"""Tools for manipulating of large commutative expressions. """
from __future__ import print_function, division
from sympy.core.add import Add
from sympy.core.compatibility import iterable, is_sequence, SYMPY_INTS, range
from sympy.core.mul import Mul, _keep_coeff
from sympy.core.power import Pow
from sympy.core.basic import Basic, preorder_traversal
from sympy.core.expr import Expr
from sympy.core.sympify import sympify
from sympy.core.numbers import Rational, Integer, Number, I
from sympy.core.singleton import S
from sympy.core.symbol import Dummy
from sympy.core.coreerrors import NonCommutativeExpression
from sympy.core.containers import Tuple, Dict
from sympy.utilities import default_sort_key
from sympy.utilities.iterables import (common_prefix, common_suffix,
variations, ordered)
from collections import defaultdict
_eps = Dummy(positive=True)
def _isnumber(i):
return isinstance(i, (SYMPY_INTS, float)) or i.is_Number
def _monotonic_sign(self):
"""Return the value closest to 0 that ``self`` may have if all symbols
are signed and the result is uniformly the same sign for all values of symbols.
If a symbol is only signed but not known to be an
integer or the result is 0 then a symbol representative of the sign of self
will be returned. Otherwise, None is returned if a) the sign could be positive
or negative or b) self is not in one of the following forms:
- L(x, y, ...) + A: a function linear in all symbols x, y, ... with an
additive constant; if A is zero then the function can be a monomial whose
sign is monotonic over the range of the variables, e.g. (x + 1)**3 if x is
nonnegative.
- A/L(x, y, ...) + B: the inverse of a function linear in all symbols x, y, ...
that does not have a sign change from positive to negative for any set
of values for the variables.
- M(x, y, ...) + A: a monomial M whose factors are all signed and a constant, A.
- A/M(x, y, ...) + B: the inverse of a monomial and constants A and B.
- P(x): a univariate polynomial
Examples
========
>>> from sympy.core.exprtools import _monotonic_sign as F
>>> from sympy import Dummy, S
>>> nn = Dummy(integer=True, nonnegative=True)
>>> p = Dummy(integer=True, positive=True)
>>> p2 = Dummy(integer=True, positive=True)
>>> F(nn + 1)
1
>>> F(p - 1)
_nneg
>>> F(nn*p + 1)
1
>>> F(p2*p + 1)
2
>>> F(nn - 1) # could be negative, zero or positive
"""
if not self.is_real:
return
if (-self).is_Symbol:
rv = _monotonic_sign(-self)
return rv if rv is None else -rv
if not self.is_Add and self.as_numer_denom()[1].is_number:
s = self
if s.is_prime:
if s.is_odd:
return S(3)
else:
return S(2)
elif s.is_positive:
if s.is_even:
return S(2)
elif s.is_integer:
return S.One
else:
return _eps
elif s.is_negative:
if s.is_even:
return S(-2)
elif s.is_integer:
return S.NegativeOne
else:
return -_eps
if s.is_zero or s.is_nonpositive or s.is_nonnegative:
return S.Zero
return None
# univariate polynomial
free = self.free_symbols
if len(free) == 1:
if self.is_polynomial():
from sympy.polys.polytools import real_roots
from sympy.polys.polyroots import roots
from sympy.polys.polyerrors import PolynomialError
x = free.pop()
x0 = _monotonic_sign(x)
if x0 == _eps or x0 == -_eps:
x0 = S.Zero
if x0 is not None:
d = self.diff(x)
if d.is_number:
roots = []
else:
try:
roots = real_roots(d)
except (PolynomialError, NotImplementedError):
roots = [r for r in roots(d, x) if r.is_real]
y = self.subs(x, x0)
if x.is_nonnegative and all(r <= x0 for r in roots):
if y.is_nonnegative and d.is_positive:
if y:
return y if y.is_positive else Dummy('pos', positive=True)
else:
return Dummy('nneg', nonnegative=True)
if y.is_nonpositive and d.is_negative:
if y:
return y if y.is_negative else Dummy('neg', negative=True)
else:
return Dummy('npos', nonpositive=True)
elif x.is_nonpositive and all(r >= x0 for r in roots):
