``gftables`` has been installed,
see ticket #11645::
sage: singular.eval("ring testgf9 = (9,x),(a,b,c,d,e,f),(M((1,2,3,0)),wp(2,3),lp);")
'ring testgf9 = (9,x),(a,b,c,d,e,f),(M((1,2,3,0)),wp(2,3),lp);'
"""
import os, re, sys
from expect import Expect, ExpectElement, FunctionElement, ExpectFunction
from sage.structure.sequence import Sequence
from sage.structure.element import RingElement
import sage.rings.integer
from sage.misc.misc import get_verbose
class SingularError(RuntimeError):
"""
Raised if Singular printed an error message
"""
pass
class Singular(Expect):
r"""
Interface to the Singular interpreter.
EXAMPLES: A Groebner basis example.
::
sage: R = singular.ring(0, '(x0,x1,x2)', 'lp')
sage: I = singular.ideal([ 'x0*x1*x2 -x0^2*x2', 'x0^2*x1*x2-x0*x1^2*x2-x0*x1*x2^2', 'x0*x1-x0*x2-x1*x2'])
sage: I.groebner()
x1^2*x2^2,
x0*x2^3-x1^2*x2^2+x1*x2^3,
x0*x1-x0*x2-x1*x2,
x0^2*x2-x0*x2^2-x1*x2^2
AUTHORS:
- David Joyner and William Stein
"""
def __init__(self, maxread=1000, script_subdirectory=None,
logfile=None, server=None,server_tmpdir=None):
"""
EXAMPLES::
sage: singular == loads(dumps(singular))
True
"""
prompt = '\n> '
Expect.__init__(self,
name = 'singular',
prompt = prompt,
command = "Singular -t --ticks-per-sec 1000",
maxread = maxread,
server = server,
server_tmpdir = server_tmpdir,
script_subdirectory = script_subdirectory,
restart_on_ctrlc = True,
verbose_start = False,
logfile = logfile,
eval_using_file_cutoff=100 if os.uname()[0]=="SunOS" else 1000)
self.__libs = []
self._prompt_wait = prompt
self.__to_clear = []
def _start(self, alt_message=None):
"""
EXAMPLES::
sage: s = Singular()
sage: s.is_running()
False
sage: s._start()
sage: s.is_running()
True
sage: s.quit()
"""
self.__libs = []
Expect._start(self, alt_message)
self.lib('general')
self.option("redTail")
self.option("redThrough")
self.option("intStrategy")
self._saved_options = self.option('get')
def __reduce__(self):
"""
EXAMPLES::
sage: singular.__reduce__()
(<function reduce_load_Singular at 0x...>, ())
"""
return reduce_load_Singular, ()
def _equality_symbol(self):
"""
EXAMPLES::
sage: singular._equality_symbol()
'=='
"""
return '=='
def _true_symbol(self):
"""
EXAMPLES::
sage: singular._true_symbol()
'1'
"""
return '1'
def _false_symbol(self):
"""
EXAMPLES::
sage: singular._false_symbol()
'0'
"""
return '0'
def _quit_string(self):
"""
EXAMPLES::
sage: singular._quit_string()
'quit'
"""
return 'quit'
def _read_in_file_command(self, filename):
r"""
EXAMPLES::
sage: singular._read_in_file_command('test')
'< "test";'
::
sage: filename = tmp_filename()
sage: f = open(filename, 'w')
sage: f.write('int x = 2;\n')
sage: f.close()
sage: singular.read(filename)
sage: singular.get('x')
'2'
"""
return '< "%s";'%filename
def eval(self, x, allow_semicolon=True, strip=True, **kwds):
r"""
Send the code x to the Singular interpreter and return the output
as a string.
INPUT:
- ``x`` - string (of code)
- ``allow_semicolon`` - default: False; if False then
raise a TypeError if the input line contains a semicolon.
- ``strip`` - ignored
EXAMPLES::
sage: singular.eval('2 > 1')
'1'
sage: singular.eval('2 + 2')
'4'
if the verbosity level is `> 1` comments are also printed
and not only returned.
::
sage: r = singular.ring(0,'(x,y,z)','dp')
sage: i = singular.ideal(['x^2','y^2','z^2'])
sage: s = i.std()
sage: singular.eval('hilb(%s)'%(s.name()))
'// 1 t^0\n// -3 t^2\n// 3 t^4\n// -1 t^6\n\n// 1 t^0\n//
3 t^1\n// 3 t^2\n// 1 t^3\n// dimension (affine) = 0\n//
degree (affine) = 8'
::
sage: set_verbose(1)
sage: o = singular.eval('hilb(%s)'%(s.name()))
// 1 t^0
// -3 t^2
// 3 t^4
// -1 t^6
// 1 t^0
// 3 t^1
// 3 t^2
// 1 t^3
// dimension (affine) = 0
// degree (affine) = 8
This is mainly useful if this method is called implicitly. Because
then intermediate results, debugging outputs and printed statements
are printed
::
sage: o = s.hilb()
// 1 t^0
// -3 t^2
// 3 t^4
// -1 t^6
// 1 t^0
// 3 t^1
// 3 t^2
// 1 t^3
// dimension (affine) = 0
// degree (affine) = 8
// ** right side is not a datum, assignment ignored
rather than ignored
::
sage: set_verbose(0)
sage: o = s.hilb()
"""
x = str(x).rstrip().rstrip(';')
x = x.replace("> ",">\t")
if not allow_semicolon and x.find(";") != -1:
raise TypeError, "singular input must not contain any semicolons:\n%s"%x
if len(x) == 0 or x[len(x) - 1] != ';':
x += ';'
s = Expect.eval(self, x, **kwds)
if s.find("error") != -1 or s.find("Segment fault") != -1:
raise SingularError('Singular error:\n%s'%s)
if get_verbose() > 0:
ret = []
for line in s.splitlines():
if line.startswith("//"):
print line
return s
else:
return s
def set(self, type, name, value):
"""
Set the variable with given name to the given value.
REMARK:
If a variable in the Singular interface was previously marked for
deletion, the actual deletion is done here, before the new variable
is created in Singular.
EXAMPLES::
sage: singular.set('int', 'x', '2')
sage: singular.get('x')
'2'
We test that an unused variable is only actually deleted if this method
is called::
sage: a = singular(3)
sage: n = a.name()
sage: del a
sage: singular.eval(n)
'3'
sage: singular.set('int', 'y', '5')
sage: singular.eval('defined(%s)'%n)
'0'
"""
cmd = ''.join('if(defined(%s)){kill %s;};'%(v,v) for v in self.__to_clear)
cmd += '%s %s=%s;'%(type, name, value)
self.__to_clear = []
self.eval(cmd)
def get(self, var):
"""
Get string representation of variable named var.
EXAMPLES::
sage: singular.set('int', 'x', '2')
sage: singular.get('x')
'2'
"""
return self.eval('print(%s);'%var)
def clear(self, var):
"""
Clear the variable named ``var``.
