def test_monomial_key():
assert monomial_key() == lex
assert monomial_key('lex') == lex
assert monomial_key('grlex') == grlex
assert monomial_key('grevlex') == grevlex
raises(ValueError, "monomial_key('foo')")
raises(ValueError, "monomial_key(1)")
def test_monomial_key():
assert monomial_key() == monomial_lex_key
assert monomial_key('lex') == monomial_lex_key
assert monomial_key('grlex') == monomial_grlex_key
assert monomial_key('grevlex') == monomial_grevlex_key
raises(ValueError, "monomial_key('foo')")
raises(ValueError, "monomial_key(1)")
def test_monomial_key():
assert monomial_key() == monomial_lex_key
assert monomial_key('lex') == monomial_lex_key
assert monomial_key('grlex') == monomial_grlex_key
assert monomial_key('grevlex') == monomial_grevlex_key
raises(ValueError, "monomial_key('foo')")
raises(ValueError, "monomial_key(1)")
def test_monomial_key():
assert monomial_key() == lex
assert monomial_key('lex') == lex
assert monomial_key('grlex') == grlex
assert monomial_key('grevlex') == grevlex
raises(ValueError, "monomial_key('foo')")
raises(ValueError, "monomial_key(1)")
def dmp_list_terms(f, u, K, order=None):
"""
List all non-zero terms from ``f`` in the given order ``order``.
Examples
========
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.densebasic import dmp_list_terms
>>> f = ZZ.map([[1, 1], [2, 3]])
>>> dmp_list_terms(f, 1, ZZ)
[((1, 1), 1), ((1, 0), 1), ((0, 1), 2), ((0, 0), 3)]
>>> dmp_list_terms(f, 1, ZZ, order='grevlex')
[((1, 1), 1), ((1, 0), 1), ((0, 1), 2), ((0, 0), 3)]
"""
terms = _rec_list_terms(f, u, ())
if not terms:
return [((0,)*(u + 1), K.zero)]
if order is None:
return terms
else:
return sdp_sort(terms, monomial_key(order))
def __init__(self, gens, container, order="lex"):
SubModule.__init__(self, gens, container)
if not isinstance(container, FreeModulePolyRing):
raise NotImplementedError('This implementation is for submodules of '
+ 'FreeModulePolyRing, got %s' % container)
self.order = ModuleOrder(monomial_key(order), self.ring.order)
self._gb = None
def __init__(self, gens, container, order="lex", TOP=True):
SubModule.__init__(self, gens, container)
if not isinstance(container, FreeModulePolyRing):
raise NotImplementedError('This implementation is for submodules of '
+ 'FreeModulePolyRing, got %s' % container)
self.order = ModuleOrder(monomial_key(order), self.ring.order, TOP)
self._gb = None
self._gbe = None
def __init__(self, dom, *gens, **opts):
if not gens:
raise GeneratorsNeeded("generators not specified")
lev = len(gens) - 1
self.zero = self.dtype.zero(lev, dom, ring=self)
self.one = self.dtype.one(lev, dom, ring=self)
self.dom = dom
self.gens = gens
# NOTE 'order' may not be set if inject was called through CompositeDomain
self.order = opts.get('order', monomial_key(self.default_order))
def __init__(self, dom, *gens, **opts):
if not gens:
raise GeneratorsNeeded("generators not specified")
lev = len(gens) - 1
self.zero = self.dtype.zero(lev, dom, ring=self)
self.one = self.dtype.one(lev, dom, ring=self)
self.dom = dom
self.gens = gens
# NOTE 'order' may not be set if inject was called through CompositeDomain
self.order = opts.get('order', monomial_key(self.default_order))
def _parse_order(cls, order):
"""Parse and configure the ordering of terms. """
from sympy.polys.monomialtools import monomial_key
try:
reverse = order.startswith('rev-')
except AttributeError:
reverse = False
else:
if reverse:
order = order[4:]
monom_key = monomial_key(order)
def key(term):
_, ((re, im), monom, ncpart) = term
ncpart = [ e.as_tuple_tree() for e in ncpart ]
return monom_key(monom), tuple(ncpart), (-re, -im)
return key, reverse
def _parse_order(cls, order):
"""Parse and configure the ordering of terms. """
from sympy.polys.monomialtools import monomial_key
try:
reverse = order.startswith('rev-')
except AttributeError:
reverse = False
else:
if reverse:
order = order[4:]
monom_key = monomial_key(order)
def key(term):
_, ((re, im), monom, ncpart) = term
ncpart = [e.as_tuple_tree() for e in ncpart]
return monom_key(monom), tuple(ncpart), (-re, -im)
return key, reverse
def PolynomialRing(dom, *gens, **opts):
r"""
Create a generalized multivariate polynomial ring.
A generalized polynomial ring is defined by a ground field `K`, a set
of generators (typically `x_1, \dots, x_n`) and a monomial order `<`.