if y.is_nonnegative and d.is_negative:
if y:
return Dummy('pos', positive=True)
else:
return Dummy('nneg', nonnegative=True)
if y.is_nonpositive and d.is_positive:
if y:
return Dummy('neg', negative=True)
else:
return Dummy('npos', nonpositive=True)
else:
n, d = self.as_numer_denom()
den = None
if n.is_number:
den = _monotonic_sign(d)
elif not d.is_number:
if _monotonic_sign(n) is not None:
den = _monotonic_sign(d)
if den is not None and (den.is_positive or den.is_negative):
v = n*den
if v.is_positive:
return Dummy('pos', positive=True)
elif v.is_nonnegative:
return Dummy('nneg', nonnegative=True)
elif v.is_negative:
return Dummy('neg', negative=True)
elif v.is_nonpositive:
return Dummy('npos', nonpositive=True)
return None
# multivariate
c, a = self.as_coeff_Add()
v = None
if not a.is_polynomial():
# F/A or A/F where A is a number and F is a signed, rational monomial
n, d = a.as_numer_denom()
if not (n.is_number or d.is_number):
return
if (
a.is_Mul or a.is_Pow) and \
a.is_rational and \
all(p.exp.is_Integer for p in a.atoms(Pow) if p.is_Pow) and \
(a.is_positive or a.is_negative):
v = S(1)
for ai in Mul.make_args(a):
if ai.is_number:
v *= ai
continue
reps = {}
for x in ai.free_symbols:
reps[x] = _monotonic_sign(x)
if reps[x] is None:
return
v *= ai.subs(reps)
elif c:
# signed linear expression
if not any(p for p in a.atoms(Pow) if not p.is_number) and (a.is_nonpositive or a.is_nonnegative):
free = list(a.free_symbols)
p = {}
for i in free:
v = _monotonic_sign(i)
if v is None:
return
p[i] = v or (_eps if i.is_nonnegative else -_eps)
v = a.xreplace(p)
if v is not None:
rv = v + c
if v.is_nonnegative and rv.is_positive:
return rv.subs(_eps, 0)
if v.is_nonpositive and rv.is_negative:
return rv.subs(_eps, 0)
def decompose_power(expr):
"""
Decompose power into symbolic base and integer exponent.
This is strictly only valid if the exponent from which
the integer is extracted is itself an integer or the
base is positive. These conditions are assumed and not
checked here.
Examples
========
>>> 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(rational=True)
if exp is S.NegativeOne:
base, exp = Pow(base, tail), -1
elif exp is not S.One:
tail = _keep_coeff(Rational(1, exp.q), tail)
base, exp = Pow(base, tail), exp.p
else:
base, exp = expr, 1
return base, exp
def decompose_power_rat(expr):
"""
Decompose power into symbolic base and rational exponent.
"""
base, exp = expr.as_base_exp()
if exp.is_Number:
if not exp.is_Rational:
base, exp = expr, 1
else:
exp, tail = exp.as_coeff_Mul(rational=True)
if exp is S.NegativeOne:
base, exp = Pow(base, tail), -1
elif exp is not S.One:
tail = _keep_coeff(Rational(1, exp.q), tail)
base, exp = Pow(base, 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): # Factors
"""Initialize Factors from dict or expr.
Examples
========
>>> from sympy.core.exprtools import Factors
>>> from sympy.abc import x
>>> from sympy import I
>>> e = 2*x**3
>>> Factors(e)
Factors({2: 1, x: 3})
>>> Factors(e.as_powers_dict())
Factors({2: 1, x: 3})
>>> f = _
>>> f.factors # underlying dictionary
{2: 1, x: 3}
>>> f.gens # base of each factor
frozenset([2, x])
>>> Factors(0)
Factors({0: 1})
>>> Factors(I)
Factors({I: 1})
Notes
=====
Although a dictionary can be passed, only minimal checking is
performed: powers of -1 and I are made canonical.
"""
if isinstance(factors, (SYMPY_INTS, float)):
factors = S(factors)
if isinstance(factors, Factors):
factors = factors.factors.copy()
elif factors is None or factors is S.One:
factors = {}
elif factors is S.Zero or factors == 0:
factors = {S.Zero: S.One}
elif isinstance(factors, Number):
n = factors
factors = {}
if n < 0:
factors[S.NegativeOne] = S.One
n = -n
if n is not S.One:
if n.is_Float or n.is_Integer or n is S.Infinity:
factors[n] = S.One
elif n.is_Rational:
