"""Tools for manipulating of large commutative expressions. """
from sympy.core.add import Add
from sympy.core.mul import Mul
from sympy.core.power import Pow
from sympy.core.basic import Basic
from sympy.core.sympify import sympify
from sympy.core.numbers import Rational
from sympy.core.singleton import S
from sympy.core.coreerrors import NonCommutativeExpression
from sympy.core.containers import Tuple
def decompose_power(expr):
"""
Decompose power into symbolic base and integer exponent.
**Example**
>>> from sympy.core.exprtools import decompose_power
>>> from sympy.abc import x, y
>>> decompose_power(x)
(x, 1)
>>> decompose_power(x**2)
(x, 2)
>>> decompose_power(x**(2*y))
(x**y, 2)
>>> decompose_power(x**(2*y/3))
(x**(y/3), 2)
"""
base, exp = expr.as_base_exp()
if exp.is_Number:
if exp.is_Rational:
if not exp.is_Integer:
base = Pow(base, Rational(1, exp.q))
exp = exp.p
else:
base, exp = expr, 1
else:
exp, tail = exp.as_coeff_mul()
if exp.is_Rational:
if not exp.is_Integer:
tail += (Rational(1, exp.q),)
base, exp = Pow(base, Mul(*tail)), exp.p
else:
base, exp = expr, 1
return base, exp
class Factors(object):
"""Efficient representation of ``f_1*f_2*...*f_n``. """
__slots__ = ['factors', 'gens']
def __init__(self, factors=None):
if factors is None:
factors = {}
self.factors = factors
self.gens = frozenset(factors.keys())
def __hash__(self):
return hash((tuple(self.factors), self.gens))
def __repr__(self):
return "Factors(%s)" % self.factors
def as_expr(self):
return Mul(*[ factor**exp for factor, exp in self.factors.iteritems() ])
def normal(self, other):
self_factors = dict(self.factors)
other_factors = dict(other.factors)
for factor, self_exp in self.factors.iteritems():
try:
other_exp = other.factors[factor]
except KeyError:
continue
exp = self_exp - other_exp
if not exp:
del self_factors[factor]
del other_factors[factor]
else:
if exp > 0:
self_factors[factor] = exp
del other_factors[factor]
else:
del self_factors[factor]
other_factors[factor] = -exp
return Factors(self_factors), Factors(other_factors)
def mul(self, other):
factors = dict(self.factors)
for factor, exp in other.factors.iteritems():
if factor in factors:
exp = factors[factor] + exp
if not exp:
del factors[factor]
continue
factors[factor] = exp
return Factors(factors)
def div(self, other):
quo, rem = dict(self.factors), {}
for factor, exp in other.factors.iteritems():
if factor in quo:
exp = quo[factor] - exp
if exp <= 0:
del quo[factor]
if exp >= 0:
if exp:
quo[factor] = exp
continue
exp = -exp
rem[factor] = exp
return Factors(quo), Factors(rem)
def quo(self, other):
return self.div(other)[0]
def rem(self, other):
return self.div(other)[1]
def pow(self, other):
if type(other) is int and other >= 0:
factors = {}
if other:
for factor, exp in self.factors.iteritems():
factors[factor] = exp*other
return Factors(factors)
else:
raise ValueError("expected non-negative integer, got %s" % other)
def gcd(self, other):
factors = {}
for factor, exp in self.factors.iteritems():
if factor in other.factors:
exp = min(exp, other.factors[factor])
factors[factor] = exp
return Factors(factors)
def lcm(self, other):
factors = dict(self.factors)
for factor, exp in other.factors.iteritems():
if factor in factors:
exp = max(exp, factors[factor])
factors[factor] = exp
return Factors(factors)
def __mul__(self, other):
if isinstance(other, Factors):
return self.mul(other)
else:
return NotImplemented
def __divmod__(self, other):
if isinstance(other, Factors):
return self.div(other)
else:
return NotImplemented
def __div__(self, other):
if isinstance(other, Factors):
return self.quo(other)
else:
return NotImplemented
__truediv__ = __div__
def __mod__(self, other):