The monomial order can be global, local or mixed. In any case it induces
a total ordering on the monomials, and there exists for every (non-zero)
polynomial `f \in K[x_1, \dots, x_n]` a well-defined "leading monomial"
`LM(f) = LM(f, >)`. One can then define a multiplicative subset
`S = S_> = \{f \in K[x_1, \dots, x_n] | LM(f) = 1\}`. The generalized
polynomial ring corresponding to the monomial order is
`R = S^{-1}K[x_1, \dots, x_n]`.
If `>` is a so-called global order, that is `1` is the smallest monomial,
then we just have `S = K` and `R = K[x_1, \dots, x_n]`.
Examples
========
A few examples may make this clearer.
>>> from sympy.abc import x, y
>>> from sympy import QQ
Our first ring uses global lexicographic order.
>>> R1 = QQ.poly_ring(x, y, order=(("lex", x, y),))
The second ring uses local lexicographic order. Note that when using a
single (non-product) order, you can just specify the name and omit the
variables:
>>> R2 = QQ.poly_ring(x, y, order="ilex")
The third and fourth rings use a mixed orders:
>>> o1 = (("ilex", x), ("lex", y))
>>> o2 = (("lex", x), ("ilex", y))
>>> R3 = QQ.poly_ring(x, y, order=o1)
>>> R4 = QQ.poly_ring(x, y, order=o2)
We will investigate what elements of `K(x, y)` are contained in the various
rings.
>>> L = [x, 1/x, y/(1 + x), 1/(1 + y), 1/(1 + x*y)]
>>> test = lambda R: [f in R for f in L]
The first ring is just `K[x, y]`:
>>> test(R1)
[True, False, False, False, False]
The second ring is R1 localised at the maximal ideal (x, y):
>>> test(R2)
[True, False, True, True, True]
The third ring is R1 localised at the prime ideal (x):
>>> test(R3)
[True, False, True, False, True]
Finally the fourth ring is R1 localised at `S = K[x, y] \setminus yK[y]`:
>>> test(R4)
[True, False, False, True, False]
"""
order = opts.get("order", GeneralizedPolynomialRing.default_order)
if iterable(order):
order = build_product_order(order, gens)
order = monomial_key(order)
opts['order'] = order
if order.is_global:
return GlobalPolynomialRing(dom, *gens, **opts)
else:
return GeneralizedPolynomialRing(dom, *gens, **opts)
def PolynomialRing(dom, *gens, **opts):
r"""
Create a generalized multivariate polynomial ring.
A generalized polynomial ring is defined by a ground field `K`, a set
of generators (typically `x_1, \dots, x_n`) and a monomial order `<`.
The monomial order can be global, local or mixed. In any case it induces
a total ordering on the monomials, and there exists for every (non-zero)
polynomial `f \in K[x_1, \dots, x_n]` a well-defined "leading monomial"
`LM(f) = LM(f, >)`. One can then define a multiplicative subset
`S = S_> = \{f \in K[x_1, \dots, x_n] | LM(f) = 1\}`. The generalized
polynomial ring corresponding to the monomial order is
`R = S^{-1}K[x_1, \dots, x_n]`.
If `>` is a so-called global order, that is `1` is the smallest monomial,
then we just have `S = K` and `R = K[x_1, \dots, x_n]`.
Examples
========
A few examples may make this clearer.
>>> from sympy.abc import x, y
>>> from sympy import QQ
Our first ring uses global lexicographic order.
>>> R1 = QQ.poly_ring(x, y, order=(("lex", x, y),))
The second ring uses local lexicographic order. Note that when using a
single (non-product) order, you can just specify the name and omit the
variables:
>>> R2 = QQ.poly_ring(x, y, order="ilex")
The third and fourth rings use a mixed orders:
>>> o1 = (("ilex", x), ("lex", y))
>>> o2 = (("lex", x), ("ilex", y))
>>> R3 = QQ.poly_ring(x, y, order=o1)
>>> R4 = QQ.poly_ring(x, y, order=o2)
We will investigate what elements of `K(x, y)` are contained in the various
rings.
>>> L = [x, 1/x, y/(1 + x), 1/(1 + y), 1/(1 + x*y)]
>>> test = lambda R: [f in R for f in L]
The first ring is just `K[x, y]`:
>>> test(R1)
[True, False, False, False, False]
The second ring is R1 localised at the maximal ideal (x, y):
>>> test(R2)
[True, False, True, True, True]
The third ring is R1 localised at the prime ideal (x):
>>> test(R3)
[True, False, True, False, True]
Finally the fourth ring is R1 localised at `S = K[x, y] \setminus yK[y]`:
>>> test(R4)
[True, False, False, True, False]
"""
order = opts.get("order", GeneralizedPolynomialRing.default_order)
if iterable(order):
order = build_product_order(order, gens)
order = monomial_key(order)
opts['order'] = order
if order.is_global:
return GlobalPolynomialRing(dom, *gens, **opts)
else:
return GeneralizedPolynomialRing(dom, *gens, **opts)