# since we're processing Numbers, the denominator is
# stored with a negative exponent; all other factors
# are left .
if n.p != 1:
factors[Integer(n.p)] = S.One
factors[Integer(n.q)] = S.NegativeOne
else:
raise ValueError('Expected Float|Rational|Integer, not %s' % n)
elif isinstance(factors, Basic) and not factors.args:
factors = {factors: S.One}
elif isinstance(factors, Expr):
c, nc = factors.args_cnc()
i = c.count(I)
for _ in range(i):
c.remove(I)
factors = dict(Mul._from_args(c).as_powers_dict())
if i:
factors[I] = S.One*i
if nc:
factors[Mul(*nc, evaluate=False)] = S.One
else:
factors = factors.copy() # /!\ should be dict-like
# tidy up -/+1 and I exponents if Rational
handle = []
for k in factors:
if k is I or k in (-1, 1):
handle.append(k)
if handle:
i1 = S.One
for k in handle:
if not _isnumber(factors[k]):
continue
i1 *= k**factors.pop(k)
if i1 is not S.One:
for a in i1.args if i1.is_Mul else [i1]: # at worst, -1.0*I*(-1)**e
if a is S.NegativeOne:
factors[a] = S.One
elif a is I:
factors[I] = S.One
elif a.is_Pow:
if S.NegativeOne not in factors:
factors[S.NegativeOne] = S.Zero
factors[S.NegativeOne] += a.exp
elif a == 1:
factors[a] = S.One
elif a == -1:
factors[-a] = S.One
factors[S.NegativeOne] = S.One
else:
raise ValueError('unexpected factor in i1: %s' % a)
self.factors = factors
try:
self.gens = frozenset(factors.keys())
except AttributeError:
raise TypeError('expecting Expr or dictionary')
def __hash__(self): # Factors
keys = tuple(ordered(self.factors.keys()))
values = [self.factors[k] for k in keys]
return hash((keys, values))
def __repr__(self): # Factors
return "Factors({%s})" % ', '.join(
['%s: %s' % (k, v) for k, v in ordered(self.factors.items())])
@property
def is_zero(self): # Factors
"""
>>> from sympy.core.exprtools import Factors
>>> Factors(0).is_zero
True
"""
f = self.factors
return len(f) == 1 and S.Zero in f
@property
def is_one(self): # Factors
"""
>>> from sympy.core.exprtools import Factors
>>> Factors(1).is_one
True
"""
return not self.factors
def as_expr(self): # Factors
"""Return the underlying expression.
Examples
========
>>> from sympy.core.exprtools import Factors
>>> from sympy.abc import x, y
>>> Factors((x*y**2).as_powers_dict()).as_expr()
x*y**2
"""
args = []
for factor, exp in self.factors.items():
if exp != 1:
b, e = factor.as_base_exp()
if isinstance(exp, int):
e = _keep_coeff(Integer(exp), e)
elif isinstance(exp, Rational):
e = _keep_coeff(exp, e)
else:
e *= exp
args.append(b**e)
else:
args.append(factor)
return Mul(*args)
def mul(self, other): # Factors
"""Return Factors of ``self * other``.
Examples
========
>>> from sympy.core.exprtools import Factors
>>> from sympy.abc import x, y, z
>>> a = Factors((x*y**2).as_powers_dict())
>>> b = Factors((x*y/z).as_powers_dict())
>>> a.mul(b)
Factors({x: 2, y: 3, z: -1})
>>> a*b
Factors({x: 2, y: 3, z: -1})
"""
if not isinstance(other, Factors):
other = Factors(other)
if any(f.is_zero for f in (self, other)):
return Factors(S.Zero)
factors = dict(self.factors)
for factor, exp in other.factors.items():
if factor in factors:
exp = factors[factor] + exp
if not exp:
del factors[factor]
continue
factors[factor] = exp
return Factors(factors)
def normal(self, other):
"""Return ``self`` and ``other`` with ``gcd`` removed from each.
The only differences between this and method ``div`` is that this
is 1) optimized for the case when there are few factors in common and
2) this does not raise an error if ``other`` is zero.
See Also
========
div
"""
if not isinstance(other, Factors):
other = Factors(other)
if other.is_zero:
return (Factors(), Factors(S.Zero))
if self.is_zero:
return (Factors(S.Zero), Factors())
self_factors = dict(self.factors)
other_factors = dict(other.factors)
for factor, self_exp in self.factors.items():
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]
elif _isnumber(exp):
if exp > 0:
self_factors[factor] = exp
del other_factors[factor]
else:
del self_factors[factor]
other_factors[factor] = -exp
else:
r = self_exp.extract_additively(other_exp)
if r is not None:
if r:
self_factors[factor] = r
del other_factors[factor]
else: # should be handled already
del self_factors[factor]
del other_factors[factor]
else:
sc, sa = self_exp.as_coeff_Add()
if sc:
oc, oa = other_exp.as_coeff_Add()
diff = sc - oc
if diff > 0:
self_factors[factor] -= oc
other_exp = oa
elif diff < 0:
self_factors[factor] -= sc
other_factors[factor] -= sc
other_exp = oa - diff
else:
self_factors[factor] = sa
other_exp = oa
if other_exp:
other_factors[factor] = other_exp
else:
del other_factors[factor]
return Factors(self_factors), Factors(other_factors)
def div(self, other): # Factors
"""Return ``self`` and ``other`` with ``gcd`` removed from each.
This is optimized for the case when there are many factors in common.
Examples
========
>>> from sympy.core.exprtools import Factors
>>> from sympy.abc import x, y, z
>>> from sympy import S
>>> a = Factors((x*y**2).as_powers_dict())
>>> a.div(a)
(Factors({}), Factors({}))
>>> a.div(x*z)
(Factors({y: 2}), Factors({z: 1}))
The ``/`` operator only gives ``quo``:
>>> a/x
Factors({y: 2})
Factors treats its factors as though they are all in the numerator, so
if you violate this assumption the results will be correct but will
not strictly correspond to the numerator and denominator of the ratio:
>>> a.div(x/z)
(Factors({y: 2}), Factors({z: -1}))
Factors is also naive about bases: it does not attempt any denesting
of Rational-base terms, for example the following does not become
2**(2*x)/2.