if isinstance(other, Factors):
return self.rem(other)
else:
return NotImplemented
def __pow__(self, other):
if type(other) is int:
return self.pow(other)
else:
return NotImplemented
def __eq__(self, other):
return self.factors == other.factors
def __ne__(self, other):
return not self.__eq__(other)
class Term(object):
"""Efficient representation of ``coeff*(numer/denom)``. """
__slots__ = ['coeff', 'numer', 'denom']
def __init__(self, term, numer=None, denom=None):
if numer is None and denom is None:
if not term.is_commutative:
raise NonCommutativeExpression('commutative expression expected')
coeff, factors = term.as_coeff_mul()
numer, denom = {}, {}
for factor in factors:
base, exp = decompose_power(factor)
if base.is_Add:
cont, base = base.primitive()
coeff *= cont**exp
if exp > 0:
numer[base] = exp
else:
denom[base] = -exp
numer = Factors(numer)
denom = Factors(denom)
else:
coeff = term
if numer is None:
numer = Factors()
if denom is None:
denom = Factors()
self.coeff = coeff
self.numer = numer
self.denom = denom
def __hash__(self):
return hash((self.coeff, self.numer, self.denom))
def __repr__(self):
return "Term(%s, %s, %s)" % (self.coeff, self.numer, self.denom)
def as_expr(self):
return self.coeff*(self.numer.as_expr()/self.denom.as_expr())
def mul(self, other):
coeff = self.coeff*other.coeff
numer = self.numer.mul(other.numer)
denom = self.denom.mul(other.denom)
numer, denom = numer.normal(denom)
return Term(coeff, numer, denom)
def inv(self):
return Term(1/self.coeff, self.denom, self.numer)
def quo(self, other):
return self.mul(other.inv())
def pow(self, other):
if other < 0:
return self.inv().pow(-other)
else:
return Term(self.coeff ** other,
self.numer.pow(other),
self.denom.pow(other))
def gcd(self, other):
return Term(self.coeff.gcd(other.coeff),
self.numer.gcd(other.numer),
self.denom.gcd(other.denom))
def lcm(self, other):
return Term(self.coeff.lcm(other.coeff),
self.numer.lcm(other.numer),
self.denom.lcm(other.denom))
def __mul__(self, other):
if isinstance(other, Term):
return self.mul(other)
else:
return NotImplemented
def __div__(self, other):
if isinstance(other, Term):
return self.quo(other)
else:
return NotImplemented
__truediv__ = __div__
def __pow__(self, other):
if type(other) is int:
return self.pow(other)
else:
return NotImplemented
def __eq__(self, other):
return (self.coeff == other.coeff and
self.numer == other.numer and
self.denom == other.denom)
def __ne__(self, other):
return not self.__eq__(other)
def _gcd_terms(terms):
"""Helper function for :func:`gcd_terms`. """
if isinstance(terms, Basic) and not isinstance(terms, Tuple):
terms = Add.make_args(terms)
if len(terms) <= 1:
if not terms:
return S.Zero, S.Zero, S.One
else:
return terms[0], S.One, S.One
terms = map(Term, terms)
cont = terms[0]
for term in terms[1:]:
cont = cont.gcd(term)
for i, term in enumerate(terms):
terms[i] = term.quo(cont)
denom = terms[0].denom
for term in terms[1:]:
denom = denom.lcm(term.denom)
numers = []
for term in terms:
numer = term.numer.mul(denom.quo(term.denom))
numers.append(term.coeff*numer.as_expr())
cont = cont.as_expr()
numer = Add(*numers)
denom = denom.as_expr()
if numer.is_Add:
_cont, numer = numer.primitive()
cont *= _cont
return cont, numer, denom
[docs]def gcd_terms(terms):
"""
Compute the GCD of ``terms`` and put them together.
**Example**
>>> from sympy.core import gcd_terms
>>> from sympy.abc import x, y
>>> gcd_terms((x + 1)**2*y + (x + 1)*y**2)
y*(x + 1)*(x + y + 1)
"""
from sympy.polys.polytools import _keep_coeff
cont, numer, denom = _gcd_terms(sympify(terms))
coeff, factors = cont.as_coeff_Mul()
return _keep_coeff(coeff, factors*numer/denom)