EXAMPLES::
sage: singular.set('int', 'x', '2')
sage: singular.get('x')
'2'
sage: singular.clear('x')
"Clearing the variable" means to allow to free the memory
that it uses in the Singular sub-process. However, the
actual deletion of the variable is only committed when
the next element in the Singular interface is created::
sage: singular.get('x')
'2'
sage: a = singular(3)
sage: singular.get('x')
'`x`'
"""
self.__to_clear.append(var)
def _create(self, value, type='def'):
"""
Creates a new variable in the Singular session and returns the name
of that variable.
EXAMPLES::
sage: singular._create('2', type='int')
'sage...'
sage: singular.get(_)
'2'
"""
name = self._next_var_name()
self.set(type, name, value)
return name
def __call__(self, x, type='def'):
"""
Create a singular object X with given type determined by the string
x. This returns var, where var is built using the Singular
statement type var = ... x ... Note that the actual name of var
could be anything, and can be recovered using X.name().
The object X returned can be used like any Sage object, and wraps
an object in self. The standard arithmetic operators work. Moreover
if foo is a function then X.foo(y,z,...) calls foo(X, y, z, ...)
and returns the corresponding object.
EXAMPLES::
sage: R = singular.ring(0, '(x0,x1,x2)', 'lp')
sage: I = singular.ideal([ 'x0*x1*x2 -x0^2*x2', 'x0^2*x1*x2-x0*x1^2*x2-x0*x1*x2^2', 'x0*x1-x0*x2-x1*x2'])
sage: I
-x0^2*x2+x0*x1*x2,
x0^2*x1*x2-x0*x1^2*x2-x0*x1*x2^2,
x0*x1-x0*x2-x1*x2
sage: type(I)
<class 'sage.interfaces.singular.SingularElement'>
sage: I.parent()
Singular
"""
if isinstance(x, SingularElement) and x.parent() is self:
return x
elif isinstance(x, ExpectElement):
return self(x.sage())
elif not isinstance(x, ExpectElement) and hasattr(x, '_singular_'):
return x._singular_(self)
if type in ("module","list") and isinstance(x,(list,tuple,Sequence)):
x = str(x)[1:-1]
return SingularElement(self, type, x, False)
def has_coerce_map_from_impl(self, S):
if hasattr(S, 'an_element'):
if hasattr(S.an_element(), '_singular_'):
return True
try:
self._coerce_(S.an_element())
except TypeError:
return False
return True
elif S is int or S is long:
return True
raise NotImplementedError
def cputime(self, t=None):
r"""
Returns the amount of CPU time that the Singular session has used.
If ``t`` is not None, then it returns the difference
between the current CPU time and ``t``.
EXAMPLES::
sage: t = singular.cputime()
sage: R = singular.ring(0, '(x0,x1,x2)', 'lp')
sage: I = singular.ideal([ 'x0*x1*x2 -x0^2*x2', 'x0^2*x1*x2-x0*x1^2*x2-x0*x1*x2^2', 'x0*x1-x0*x2-x1*x2'])
sage: gb = I.groebner()
sage: singular.cputime(t) #random
0.02
"""
if t:
return float(self.eval('timer-(%d)'%(int(1000*t))))/1000.0
else:
return float(self.eval('timer'))/1000.0
def lib(self, lib, reload=False):
"""
Load the Singular library named lib.
Note that if the library was already loaded during this session it
is not reloaded unless the optional reload argument is True (the
default is False).
EXAMPLES::
sage: singular.lib('sing.lib')
sage: singular.lib('sing.lib', reload=True)
"""
if lib[-4:] != ".lib":
lib += ".lib"
if not reload and lib in self.__libs:
return
self.eval('LIB "%s"'%lib)
self.__libs.append(lib)
LIB = lib
load = lib
def ideal(self, *gens):
"""
Return the ideal generated by gens.
INPUT:
- ``gens`` - list or tuple of Singular objects (or
objects that can be made into Singular objects via evaluation)
OUTPUT: the Singular ideal generated by the given list of gens
EXAMPLES: A Groebner basis example done in a different way.
::
sage: _ = singular.eval("ring R=0,(x0,x1,x2),lp")
sage: i1 = singular.ideal([ 'x0*x1*x2 -x0^2*x2', 'x0^2*x1*x2-x0*x1^2*x2-x0*x1*x2^2', 'x0*x1-x0*x2-x1*x2'])
sage: i1
-x0^2*x2+x0*x1*x2,
x0^2*x1*x2-x0*x1^2*x2-x0*x1*x2^2,
x0*x1-x0*x2-x1*x2
::
sage: i2 = singular.ideal('groebner(%s);'%i1.name())
sage: i2
x1^2*x2^2,
x0*x2^3-x1^2*x2^2+x1*x2^3,
x0*x1-x0*x2-x1*x2,
x0^2*x2-x0*x2^2-x1*x2^2
"""
if isinstance(gens, str):
gens = self(gens)
if isinstance(gens, SingularElement):
return self(gens.name(), 'ideal')
if not isinstance(gens, (list, tuple)):
raise TypeError, "gens (=%s) must be a list, tuple, string, or Singular element"%gens
if len(gens) == 1 and isinstance(gens[0], (list, tuple)):
gens = gens[0]
gens2 = []
for g in gens:
if not isinstance(g, SingularElement):
gens2.append(self.new(g))
else:
gens2.append(g)
return self(",".join([g.name() for g in gens2]), 'ideal')
def list(self, x):
r"""
Creates a list in Singular from a Sage list ``x``.
EXAMPLES::
sage: singular.list([1,2])
[1]:
1
[2]:
2
"""
return self(x, 'list')
def matrix(self, nrows, ncols, entries=None):
"""
EXAMPLES::
sage: singular.lib("matrix")
sage: R = singular.ring(0, '(x,y,z)', 'dp')
sage: A = singular.matrix(3,2,'1,2,3,4,5,6')
sage: A
1,2,
3,4,
5,6
sage: A.gauss_col()
2,-1,
1,0,
0,1
AUTHORS:
- Martin Albrecht (2006-01-14)
"""
name = self._next_var_name()
if entries is None:
s = self.eval('matrix %s[%s][%s]'%(name, nrows, ncols))
else:
s = self.eval('matrix %s[%s][%s] = %s'%(name, nrows, ncols, entries))
return SingularElement(self, None, name, True)
def ring(self, char=0, vars='(x)', order='lp', check=True):
r"""
Create a Singular ring and makes it the current ring.
INPUT:
- ``char`` - characteristic of the base ring (see
examples below), which must be either 0, prime (!), or one of
several special codes (see examples below).
- ``vars`` - a tuple or string that defines the
variable names
- ``order`` - string - the monomial order (default:
'lp')
- ``check`` - if True, check primality of the
characteristic if it is an integer.
OUTPUT: a Singular ring
.. note::
This function is *not* identical to calling the Singular
``ring`` function. In particular, it also attempts to
"kill" the variable names, so they can actually be used
without getting errors, and it sets printing of elements
for this range to short (i.e., with \*'s and carets).