>>> Factors(2**(2*x + 2)).div(S(8))
(Factors({2: 2*x + 2}), Factors({8: 1}))
factor_terms can clean up such Rational-bases powers:
>>> from sympy.core.exprtools import factor_terms
>>> n, d = Factors(2**(2*x + 2)).div(S(8))
>>> n.as_expr()/d.as_expr()
2**(2*x + 2)/8
>>> factor_terms(_)
2**(2*x)/2
"""
quo, rem = dict(self.factors), {}
if not isinstance(other, Factors):
other = Factors(other)
if other.is_zero:
raise ZeroDivisionError
if self.is_zero:
return (Factors(S.Zero), Factors())
for factor, exp in other.factors.items():
if factor in quo:
d = quo[factor] - exp
if _isnumber(d):
if d <= 0:
del quo[factor]
if d >= 0:
if d:
quo[factor] = d
continue
exp = -d
else:
r = quo[factor].extract_additively(exp)
if r is not None:
if r:
quo[factor] = r
else: # should be handled already
del quo[factor]
else:
other_exp = exp
sc, sa = quo[factor].as_coeff_Add()
if sc:
oc, oa = other_exp.as_coeff_Add()
diff = sc - oc
if diff > 0:
quo[factor] -= oc
other_exp = oa
elif diff < 0:
quo[factor] -= sc
other_exp = oa - diff
else:
quo[factor] = sa
other_exp = oa
if other_exp:
rem[factor] = other_exp
else:
assert factor not in rem
continue
rem[factor] = exp
return Factors(quo), Factors(rem)
def quo(self, other): # Factors
"""Return numerator Factor of ``self / other``.
Examples
========
>>> from sympy.core.exprtools import Factors
>>> from sympy.abc import x, y, z
>>> a = Factors((x*y**2).as_powers_dict())
>>> b = Factors((x*y/z).as_powers_dict())
>>> a.quo(b) # same as a/b
Factors({y: 1})
"""
return self.div(other)[0]
def rem(self, other): # Factors
"""Return denominator Factors of ``self / other``.
Examples
========
>>> from sympy.core.exprtools import Factors
>>> from sympy.abc import x, y, z
>>> a = Factors((x*y**2).as_powers_dict())
>>> b = Factors((x*y/z).as_powers_dict())
>>> a.rem(b)
Factors({z: -1})
>>> a.rem(a)
Factors({})
"""
return self.div(other)[1]
def pow(self, other): # Factors
"""Return self raised to a non-negative integer power.
Examples
========
>>> from sympy.core.exprtools import Factors
>>> from sympy.abc import x, y
>>> a = Factors((x*y**2).as_powers_dict())
>>> a**2
Factors({x: 2, y: 4})
"""
if isinstance(other, Factors):
other = other.as_expr()
if other.is_Integer:
other = int(other)
if isinstance(other, SYMPY_INTS) and other >= 0:
factors = {}
if other:
for factor, exp in self.factors.items():
factors[factor] = exp*other
return Factors(factors)
else:
raise ValueError("expected non-negative integer, got %s" % other)
def gcd(self, other): # Factors
"""Return Factors of ``gcd(self, other)``. The keys are
the intersection of factors with the minimum exponent for
each factor.
Examples
========
>>> from sympy.core.exprtools import Factors
>>> from sympy.abc import x, y, z
>>> a = Factors((x*y**2).as_powers_dict())
>>> b = Factors((x*y/z).as_powers_dict())
>>> a.gcd(b)
Factors({x: 1, y: 1})
"""
if not isinstance(other, Factors):
other = Factors(other)
if other.is_zero:
return Factors(self.factors)
factors = {}
for factor, exp in self.factors.items():
factor, exp = sympify(factor), sympify(exp)
if factor in other.factors:
lt = (exp - other.factors[factor]).is_negative
if lt == True:
factors[factor] = exp
elif lt == False:
factors[factor] = other.factors[factor]
return Factors(factors)
def lcm(self, other): # Factors
"""Return Factors of ``lcm(self, other)`` which are
the union of factors with the maximum exponent for
each factor.
Examples
========
>>> from sympy.core.exprtools import Factors
>>> from sympy.abc import x, y, z
>>> a = Factors((x*y**2).as_powers_dict())
>>> b = Factors((x*y/z).as_powers_dict())
>>> a.lcm(b)
Factors({x: 1, y: 2, z: -1})
"""
if not isinstance(other, Factors):
other = Factors(other)
if any(f.is_zero for f in (self, other)):
return Factors(S.Zero)
factors = dict(self.factors)
for factor, exp in other.factors.items():
if factor in factors:
exp = max(exp, factors[factor])
factors[factor] = exp
return Factors(factors)
def __mul__(self, other): # Factors
return self.mul(other)
def __divmod__(self, other): # Factors
return self.div(other)
def __div__(self, other): # Factors
return self.quo(other)
__truediv__ = __div__
def __mod__(self, other): # Factors
return self.rem(other)
def __pow__(self, other): # Factors
return self.pow(other)
def __eq__(self, other): # Factors
if not isinstance(other, Factors):
other = Factors(other)
return self.factors == other.factors
def __ne__(self, other): # Factors
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): # Term
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 = defaultdict(int), defaultdict(int)
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): # Term
return hash((self.coeff, self.numer, self.denom))
def __repr__(self): # Term
return "Term(%s, %s, %s)" % (self.coeff, self.numer, self.denom)
def as_expr(self): # Term
return self.coeff*(self.numer.as_expr()/self.denom.as_expr())
def mul(self, other): # Term
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): # Term
return Term(1/self.coeff, self.denom, self.numer)
def quo(self, other): # Term
return self.mul(other.inv())
def pow(self, other): # Term
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): # Term
return Term(self.coeff.gcd(other.coeff),
self.numer.gcd(other.numer),
self.denom.gcd(other.denom))
def lcm(self, other): # Term
return Term(self.coeff.lcm(other.coeff),
self.numer.lcm(other.numer),
self.denom.lcm(other.denom))
def __mul__(self, other): # Term
if isinstance(other, Term):
return self.mul(other)
else:
return NotImplemented
def __div__(self, other): # Term
if isinstance(other, Term):
return self.quo(other)
else:
return NotImplemented
__truediv__ = __div__
def __pow__(self, other): # Term
if isinstance(other, SYMPY_INTS):
return self.pow(other)
else:
return NotImplemented
def __eq__(self, other): # Term
return (self.coeff == other.coeff and
self.numer == other.numer and
self.denom == other.denom)
def __ne__(self, other): # Term
return not self.__eq__(other)
def _gcd_terms(terms, isprimitive=False, fraction=True):
"""Helper function for :func:`gcd_terms`.
If ``isprimitive`` is True then the call to primitive
for an Add will be skipped. This is useful when the
content has already been extrated.
If ``fraction`` is True then the expression will appear over a common
denominator, the lcm of all term denominators.
"""
if isinstance(terms, Basic) and not isinstance(terms, Tuple):
terms = Add.make_args(terms)
terms = list(map(Term, [t for t in terms if t]))
# there is some simplification that may happen if we leave this
# here rather than duplicate it before the mapping of Term onto
# the terms
if len(terms) == 0:
return S.Zero, S.Zero, S.One
if len(terms) == 1:
cont = terms[0].coeff
numer = terms[0].numer.as_expr()
denom = terms[0].denom.as_expr()
else:
cont = terms[0]
for term in terms[1:]:
cont = cont.gcd(term)
for i, term in enumerate(terms):
terms[i] = term.quo(cont)
if fraction:
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())
else:
numers = [t.as_expr() for t in terms]
denom = Term(S(1)).numer
cont = cont.as_expr()
numer = Add(*numers)
denom = denom.as_expr()
if not isprimitive and numer.is_Add:
_cont, numer = numer.primitive()
cont *= _cont
return cont, numer, denom
[docs]def gcd_terms(terms, isprimitive=False, clear=True, fraction=True):
"""Compute the GCD of ``terms`` and put them together.
``terms`` can be an expression or a non-Basic sequence of expressions
which will be handled as though they are terms from a sum.
If ``isprimitive`` is True the _gcd_terms will not run the primitive
method on the terms.
``clear`` controls the removal of integers from the denominator of an Add
expression. When True (default), all numerical denominator will be cleared;
when False the denominators will be cleared only if all terms had numerical
denominators other than 1.
``fraction``, when True (default), will put the expression over a common
denominator.
Examples
========
>>> 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)
>>> gcd_terms(x/2 + 1)
(x + 2)/2
>>> gcd_terms(x/2 + 1, clear=False)
x/2 + 1
>>> gcd_terms(x/2 + y/2, clear=False)
(x + y)/2
>>> gcd_terms(x/2 + 1/x)
(x**2 + 2)/(2*x)
>>> gcd_terms(x/2 + 1/x, fraction=False)
(x + 2/x)/2
>>> gcd_terms(x/2 + 1/x, fraction=False, clear=False)
x/2 + 1/x
>>> gcd_terms(x/2/y + 1/x/y)
(x**2 + 2)/(2*x*y)
>>> gcd_terms(x/2/y + 1/x/y, clear=False)
(x**2/2 + 1)/(x*y)
>>> gcd_terms(x/2/y + 1/x/y, clear=False, fraction=False)
(x/2 + 1/x)/y
The ``clear`` flag was ignored in this case because the returned
expression was a rational expression, not a simple sum.