EXAMPLES: We first declare `\QQ[x,y,z]` with degree reverse
lexicographic ordering.
::
sage: R = singular.ring(0, '(x,y,z)', 'dp')
sage: R
// characteristic : 0
// number of vars : 3
// block 1 : ordering dp
// : names x y z
// block 2 : ordering C
::
sage: R1 = singular.ring(32003, '(x,y,z)', 'dp')
sage: R2 = singular.ring(32003, '(a,b,c,d)', 'lp')
This is a ring in variables named x(1) through x(10) over the
finite field of order `7`::
sage: R3 = singular.ring(7, '(x(1..10))', 'ds')
This is a polynomial ring over the transcendental extension
`\QQ(a)` of `\QQ`::
sage: R4 = singular.ring('(0,a)', '(mu,nu)', 'lp')
This is a ring over the field of single-precision floats::
sage: R5 = singular.ring('real', '(a,b)', 'lp')
This is over 50-digit floats::
sage: R6 = singular.ring('(real,50)', '(a,b)', 'lp')
sage: R7 = singular.ring('(complex,50,i)', '(a,b)', 'lp')
To use a ring that you've defined, use the set_ring() method on
the ring. This sets the ring to be the "current ring". For
example,
::
sage: R = singular.ring(7, '(a,b)', 'ds')
sage: S = singular.ring('real', '(a,b)', 'lp')
sage: singular.new('10*a')
1.000e+01*a
sage: R.set_ring()
sage: singular.new('10*a')
3*a
"""
if len(vars) > 2:
s = '; '.join(['if(defined(%s)>0){kill %s;};'%(x,x)
for x in vars[1:-1].split(',')])
self.eval(s)
if check and isinstance(char, (int, long, sage.rings.integer.Integer)):
if char != 0:
n = sage.rings.integer.Integer(char)
if not n.is_prime():
raise ValueError, "the characteristic must be 0 or prime"
R = self('%s,%s,%s'%(char, vars, order), 'ring')
self.eval('short=0')
return R
def string(self, x):
"""
Creates a Singular string from a Sage string. Note that the Sage
string has to be "double-quoted".
EXAMPLES::
sage: singular.string('"Sage"')
Sage
"""
return self(x, 'string')
def set_ring(self, R):
"""
Sets the current Singular ring to R.
EXAMPLES::
sage: R = singular.ring(7, '(a,b)', 'ds')
sage: S = singular.ring('real', '(a,b)', 'lp')
sage: singular.current_ring()
// characteristic : 0 (real)
// number of vars : 2
// block 1 : ordering lp
// : names a b
// block 2 : ordering C
sage: singular.set_ring(R)
sage: singular.current_ring()
// characteristic : 7
// number of vars : 2
// block 1 : ordering ds
// : names a b
// block 2 : ordering C
"""
if not isinstance(R, SingularElement):
raise TypeError, "R must be a singular ring"
self.eval("setring %s; short=0"%R.name(), allow_semicolon=True)
setring = set_ring
def current_ring_name(self):
"""
Returns the Singular name of the currently active ring in
Singular.
OUTPUT: currently active ring's name
EXAMPLES::
sage: r = PolynomialRing(GF(127),3,'xyz')
sage: r._singular_().name() == singular.current_ring_name()
True
"""
ringlist = self.eval("listvar(ring)").splitlines()
p = re.compile("// ([a-zA-Z0-9_]*).*\[.*\].*\*.*")
for line in ringlist:
m = p.match(line)
if m:
return m.group(int(1))
return None
def current_ring(self):
"""
Returns the current ring of the running Singular session.
EXAMPLES::
sage: r = PolynomialRing(GF(127),3,'xyz', order='invlex')
sage: r._singular_()
// characteristic : 127
// number of vars : 3
// block 1 : ordering rp
// : names x y z
// block 2 : ordering C
sage: singular.current_ring()
// characteristic : 127
// number of vars : 3
// block 1 : ordering rp
// : names x y z
// block 2 : ordering C
"""
name = self.current_ring_name()
if name:
return self(name)
else:
return None
def trait_names(self):
"""
Return a list of all Singular commands.
EXAMPLES::
sage: singular.trait_names()
['exteriorPower',
...
'stdfglm']
"""
p = re.compile("// *([a-z0-9A-Z_]*).*")
proclist = self.eval("listvar(proc)").splitlines()
return [p.match(line).group(int(1)) for line in proclist]
def console(self):
"""
EXAMPLES::
sage: singular_console() #not tested
SINGULAR / Development
A Computer Algebra System for Polynomial Computations / version 3-0-4
0<
by: G.-M. Greuel, G. Pfister, H. Schoenemann \ Nov 2007
FB Mathematik der Universitaet, D-67653 Kaiserslautern \
"""
singular_console()
def version(self):
"""
EXAMPLES:
"""
return singular_version()
def _function_class(self):
"""
EXAMPLES::
sage: singular._function_class()
<class 'sage.interfaces.singular.SingularFunction'>
"""
return SingularFunction
def _function_element_class(self):
"""
EXAMPLES::
sage: singular._function_element_class()
<class 'sage.interfaces.singular.SingularFunctionElement'>
"""
return SingularFunctionElement
def option(self, cmd=None, val=None):
"""
Access to Singular's options as follows:
Syntax: option() Returns a string of all defined options.
Syntax: option( 'option_name' ) Sets an option. Note to disable an
option, use the prefix no.
Syntax: option( 'get' ) Returns an intvec of the state of all
options.
Syntax: option( 'set', intvec_expression ) Restores the state of
all options from an intvec (produced by option('get')).
EXAMPLES::
sage: singular.option()
//options: redefine loadLib usage prompt
sage: singular.option('get')
0,
10321
sage: old_options = _
sage: singular.option('noredefine')
sage: singular.option()
//options: loadLib usage prompt
sage: singular.option('set', old_options)
sage: singular.option('get')
0,
10321
"""
if cmd is None:
return SingularFunction(self,"option")()
elif cmd == "get":
return self(self.eval("option(get)"),"intvec")
elif cmd == "set":
if not isinstance(val,SingularElement):
raise TypeError, "singular.option('set') needs SingularElement as second parameter"
self.eval("option(set,%s)"%val.name())
else:
SingularFunction(self,"option")("\""+str(cmd)+"\"")
def _keyboard_interrupt(self):
print "Interrupting %s..."%self
try:
self._expect.sendline(chr(4))
except pexpect.ExceptionPexpect, msg:
raise pexcept.ExceptionPexpect("THIS IS A BUG -- PLEASE REPORT. This should never happen.\n" + msg)
self._start()
raise KeyboardInterrupt, "Restarting %s (WARNING: all variables defined in previous session are now invalid)"%self
class SingularElement(ExpectElement):
def __init__(self, parent, type, value, is_name=False):
"""
EXAMPLES::
sage: a = singular(2)
sage: loads(dumps(a))
(invalid object -- defined in terms of closed session)
"""
RingElement.__init__(self, parent)
if parent is None: return
if not is_name:
try:
self._name = parent._create( value, type)
except SingularError, x:
self._session_number = -1
raise TypeError, x, sys.exc_info()[2]
except BaseException:
self._session_number = -1
raise
else:
self._name = value
self._session_number = parent._session_number
def __repr__(self):
r"""
Return string representation of ``self``.