See Also
========
factor_terms, sympy.polys.polytools.terms_gcd
"""
def mask(terms):
"""replace nc portions of each term with a unique Dummy symbols
and return the replacements to restore them"""
args = [(a, []) if a.is_commutative else a.args_cnc() for a in terms]
reps = []
for i, (c, nc) in enumerate(args):
if nc:
nc = Mul._from_args(nc)
d = Dummy()
reps.append((d, nc))
c.append(d)
args[i] = Mul._from_args(c)
else:
args[i] = c
return args, dict(reps)
isadd = isinstance(terms, Add)
addlike = isadd or not isinstance(terms, Basic) and \
is_sequence(terms, include=set) and \
not isinstance(terms, Dict)
if addlike:
if isadd: # i.e. an Add
terms = list(terms.args)
else:
terms = sympify(terms)
terms, reps = mask(terms)
cont, numer, denom = _gcd_terms(terms, isprimitive, fraction)
numer = numer.xreplace(reps)
coeff, factors = cont.as_coeff_Mul()
if not clear:
c, _coeff = coeff.as_coeff_Mul()
if not c.is_Integer and not clear and numer.is_Add:
n, d = c.as_numer_denom()
_numer = numer/d
if any(a.as_coeff_Mul()[0].is_Integer
for a in _numer.args):
numer = _numer
coeff = n*_coeff
return _keep_coeff(coeff, factors*numer/denom, clear=clear)
if not isinstance(terms, Basic):
return terms
if terms.is_Atom:
return terms
if terms.is_Mul:
c, args = terms.as_coeff_mul()
return _keep_coeff(c, Mul(*[gcd_terms(i, isprimitive, clear, fraction)
for i in args]), clear=clear)
def handle(a):
# don't treat internal args like terms of an Add
if not isinstance(a, Expr):
if isinstance(a, Basic):
return a.func(*[handle(i) for i in a.args])
return type(a)([handle(i) for i in a])
return gcd_terms(a, isprimitive, clear, fraction)
if isinstance(terms, Dict):
return Dict(*[(k, handle(v)) for k, v in terms.args])
return terms.func(*[handle(i) for i in terms.args])
[docs]def factor_terms(expr, radical=False, clear=False, fraction=False, sign=True):
"""Remove common factors from terms in all arguments without
changing the underlying structure of the expr. No expansion or
simplification (and no processing of non-commutatives) is performed.
If radical=True then a radical common to all terms will be factored
out of any Add sub-expressions of the expr.
If clear=False (default) then coefficients will not be separated
from a single Add if they can be distributed to leave one or more
terms with integer coefficients.
If fraction=True (default is False) then a common denominator will be
constructed for the expression.
If sign=True (default) then even if the only factor in common is a -1,
it will be factored out of the expression.
Examples
========
>>> from sympy import factor_terms, Symbol
>>> from sympy.abc import x, y
>>> factor_terms(x + x*(2 + 4*y)**3)
x*(8*(2*y + 1)**3 + 1)
>>> A = Symbol('A', commutative=False)
>>> factor_terms(x*A + x*A + x*y*A)
x*(y*A + 2*A)
When ``clear`` is False, a rational will only be factored out of an
Add expression if all terms of the Add have coefficients that are
fractions:
>>> factor_terms(x/2 + 1, clear=False)
x/2 + 1
>>> factor_terms(x/2 + 1, clear=True)
(x + 2)/2
If a -1 is all that can be factored out, to *not* factor it out, the
flag ``sign`` must be False:
>>> factor_terms(-x - y)
-(x + y)
>>> factor_terms(-x - y, sign=False)
-x - y
>>> factor_terms(-2*x - 2*y, sign=False)
-2*(x + y)
See Also
========
gcd_terms, sympy.polys.polytools.terms_gcd
"""
def do(expr):
is_iterable = iterable(expr)
if not isinstance(expr, Basic) or expr.is_Atom:
if is_iterable:
return type(expr)([do(i) for i in expr])
return expr
if expr.is_Pow or expr.is_Function or \
is_iterable or not hasattr(expr, 'args_cnc'):
args = expr.args
newargs = tuple([do(i) for i in args])
if newargs == args:
return expr
return expr.func(*newargs)
cont, p = expr.as_content_primitive(radical=radical, clear=clear)
if p.is_Add:
list_args = [do(a) for a in Add.make_args(p)]
# get a common negative (if there) which gcd_terms does not remove
if all(a.as_coeff_Mul()[0] < 0 for a in list_args):
cont = -cont
list_args = [-a for a in list_args]
# watch out for exp(-(x+2)) which gcd_terms will change to exp(-x-2)
special = {}
for i, a in enumerate(list_args):
b, e = a.as_base_exp()
if e.is_Mul and e != Mul(*e.args):
list_args[i] = Dummy()
special[list_args[i]] = a
# rebuild p not worrying about the order which gcd_terms will fix
p = Add._from_args(list_args)
p = gcd_terms(p,
isprimitive=True,
clear=clear,
fraction=fraction).xreplace(special)
elif p.args:
p = p.func(
*[do(a) for a in p.args])
rv = _keep_coeff(cont, p, clear=clear, sign=sign)
return rv
expr = sympify(expr)
return do(expr)
def _mask_nc(eq, name=None):
"""
Return ``eq`` with non-commutative objects replaced with Dummy
symbols. A dictionary that can be used to restore the original
values is returned: if it is None, the expression is noncommutative
and cannot be made commutative. The third value returned is a list
of any non-commutative symbols that appear in the returned equation.