EXAMPLE::
sage: r = singular.ring(0,'(x,y)','dp')
sage: singular(0)
0
sage: singular('x') # indirect doctest
x
sage: singular.matrix(2,2)
0,0,
0,0
sage: singular.matrix(2,2,"(25/47*x^2*y^4 + 63/127*x + 27)^3,y,0,1")
15625/103823*x^6*y.., y,
0, 1
Note that the output is truncated
::
sage: M= singular.matrix(2,2,"(25/47*x^2*y^4 + 63/127*x + 27)^3,y,0,1")
sage: M.rename('T')
sage: M
T[1,1],y,
0, 1
if ``self`` has a custom name, it is used to print the
matrix, rather than abbreviating its contents
"""
try:
self._check_valid()
except ValueError:
return '(invalid object -- defined in terms of closed session)'
try:
if self._get_using_file:
s = self.parent().get_using_file(self._name)
except AttributeError:
s = self.parent().get(self._name)
if s.__contains__(self._name):
if hasattr(self, '__custom_name'):
s = s.replace(self._name, self.__dict__['__custom_name'])
elif self.type() == 'matrix':
s = self.parent().eval('pmat(%s,20)'%(self.name()))
return s
def __copy__(self):
r"""
Returns a copy of ``self``.
EXAMPLES::
sage: R=singular.ring(0,'(x,y)','dp')
sage: M=singular.matrix(3,3,'0,0,-x, 0,y,0, x*y,0,0')
sage: N=copy(M)
sage: N[1,1]=singular('x+y')
sage: N
x+y,0,-x,
0, y,0,
x*y,0,0
sage: M
0, 0,-x,
0, y,0,
x*y,0,0
sage: L=R.ringlist()
sage: L[4]=singular.ideal('x**2-5')
sage: Q=L.ring()
sage: otherR=singular.ring(5,'(x)','dp')
sage: cpQ=copy(Q)
sage: cpQ.set_ring()
sage: cpQ
// characteristic : 0
// number of vars : 2
// block 1 : ordering dp
// : names x y
// block 2 : ordering C
// quotient ring from ideal
_[1]=x^2-5
sage: R.fetch(M)
0, 0,-x,
0, y,0,
x*y,0,0
"""
if (self.type()=='ring') or (self.type()=='qring'):
br=self.parent().current_ring()
self.set_ring()
OUT = (self.ringlist()).ring()
br.set_ring()
return OUT
else:
return self.parent()(self.name())
def __len__(self):
"""
Returns the size of this Singular element.
EXAMPLES::
sage: R = singular.ring(0, '(x,y,z)', 'dp')
sage: A = singular.matrix(2,2)
sage: len(A)
4
"""
return int(self.size())
def __reduce__(self):
"""
Note that the result of the returned reduce_load is an invalid
Singular object.
EXAMPLES::
sage: singular(2).__reduce__()
(<function reduce_load at 0x...>, ())
"""
return reduce_load, ()
def __setitem__(self, n, value):
"""
Set the n-th element of self to x.
INPUT:
- ``n`` - an integer *or* a 2-tuple (for setting
matrix elements)
- ``value`` - anything (is coerced to a Singular
object if it is not one already)
OUTPUT: Changes elements of self.
EXAMPLES::
sage: R = singular.ring(0, '(x,y,z)', 'dp')
sage: A = singular.matrix(2,2)
sage: A
0,0,
0,0
sage: A[1,1] = 5
sage: A
5,0,
0,0
sage: A[1,2] = '5*x + y + z3'
sage: A
5,z^3+5*x+y,
0,0
"""
P = self.parent()
if not isinstance(value, SingularElement):
value = P(value)
if isinstance(n, tuple):
if len(n) != 2:
raise ValueError, "If n (=%s) is a tuple, it must be a 2-tuple"%n
x, y = n
P.eval('%s[%s,%s] = %s'%(self.name(), x, y, value.name()))
else:
P.eval('%s[%s] = %s'%(self.name(), n, value.name()))
def __nonzero__(self):
"""
Returns True if this Singular element is not zero.
EXAMPLES::
sage: singular(0).__nonzero__()
False
sage: singular(1).__nonzero__()
True
"""
P = self.parent()
return P.eval('%s == 0'%self.name()) == '0'
def sage_polystring(self):
r"""
If this Singular element is a polynomial, return a string
representation of this polynomial that is suitable for evaluation
in Python. Thus \* is used for multiplication and \*\* for
exponentiation. This function is primarily used internally.
The short=0 option *must* be set for the parent ring or this
function will not work as expected. This option is set by default
for rings created using ``singular.ring`` or set using
``ring_name.set_ring()``.
EXAMPLES::
sage: R = singular.ring(0,'(x,y)')
sage: f = singular('x^3 + 3*y^11 + 5')
sage: f
x^3+3*y^11+5
sage: f.sage_polystring()
'x**3+3*y**11+5'
"""
return str(self).replace('^','**')
def sage_global_ring(self):
"""
Return the current basering in Singular as a polynomial ring or quotient ring.
EXAMPLE::
sage: singular.eval('ring r1 = (9,x),(a,b,c,d,e,f),(M((1,2,3,0)),wp(2,3),lp)')
'ring r1 = (9,x),(a,b,c,d,e,f),(M((1,2,3,0)),wp(2,3),lp);'
sage: R = singular('r1').sage_global_ring()
sage: R
Multivariate Polynomial Ring in a, b, c, d, e, f over Finite Field in x of size 3^2
sage: R.term_order()
Block term order with blocks:
(Matrix term order with matrix
[1 2]
[3 0],
Weighted degree reverse lexicographic term order with weights (2, 3),
Lexicographic term order of length 2)
::
sage: singular.eval('ring r2 = (0,x),(a,b,c),dp')
'ring r2 = (0,x),(a,b,c),dp;'
sage: singular('r2').sage_global_ring()
Multivariate Polynomial Ring in a, b, c over Fraction Field of Univariate Polynomial Ring in x over Rational Field
::
sage: singular.eval('ring r3 = (3,z),(a,b,c),dp')
'ring r3 = (3,z),(a,b,c),dp;'
sage: singular.eval('minpoly = 1+z+z2+z3+z4')
'minpoly = 1+z+z2+z3+z4;'
sage: singular('r3').sage_global_ring()
Multivariate Polynomial Ring in a, b, c over Finite Field in z of size 3^4
Real and complex fields in both Singular and Sage are defined with a precision.