``name``, if given, is the name that will be used with numered Dummy
variables that will replace the non-commutative objects and is mainly
used for doctesting purposes.
Notes
=====
All non-commutative objects other than Symbols are replaced with
a non-commutative Symbol. Identical objects will be identified
by identical symbols.
If there is only 1 non-commutative object in an expression it will
be replaced with a commutative symbol. Otherwise, the non-commutative
entities are retained and the calling routine should handle
replacements in this case since some care must be taken to keep
track of the ordering of symbols when they occur within Muls.
Examples
========
>>> from sympy.physics.secondquant import Commutator, NO, F, Fd
>>> from sympy import symbols, Mul
>>> from sympy.core.exprtools import _mask_nc
>>> from sympy.abc import x, y
>>> A, B, C = symbols('A,B,C', commutative=False)
One nc-symbol:
>>> _mask_nc(A**2 - x**2, 'd')
(_d0**2 - x**2, {_d0: A}, [])
Multiple nc-symbols:
>>> _mask_nc(A**2 - B**2, 'd')
(A**2 - B**2, None, [A, B])
An nc-object with nc-symbols but no others outside of it:
>>> _mask_nc(1 + x*Commutator(A, B), 'd')
(_d0*x + 1, {_d0: Commutator(A, B)}, [])
>>> _mask_nc(NO(Fd(x)*F(y)), 'd')
(_d0, {_d0: NO(CreateFermion(x)*AnnihilateFermion(y))}, [])
Multiple nc-objects:
>>> eq = x*Commutator(A, B) + x*Commutator(A, C)*Commutator(A, B)
>>> _mask_nc(eq, 'd')
(x*_d0 + x*_d1*_d0, {_d0: Commutator(A, B), _d1: Commutator(A, C)}, [_d0, _d1])
Multiple nc-objects and nc-symbols:
>>> eq = A*Commutator(A, B) + B*Commutator(A, C)
>>> _mask_nc(eq, 'd')
(A*_d0 + B*_d1, {_d0: Commutator(A, B), _d1: Commutator(A, C)}, [_d0, _d1, A, B])
If there is an object that:
- doesn't contain nc-symbols
- but has arguments which derive from Basic, not Expr
- and doesn't define an _eval_is_commutative routine
then it will give False (or None?) for the is_commutative test. Such
objects are also removed by this routine:
>>> from sympy import Basic
>>> eq = (1 + Mul(Basic(), Basic(), evaluate=False))
>>> eq.is_commutative
False
>>> _mask_nc(eq, 'd')
(_d0**2 + 1, {_d0: Basic()}, [])
"""
name = name or 'mask'
# Make Dummy() append sequential numbers to the name
def numbered_names():
i = 0
while True:
yield name + str(i)
i += 1
names = numbered_names()
def Dummy(*args, **kwargs):
from sympy import Dummy
return Dummy(next(names), *args, **kwargs)
expr = eq
if expr.is_commutative:
return eq, {}, []
# identify nc-objects; symbols and other
rep = []
nc_obj = set()
nc_syms = set()
pot = preorder_traversal(expr, keys=default_sort_key)
for i, a in enumerate(pot):
if any(a == r[0] for r in rep):
pot.skip()
elif not a.is_commutative:
if a.is_Symbol:
nc_syms.add(a)
elif not (a.is_Add or a.is_Mul or a.is_Pow):
if all(s.is_commutative for s in a.free_symbols):
rep.append((a, Dummy()))
else:
nc_obj.add(a)
pot.skip()
# If there is only one nc symbol or object, it can be factored regularly
# but polys is going to complain, so replace it with a Dummy.
if len(nc_obj) == 1 and not nc_syms:
rep.append((nc_obj.pop(), Dummy()))
elif len(nc_syms) == 1 and not nc_obj:
rep.append((nc_syms.pop(), Dummy()))
# Any remaining nc-objects will be replaced with an nc-Dummy and
# identified as an nc-Symbol to watch out for
nc_obj = sorted(nc_obj, key=default_sort_key)
for n in nc_obj:
nc = Dummy(commutative=False)
rep.append((n, nc))
nc_syms.add(nc)
expr = expr.subs(rep)
nc_syms = list(nc_syms)
nc_syms.sort(key=default_sort_key)
return expr, {v: k for k, v in rep} or None, nc_syms
def factor_nc(expr):
"""Return the factored form of ``expr`` while handling non-commutative
expressions.