The precision in Singular is given in terms of digits, but in Sage it is given
in terms of bits. So, the digit precision is internally converted to a reasonable
bit precision::
sage: singular.eval('ring r4 = (real,20),(a,b,c),dp')
'ring r4 = (real,20),(a,b,c),dp;'
sage: singular('r4').sage_global_ring()
Multivariate Polynomial Ring in a, b, c over Real Field with 70 bits of precision
The case of complex coefficients is not fully supported, yet, since
the generator of a complex field in Sage is always called "I"::
sage: singular.eval('ring r5 = (complex,15,j),(a,b,c),dp')
'ring r5 = (complex,15,j),(a,b,c),dp;'
sage: R = singular('r5').sage_global_ring(); R
Multivariate Polynomial Ring in a, b, c over Complex Field with 54 bits of precision
sage: R.base_ring()('j')
Traceback (most recent call last):
...
NameError: name 'j' is not defined
sage: R.base_ring()('I')
1.00000000000000*I
In our last example, the base ring is a quotient ring::
sage: singular.eval('ring r6 = (9,a), (x,y,z),lp')
'ring r6 = (9,a), (x,y,z),lp;'
sage: Q = singular('std(ideal(x^2,x+y^2+z^3))', type='qring')
sage: Q.sage_global_ring()
Quotient of Multivariate Polynomial Ring in x, y, z over Finite Field in a of size 3^2 by the ideal (y^4 - y^2*z^3 + z^6, x + y^2 + z^3)
AUTHOR:
- Simon King (2011-06-06)
"""
singular = self.parent()
charstr = singular.eval('charstr(basering)').split(',',1)
from sage.all import ZZ
is_extension = len(charstr)==2
if charstr[0]=='integer':
br = ZZ
is_extension = False
elif charstr[0]=='0':
from sage.all import QQ
br = QQ
elif charstr[0]=='real':
from sage.all import RealField, ceil, log
prec = singular.eval('ringlist(basering)[1][2][1]')
br = RealField(ceil((ZZ(prec)+1)/log(2,10)))
is_extension = False
elif charstr[0]=='complex':
from sage.all import ComplexField, ceil, log
prec = singular.eval('ringlist(basering)[1][2][1]')
I = singular.eval('ringlist(basering)[1][3]')
br = ComplexField(ceil((ZZ(prec)+1)/log(2,10)))
is_extension = False
else:
q = ZZ(charstr[0])
from sage.all import GF
if q.is_prime():
br = GF(q)
else:
br = GF(q,charstr[1])
is_extension = False
if is_extension:
minpoly = singular.eval('minpoly')
if minpoly == '0':
from sage.all import Frac
BR = Frac(br[charstr[1]])
else:
is_short = singular.eval('short')
if is_short!='0':
singular.eval('short=0')
minpoly = ZZ[charstr[1]](singular.eval('minpoly'))
singular.eval('short=%s'%is_short)
else:
minpoly = ZZ[charstr[1]](minpoly)
BR = br.extension(minpoly,name=charstr[1])
else:
BR = br
from sage.rings.polynomial.term_order import termorder_from_singular
from sage.all import PolynomialRing
if singular.eval('typeof(basering)')=='ring':
return PolynomialRing(BR, names=singular.eval('varstr(basering)'), order=termorder_from_singular(singular))
P = PolynomialRing(BR, names=singular.eval('varstr(basering)'), order=termorder_from_singular(singular))
return P.quotient(singular('ringlist(basering)[4]')._sage_(P), names=singular.eval('varstr(basering)'))
def sage_poly(self, R=None, kcache=None):
"""
Returns a Sage polynomial in the ring r matching the provided poly
which is a singular polynomial.
INPUT:
- ``R`` - (default: None); an optional polynomial ring.
If it is provided, then you have to make sure that it
matches the current singular ring as, e.g., returned by
singular.current_ring(). By default, the output of
:meth:`sage_global_ring` is used.
- ``kcache`` - (default: None); an optional dictionary
for faster finite field lookups, this is mainly useful for finite
extension fields
OUTPUT: MPolynomial
EXAMPLES::
sage: R = PolynomialRing(GF(2^8,'a'),2,'xy')
sage: f=R('a^20*x^2*y+a^10+x')
sage: f._singular_().sage_poly(R)==f
True
sage: R = PolynomialRing(GF(2^8,'a'),1,'x')
sage: f=R('a^20*x^3+x^2+a^10')
sage: f._singular_().sage_poly(R)==f
True
::
sage: P.<x,y> = PolynomialRing(QQ, 2)
sage: f = x*y**3 - 1/9 * x + 1; f
x*y^3 - 1/9*x + 1
sage: singular(f)
x*y^3-1/9*x+1
sage: P(singular(f))
x*y^3 - 1/9*x + 1
TESTS::
sage: singular.eval('ring r = (3,z),(a,b,c),dp')
'ring r = (3,z),(a,b,c),dp;'
sage: singular.eval('minpoly = 1+z+z2+z3+z4')
'minpoly = 1+z+z2+z3+z4;'
sage: p = singular('z^4*a^3+z^2*a*b*c')
sage: p.sage_poly()
(-z^3 - z^2 - z - 1)*a^3 + (z^2)*a*b*c
sage: singular('z^4')
(-z3-z2-z-1)
AUTHORS:
- Martin Albrecht (2006-05-18)
- Simon King (2011-06-06): Deal with Singular's short polynomial representation,
automatic construction of a polynomial ring, if it is not explicitly given.
.. note::
For very simple polynomials
``eval(SingularElement.sage_polystring())`` is faster than
SingularElement.sage_poly(R), maybe we should detect the
crossover point (in dependence of the string length) and
choose an appropriate conversion strategy
"""
from sage.rings.polynomial.multi_polynomial_ring import MPolynomialRing_polydict
from sage.rings.polynomial.multi_polynomial_libsingular import MPolynomialRing_libsingular
from sage.rings.polynomial.multi_polynomial_element import MPolynomial_polydict
from sage.rings.polynomial.polynomial_ring import is_PolynomialRing
from sage.rings.polynomial.polydict import PolyDict,ETuple
from sage.rings.polynomial.polynomial_singular_interface import can_convert_to_singular
from sage.rings.quotient_ring import QuotientRing_generic
from sage.rings.quotient_ring_element import QuotientRingElement
ring_is_fine = False
if R is None:
ring_is_fine = True
R = self.sage_global_ring()
sage_repr = {}
k = R.base_ring()
variable_str = "*".join(R.variable_names())
is_short = self.parent().eval('short')
if is_short!='0':
self.parent().eval('short=0')
if isinstance(R, MPolynomialRing_libsingular):
out = R(self)
self.parent().eval('short=%s'%is_short)
return out
singular_poly_list = self.parent().eval("string(coef(%s,%s))"%(\
self.name(),variable_str)).split(",")
self.parent().eval('short=%s'%is_short)
else:
if isinstance(R, MPolynomialRing_libsingular):
return R(self)
singular_poly_list = self.parent().eval("string(coef(%s,%s))"%(\
self.name(),variable_str)).split(",")
if singular_poly_list == ['1','0'] :
return R(0)
coeff_start = int(len(singular_poly_list)/2)
if isinstance(R,(MPolynomialRing_polydict,QuotientRing_generic)) and (ring_is_fine or can_convert_to_singular(R)):
var_dict = dict(zip(R.variable_names(),range(R.ngens())))
ngens = R.ngens()
for i in range(coeff_start):
exp = dict()
monomial = singular_poly_list[i]
if monomial!="1":
variables = [var.split("^") for var in monomial.split("*") ]
for e in variables:
var = e[0]
if len(e)==int(2):
power = int(e[1])
else:
power=1
exp[var_dict[var]]=power
if kcache==None:
sage_repr[ETuple(exp,ngens)]=k(singular_poly_list[coeff_start+i])
else:
elem = singular_poly_list[coeff_start+i]
if not kcache.has_key(elem):
kcache[elem] = k( elem )
sage_repr[ETuple(exp,ngens)]= kcache[elem]
p = MPolynomial_polydict(R,PolyDict(sage_repr,force_int_exponents=False,force_etuples=False))
if isinstance(R, MPolynomialRing_polydict):
return p
else:
return QuotientRingElement(R,p,reduce=False)
elif is_PolynomialRing(R) and (ring_is_fine or can_convert_to_singular(R)):
sage_repr = [0]*int(self.deg()+1)
for i in range(coeff_start):
monomial = singular_poly_list[i]
exp = int(0)
if monomial!="1":
term = monomial.split("^")
if len(term)==int(2):
exp = int(term[1])
else:
exp = int(1)
if kcache==None:
sage_repr[exp]=k(singular_poly_list[coeff_start+i])
else:
elem = singular_poly_list[coeff_start+i]
if not kcache.has_key(elem):
kcache[elem] = k( elem )
sage_repr[ exp ]= kcache[elem]
return R(sage_repr)
else:
raise TypeError, "Cannot coerce %s into %s"%(self,R)
def sage_matrix(self, R, sparse=True):
"""
Returns Sage matrix for self
INPUT:
- ``R`` - (default: None); an optional ring, over which
the resulting matrix is going to be defined.
By default, the output of :meth:`sage_global_ring` is used.
- ``sparse`` - (default: True); determines whether the
resulting matrix is sparse or not.
EXAMPLES::
sage: R = singular.ring(0, '(x,y,z)', 'dp')
sage: A = singular.matrix(2,2)
sage: A.sage_matrix(ZZ)
[0 0]
[0 0]
sage: A.sage_matrix(RDF)
[0.0 0.0]
[0.0 0.0]
"""
from sage.matrix.constructor import Matrix
nrows, ncols = int(self.nrows()),int(self.ncols())
if R is None:
R = self.sage_global_ring()
A = Matrix(R, nrows, ncols, sparse=sparse)
for x in range(nrows):
for y in range(ncols):
A[x,y]=self[x+1,y+1].sage_poly(R)
return A
A = Matrix(R, nrows, ncols, sparse=sparse)
for x in range(nrows):
for y in range(ncols):
A[x,y]=R(self[x+1,y+1])
return A
def _sage_(self, R=None):
r"""
Coerces self to Sage.
EXAMPLES::
sage: R = singular.ring(0, '(x,y,z)', 'dp')
sage: A = singular.matrix(2,2)
sage: A._sage_(ZZ)
[0 0]
[0 0]
sage: A = random_matrix(ZZ,3,3); A
[ -8 2 0]
[ 0 1 -1]
[ 2 1 -95]
sage: As = singular(A); As
-8 2 0
0 1 -1
2 1 -95
sage: As._sage_()
[ -8 2 0]
[ 0 1 -1]
[ 2 1 -95]
::
sage: singular.eval('ring R = integer, (x,y,z),lp')
'// ** redefining R **'
sage: I = singular.ideal(['x^2','y*z','z+x'])
sage: I.sage() # indirect doctest
Ideal (x^2, y*z, x + z) of Multivariate Polynomial Ring in x, y, z over Integer Ring
::
sage: singular('ringlist(basering)').sage()
[['integer'], ['x', 'y', 'z'], [['lp', (1, 1, 1)], ['C', (0)]], Ideal (0) of Multivariate Polynomial Ring in x, y, z over Integer Ring]
::
sage: singular.eval('ring r10 = (9,a), (x,y,z),lp')
'ring r10 = (9,a), (x,y,z),lp;'
sage: singular.eval('setring R')
'setring R;'
sage: singular('r10').sage()
Multivariate Polynomial Ring in x, y, z over Finite Field in a of size 3^2
Note that the current base ring has not been changed by asking for another ring::
sage: singular('basering')
// coeff. ring is : Integers
// number of vars : 3
// block 1 : ordering lp
// : names x y z
// block 2 : ordering C
::
sage: singular.eval('setring r10')
'setring r10;'
sage: Q = singular('std(ideal(x^2,x+y^2+z^3))', type='qring')
sage: Q.sage()
Quotient of Multivariate Polynomial Ring in x, y, z over Finite Field in a of size 3^2 by the ideal (y^4 - y^2*z^3 + z^6, x + y^2 + z^3)
sage: singular('x^2+y').sage()
x^2 + y
sage: singular('x^2+y').sage().parent()
Quotient of Multivariate Polynomial Ring in x, y, z over Finite Field in a of size 3^2 by the ideal (y^4 - y^2*z^3 + z^6, x + y^2 + z^3)
"""
typ = self.type()
if typ=='poly':
return self.sage_poly(R)
elif typ == 'module':
return self.sage_matrix(R,sparse=True)
elif typ == 'matrix':
return self.sage_matrix(R,sparse=False)
elif typ == 'list':
return [ f._sage_(R) for f in self ]
elif typ == 'intvec':
from sage.modules.free_module_element import vector
return vector([sage.rings.integer.Integer(str(e)) for e in self])
elif typ == 'intmat':
from sage.matrix.constructor import matrix
from sage.rings.integer_ring import ZZ
A = matrix(ZZ, int(self.nrows()), int(self.ncols()))
for i in xrange(A.nrows()):
for j in xrange(A.ncols()):
A[i,j] = sage.rings.integer.Integer(str(self[i+1,j+1]))
return A
elif typ == 'string':
return repr(self)
elif typ == 'ideal':
R = R or self.sage_global_ring()
return R.ideal([p.sage_poly(R) for p in self])
elif typ in ['ring', 'qring']:
br = singular('basering')
self.set_ring()
R = self.sage_global_ring()
br.set_ring()
return R
raise NotImplementedError, "Coercion of this datatype not implemented yet"
def set_ring(self):
"""
Sets the current ring in Singular to be self.