Examples
========
>>> from sympy.core.exprtools import factor_nc
>>> from sympy import Symbol
>>> from sympy.abc import x
>>> A = Symbol('A', commutative=False)
>>> B = Symbol('B', commutative=False)
>>> factor_nc((x**2 + 2*A*x + A**2).expand())
(x + A)**2
>>> factor_nc(((x + A)*(x + B)).expand())
(x + A)*(x + B)
"""
from sympy.simplify.simplify import powsimp
from sympy.polys import gcd, factor
def _pemexpand(expr):
"Expand with the minimal set of hints necessary to check the result."
return expr.expand(deep=True, mul=True, power_exp=True,
power_base=False, basic=False, multinomial=True, log=False)
expr = sympify(expr)
if not isinstance(expr, Expr) or not expr.args:
return expr
if not expr.is_Add:
return expr.func(*[factor_nc(a) for a in expr.args])
expr, rep, nc_symbols = _mask_nc(expr)
if rep:
return factor(expr).subs(rep)
else:
args = [a.args_cnc() for a in Add.make_args(expr)]
c = g = l = r = S.One
hit = False
# find any commutative gcd term
for i, a in enumerate(args):
if i == 0:
c = Mul._from_args(a[0])
elif a[0]:
c = gcd(c, Mul._from_args(a[0]))
else:
c = S.One
if c is not S.One:
hit = True
c, g = c.as_coeff_Mul()
if g is not S.One:
for i, (cc, _) in enumerate(args):
cc = list(Mul.make_args(Mul._from_args(list(cc))/g))
args[i][0] = cc
for i, (cc, _) in enumerate(args):
cc[0] = cc[0]/c
args[i][0] = cc
# find any noncommutative common prefix
for i, a in enumerate(args):
if i == 0:
n = a[1][:]
else:
n = common_prefix(n, a[1])
if not n:
# is there a power that can be extracted?
if not args[0][1]:
break
b, e = args[0][1][0].as_base_exp()
ok = False
if e.is_Integer:
for t in args:
if not t[1]:
break
bt, et = t[1][0].as_base_exp()
if et.is_Integer and bt == b:
e = min(e, et)
else:
break
else:
ok = hit = True
l = b**e
il = b**-e
for i, a in enumerate(args):
args[i][1][0] = il*args[i][1][0]
break
if not ok:
break
else:
hit = True
lenn = len(n)
l = Mul(*n)
for i, a in enumerate(args):
args[i][1] = args[i][1][lenn:]
# find any noncommutative common suffix
for i, a in enumerate(args):
if i == 0:
n = a[1][:]
else:
n = common_suffix(n, a[1])
if not n:
# is there a power that can be extracted?
if not args[0][1]:
break
b, e = args[0][1][-1].as_base_exp()
ok = False
if e.is_Integer:
for t in args:
if not t[1]:
break
bt, et = t[1][-1].as_base_exp()
if et.is_Integer and bt == b:
e = min(e, et)
else:
break
else:
ok = hit = True
r = b**e
il = b**-e
for i, a in enumerate(args):
args[i][1][-1] = args[i][1][-1]*il
break
if not ok:
break
else:
hit = True
lenn = len(n)
r = Mul(*n)
for i, a in enumerate(args):
args[i][1] = a[1][:len(a[1]) - lenn]
if hit:
mid = Add(*[Mul(*cc)*Mul(*nc) for cc, nc in args])
else:
mid = expr
# sort the symbols so the Dummys would appear in the same
# order as the original symbols, otherwise you may introduce
# a factor of -1, e.g. A**2 - B**2) -- {A:y, B:x} --> y**2 - x**2
# and the former factors into two terms, (A - B)*(A + B) while the
# latter factors into 3 terms, (-1)*(x - y)*(x + y)
rep1 = [(n, Dummy()) for n in sorted(nc_symbols, key=default_sort_key)]
unrep1 = [(v, k) for k, v in rep1]
unrep1.reverse()
new_mid, r2, _ = _mask_nc(mid.subs(rep1))
new_mid = powsimp(factor(new_mid))
new_mid = new_mid.subs(r2).subs(unrep1)
if new_mid.is_Pow:
return _keep_coeff(c, g*l*new_mid*r)
if new_mid.is_Mul:
# XXX TODO there should be a way to inspect what order the terms
# must be in and just select the plausible ordering without
# checking permutations
cfac = []
ncfac = []
for f in new_mid.args:
if f.is_commutative:
cfac.append(f)
else:
b, e = f.as_base_exp()
if e.is_Integer:
ncfac.extend([b]*e)
else:
ncfac.append(f)
pre_mid = g*Mul(*cfac)*l
target = _pemexpand(expr/c)
for s in variations(ncfac, len(ncfac)):
ok = pre_mid*Mul(*s)*r
if _pemexpand(ok) == target:
return _keep_coeff(c, ok)
# mid was an Add that didn't factor successfully
return _keep_coeff(c, g*l*mid*r)