EXAMPLES::
sage: R = singular.ring(7, '(a,b)', 'ds')
sage: S = singular.ring('real', '(a,b)', 'lp')
sage: singular.current_ring()
// characteristic : 0 (real)
// number of vars : 2
// block 1 : ordering lp
// : names a b
// block 2 : ordering C
sage: R.set_ring()
sage: singular.current_ring()
// characteristic : 7
// number of vars : 2
// block 1 : ordering ds
// : names a b
// block 2 : ordering C
"""
self.parent().set_ring(self)
def sage_flattened_str_list(self):
"""
EXAMPLES::
sage: R=singular.ring(0,'(x,y)','dp')
sage: RL = R.ringlist()
sage: RL.sage_flattened_str_list()
['0', 'x', 'y', 'dp', '1,1', 'C', '0', '_[1]=0']
"""
s = str(self)
c = '\[[0-9]*\]:'
r = re.compile(c)
s = r.sub('',s).strip()
return s.split()
def sage_structured_str_list(self):
r"""
If self is a Singular list of lists of Singular elements, returns
corresponding Sage list of lists of strings.
EXAMPLES::
sage: R=singular.ring(0,'(x,y)','dp')
sage: RL=R.ringlist()
sage: RL
[1]:
0
[2]:
[1]:
x
[2]:
y
[3]:
[1]:
[1]:
dp
[2]:
1,1
[2]:
[1]:
C
[2]:
0
[4]:
_[1]=0
sage: RL.sage_structured_str_list()
['0', ['x', 'y'], [['dp', '1,\n1 '], ['C', '0 ']], '0']
"""
if not (self.type()=='list'):
return str(self)
return [X.sage_structured_str_list() for X in self]
def trait_names(self):
"""
Returns the possible tab-completions for self. In this case, we
just return all the tab completions for the Singular object.
EXAMPLES::
sage: R = singular.ring(0,'(x,y)','dp')
sage: R.trait_names()
['exteriorPower',
...
'stdfglm']
"""
return self.parent().trait_names()
def type(self):
"""
Returns the internal type of this element.
EXAMPLES::
sage: R = PolynomialRing(GF(2^8,'a'),2,'x')
sage: R._singular_().type()
'ring'
sage: fs = singular('x0^2','poly')
sage: fs.type()
'poly'
"""
p = re.compile("// [\w]+ (\w+) [\w]*")
m = p.match(self.parent().eval("type(%s)"%self.name()))
return m.group(1)
def __iter__(self):
"""
EXAMPLES::
sage: R = singular.ring(0, '(x,y,z)', 'dp')
sage: A = singular.matrix(2,2)
sage: list(iter(A))
[[0], [0]]
sage: A[1,1] = 1; A[1,2] = 2
sage: A[2,1] = 3; A[2,2] = 4
sage: list(iter(A))
[[1,3], [2,4]]
"""
if self.type()=='matrix':
l = self.ncols()
else:
l = len(self)
for i in range(1, l+1):
yield self[i]
def _singular_(self):
"""
EXAMPLES::
sage: R = singular.ring(0, '(x,y,z)', 'dp')
sage: A = singular.matrix(2,2)
sage: A._singular_() is A
True
"""
return self
def attrib(self, name, value=None):
"""
Get and set attributes for self.
INPUT:
- ``name`` - string to choose the attribute
- ``value`` - boolean value or None for reading,
(default:None)
VALUES: isSB - the standard basis property is set by all commands
computing a standard basis like groebner, std, stdhilb etc.; used
by lift, dim, degree, mult, hilb, vdim, kbase isHomog - the weight
vector for homogeneous or quasihomogeneous ideals/modules isCI -
complete intersection property isCM - Cohen-Macaulay property rank
- set the rank of a module (see nrows) withSB - value of type
ideal, resp. module, is std withHilb - value of type intvec is
hilb(_,1) (see hilb) withRes - value of type list is a free
resolution withDim - value of type int is the dimension (see dim)
withMult - value of type int is the multiplicity (see mult)
EXAMPLE::
sage: P.<x,y,z> = PolynomialRing(QQ)
sage: I = Ideal([z^2, y*z, y^2, x*z, x*y, x^2])
sage: Ibar = I._singular_()
sage: Ibar.attrib('isSB')
0
sage: singular.eval('vdim(%s)'%Ibar.name()) # sage7 name is random
// ** sage7 is no standard basis
4
sage: Ibar.attrib('isSB',1)
sage: singular.eval('vdim(%s)'%Ibar.name())
'4'
"""
if value is None:
return int(self.parent().eval('attrib(%s,"%s")'%(self.name(),name)))
else:
self.parent().eval('attrib(%s,"%s",%d)'%(self.name(),name,value))
class SingularFunction(ExpectFunction):
def _sage_doc_(self):
"""
EXAMPLES::
sage: 'groebner' in singular.groebner._sage_doc_()
True
"""
if not nodes:
generate_docstring_dictionary()
prefix = \
"""
This function is an automatically generated pexpect wrapper around the Singular
function '%s'.
EXAMPLE::
sage: groebner = singular.groebner
sage: P.<x, y> = PolynomialRing(QQ)
sage: I = P.ideal(x^2-y, y+x)
sage: groebner(singular(I))
x+y,
y^2-y
"""%(self._name,)
prefix2 = \
"""
The Singular documentation for '%s' is given below.
"""%(self._name,)
try:
return prefix + prefix2 + nodes[node_names[self._name]]
except KeyError:
return prefix
class SingularFunctionElement(FunctionElement):
def _sage_doc_(self):
r"""
EXAMPLES::
sage: R = singular.ring(0, '(x,y,z)', 'dp')
sage: A = singular.matrix(2,2)
sage: 'matrix_expression' in A.nrows._sage_doc_()
True
"""
if not nodes:
generate_docstring_dictionary()
try:
return nodes[node_names[self._name]]
except KeyError:
return ""
def is_SingularElement(x):
r"""
Returns True is x is of type ``SingularElement``.
EXAMPLES::
sage: from sage.interfaces.singular import is_SingularElement
sage: is_SingularElement(singular(2))
True
sage: is_SingularElement(2)
False
"""
return isinstance(x, SingularElement)
def reduce_load():
"""
Note that this returns an invalid Singular object!
EXAMPLES::
sage: from sage.interfaces.singular import reduce_load
sage: reduce_load()
(invalid object -- defined in terms of closed session)
"""
return SingularElement(None, None, None)
nodes = {}
node_names = {}
def generate_docstring_dictionary():
"""
Generate global dictionaries which hold the docstrings for
Singular functions.
EXAMPLE::
sage: from sage.interfaces.singular import generate_docstring_dictionary
sage: generate_docstring_dictionary()
"""
global nodes
global node_names
nodes.clear()
node_names.clear()
singular_docdir = os.environ["SAGE_LOCAL"]+"/share/singular/"
new_node = re.compile("File: singular\.hlp, Node: ([^,]*),.*")
new_lookup = re.compile("\* ([^:]*):*([^.]*)\..*")
L, in_node, curr_node = [], False, None
for line in open(singular_docdir + "singular.hlp"):
m = re.match(new_node,line)
if m:
in_node = True
nodes[curr_node] = "".join(L)
L = []
curr_node, = m.groups()
elif in_node:
L.append(line)
else:
m = re.match(new_lookup, line)
if m:
a,b = m.groups()
node_names[a] = b.strip()
if line == "6 Index\n":
in_node = False
nodes[curr_node] = "".join(L)
def get_docstring(name):
"""
Return the docstring for the function ``name``.
INPUT:
- ``name`` - a Singular function name
EXAMPLE::
sage: from sage.interfaces.singular import get_docstring
sage: 'groebner' in get_docstring('groebner')
True
sage: 'standard.lib' in get_docstring('groebner')
True
"""
if not nodes:
generate_docstring_dictionary()
try:
return nodes[node_names[name]]
except KeyError:
return ""
singular = Singular()
def reduce_load_Singular():
"""
EXAMPLES::
sage: from sage.interfaces.singular import reduce_load_Singular
sage: reduce_load_Singular()
Singular
"""
return singular
import os
def singular_console():
"""
Spawn a new Singular command-line session.
EXAMPLES::
sage: singular_console() #not tested
SINGULAR / Development
A Computer Algebra System for Polynomial Computations / version 3-0-4
0<
by: G.-M. Greuel, G. Pfister, H. Schoenemann \ Nov 2007
FB Mathematik der Universitaet, D-67653 Kaiserslautern \
"""
os.system('Singular')
def singular_version():
"""
Returns the version of Singular being used.
EXAMPLES:
"""
return singular.eval('system("--version");')
class SingularGBLogPrettyPrinter:
"""
A device which prints Singular Groebner basis computation logs
more verbatim.
"""
rng_chng = re.compile("\[\d+:\d+\]")
new_elem = re.compile("s")
red_zero = re.compile("-")
red_post = re.compile("\.")
cri_hilb = re.compile("h")
hig_corn = re.compile("H\(\d+\)")
num_crit = re.compile("\(\d+\)")
red_num = re.compile("\(S:\d+\)")
deg_lead = re.compile("\d+")
red_para = re.compile("M\[(\d+),(\d+)\]")
red_betr = re.compile("b")
non_mini = re.compile("e")
crt_lne1 = re.compile("product criterion:(\d+) chain criterion:(\d+)")
crt_lne2 = re.compile("NF:(\d+) product criterion:(\d+), ext_product criterion:(\d+)")
pat_sync = re.compile("1\+(\d+);")
global_pattern = re.compile("(\[\d+:\d+\]|s|-|\.|h|H\(\d+\)|\(\d+\)|\(S:\d+\)|\d+|M\[\d+,[b,e]*\d+\]|b|e).*")
def __init__(self, verbosity=1):
"""
Construct a new Singular Groebner Basis log pretty printer.
INPUT:
- ``verbosity`` - how much information should be printed
(between 0 and 3)
EXAMPLE::
sage: from sage.interfaces.singular import SingularGBLogPrettyPrinter
sage: s0 = SingularGBLogPrettyPrinter(verbosity=0)
sage: s1 = SingularGBLogPrettyPrinter(verbosity=1)
sage: s0.write("[1:2]12")
sage: s1.write("[1:2]12")
Leading term degree: 12.
"""
self.verbosity = verbosity
self.curr_deg = 0
self.max_deg = 0
self.nf = 0
self.prod = 0
self.ext_prod = 0
self.chain = 0
self.storage = ""
self.sync = None
def write(self, s):
"""
EXAMPLE::
sage: from sage.interfaces.singular import SingularGBLogPrettyPrinter
sage: s3 = SingularGBLogPrettyPrinter(verbosity=3)
sage: s3.write("(S:1337)")
Performing complete reduction of 1337 elements.
sage: s3.write("M[389,12]")
Parallel reduction of 389 elements with 12 non-zero output elements.
"""
verbosity = self.verbosity
if self.storage:
s = self.storage + s
self.storage = ""
for line in s.splitlines():
match = re.match(SingularGBLogPrettyPrinter.pat_sync,line)
if match:
self.sync = int(match.groups()[0])
continue
if self.sync and line == "%d"%(self.sync+1):
self.sync = None
continue
if line.endswith(";"):
continue
if line.startswith(">"):
continue
if line.startswith("std") or line.startswith("slimgb"):
continue
match = re.match(SingularGBLogPrettyPrinter.crt_lne1,line)
if match:
self.prod,self.chain = map(int,re.match(SingularGBLogPrettyPrinter.crt_lne1,line).groups())
self.storage = ""
continue
match = re.match(SingularGBLogPrettyPrinter.crt_lne2,line)
if match:
self.nf,self.prod,self.ext_prod = map(int,re.match(SingularGBLogPrettyPrinter.crt_lne2,line).groups())
self.storage = ""
continue
while line:
match = re.match(SingularGBLogPrettyPrinter.global_pattern, line)
if not match:
self.storage = line
line = None
continue
token, = match.groups()
line = line[len(token):]
if re.match(SingularGBLogPrettyPrinter.rng_chng,token):
continue
elif re.match(SingularGBLogPrettyPrinter.new_elem,token) and verbosity >= 3:
print "New element found."
elif re.match(SingularGBLogPrettyPrinter.red_zero,token) and verbosity >= 2:
print "Reduction to zero."
elif re.match(SingularGBLogPrettyPrinter.red_post, token) and verbosity >= 2:
print "Reduction postponed."
elif re.match(SingularGBLogPrettyPrinter.cri_hilb, token) and verbosity >= 2:
print "Hilber series criterion applied."
elif re.match(SingularGBLogPrettyPrinter.hig_corn, token) and verbosity >= 1:
print "Maximal degree found: %s"%token
elif re.match(SingularGBLogPrettyPrinter.num_crit, token) and verbosity >= 1:
print "Leading term degree: %2d. Critical pairs: %s."%(self.curr_deg,token[1:-1])
elif re.match(SingularGBLogPrettyPrinter.red_num, token) and verbosity >= 3:
print "Performing complete reduction of %s elements."%token[3:-1]
elif re.match(SingularGBLogPrettyPrinter.deg_lead, token):
if verbosity >= 1:
print "Leading term degree: %2d."%int(token)
self.curr_deg = int(token)
if self.max_deg < self.curr_deg:
self.max_deg = self.curr_deg
elif re.match(SingularGBLogPrettyPrinter.red_para, token) and verbosity >= 3:
m,n = re.match(SingularGBLogPrettyPrinter.red_para,token).groups()
print "Parallel reduction of %s elements with %s non-zero output elements."%(m,n)
elif re.match(SingularGBLogPrettyPrinter.red_betr, token) and verbosity >= 3:
print "Replaced reductor by 'better' one."
elif re.match(SingularGBLogPrettyPrinter.non_mini, token) and verbosity >= 2:
print "New reductor with non-minimal leading term found."
def flush(self):
"""
EXAMPLE::
sage: from sage.interfaces.singular import SingularGBLogPrettyPrinter
sage: s3 = SingularGBLogPrettyPrinter(verbosity=3)
sage: s3.flush()
"""
sys.stdout.flush()
