def search(expr, test):
if not isinstance(expr, Basic):
try:
return any(search(i, test) for i in expr)
except TypeError:
return False
elif test(expr):
return True
else:
return any(search(i, test) for i in expr.iter_basic_args())
def search(expr, test):
if not isinstance(expr, Basic):
try:
return any(search(i, test) for i in expr)
except TypeError:
return False
elif test(expr):
return True
else:
return any(search(i, test) for i in expr.iter_basic_args())
def has(self, *patterns):
"""
Test whether any subexpression matches any of the patterns.
Examples:
>>> from sympy import sin, S
>>> from sympy.abc import x, y, z
>>> (x**2 + sin(x*y)).has(z)
False
>>> (x**2 + sin(x*y)).has(x, y, z)
True
>>> x.has(x)
True
Note that ``expr.has(*patterns)`` is exactly equivalent to
``any(expr.has(p) for p in patterns)``. In particular, ``False`` is
returned when the list of patterns is empty.
>>> x.has()
False
"""
def search(expr, test):
if not isinstance(expr, Basic):
try:
return any(search(i, test) for i in expr)
except TypeError:
return False
elif test(expr):
return True
else:
return any(search(i, test) for i in expr.iter_basic_args())
def _match(p):
if isinstance(p, BasicType):
return lambda w: isinstance(w, p)
else:
return lambda w: p.matches(w) is not None
patterns = map(sympify, patterns)
return any(search(self, _match(p)) for p in patterns)
def has(self, *patterns):
"""
Test whether any subexpression matches any of the patterns.
Examples:
>>> from sympy import sin, S
>>> from sympy.abc import x, y, z
>>> (x**2 + sin(x*y)).has(z)
False
>>> (x**2 + sin(x*y)).has(x, y, z)
True
>>> x.has(x)
True
Note that ``expr.has(*patterns)`` is exactly equivalent to
``any(expr.has(p) for p in patterns)``. In particular, ``False`` is
returned when the list of patterns is empty.
>>> x.has()
False
"""
def search(expr, test):
if not isinstance(expr, Basic):
try:
return any(search(i, test) for i in expr)
except TypeError:
return False
elif test(expr):
return True
else:
return any(search(i, test) for i in expr.iter_basic_args())
def _match(p):
if isinstance(p, BasicType):
return lambda w: isinstance(w, p)
else:
return lambda w: p.matches(w) is not None
patterns = map(sympify, patterns)
return any(search(self, _match(p)) for p in patterns)
def as_ordered_terms(self, order=None, data=False):
"""
Transform an expression to an ordered list of terms.
**Examples**
>>> from sympy import sin, cos
>>> from sympy.abc import x, y
>>> (sin(x)**2*cos(x) + sin(x)**2 + 1).as_ordered_terms()
[sin(x)**2*cos(x), sin(x)**2, 1]
"""
from sympy.utilities import any
key, reverse = self._parse_order(order)
terms, gens = self.as_terms()
if not any(term.is_Order for term, _ in terms):
ordered = sorted(terms, key=key, reverse=not reverse)
else:
_terms, _order = [], []
for term, repr in terms:
if not term.is_Order:
_terms.append((term, repr))
else:
_order.append((term, repr))
ordered = sorted(_terms, key=key) \
+ sorted(_order, key=key)
if data:
return ordered, gens
else:
return [ term for term, _ in ordered ]
def as_ordered_terms(self, order=None, data=False):
"""
Transform an expression to an ordered list of terms.
**Examples**
>>> from sympy import sin, cos
>>> from sympy.abc import x, y
>>> (sin(x)**2*cos(x) + sin(x)**2 + 1).as_ordered_terms()
[sin(x)**2*cos(x), sin(x)**2, 1]
"""
from sympy.utilities import any
key, reverse = self._parse_order(order)
terms, gens = self.as_terms()
if not any(term.is_Order for term, _ in terms):
ordered = sorted(terms, key=key, reverse=not reverse)
else:
_terms, _order = [], []
for term, repr in terms:
if not term.is_Order:
_terms.append((term, repr))
else:
_order.append((term, repr))
ordered = sorted(_terms, key=key) \
+ sorted(_order, key=key)
if data:
return ordered, gens
else:
return [term for term, _ in ordered]
def _eval_subs(self, old, new):
from sympy import sign
from sympy.simplify.simplify import powdenest
if self == old:
return new
def fallback():
"""Return this value when partial subs has failed."""
return self.__class__(*[s._eval_subs(old, new) for s in self.args])
def breakup(eq):
"""break up powers assuming (not checking) that eq is a Mul:
b**(Rational*e) -> b**e, Rational
commutatives come back as a dictionary {b**e: Rational}
noncommutatives come back as a list [(b**e, Rational)]
"""
(c, nc) = (dict(), list())
for (i, a) in enumerate(
Mul.make_args(eq)
or [eq]): # remove or [eq] after 2114 accepted
a = powdenest(a)
(b, e) = a.as_base_exp()
if not e is S.One:
(co, _) = e.as_coeff_mul()
b = Pow(b, e / co)
e = co
if a.is_commutative:
if b in c: # handle I and -1 like things where b, e for I is -1, 1/2
c[b] += e
else:
c[b] = e
else:
nc.append([b, e])
return (c, nc)
def rejoin(b, co):
"""
Put rational back with exponent; in general this is not ok, but
since we took it from the exponent for analysis, it's ok to put
it back.
"""
(b, e) = b.as_base_exp()
return Pow(b, e * co)
def ndiv(a, b):
"""if b divides a in an extractive way (like 1/4 divides 1/2
but not vice versa, and 2/5 does not divide 1/3) then return
the integer number of times it divides, else return 0.
"""
if not b.q % a.q or not a.q % b.q:
return int(a / b)
return 0
if not old.is_Mul:
return fallback()
# handle the leading coefficient and use it to decide if anything
# should even be started; we always know where to find the Rational
# so it's a quick test
coeff = S.One
co_self = self.args[0]
co_old = old.args[0]
if co_old.is_Rational and co_self.is_Rational:
co_xmul = co_self.extract_multiplicatively(co_old)
elif co_old.is_Rational:
co_xmul = None
else:
co_xmul = True
if not co_xmul:
return fallback()
(c, nc) = breakup(self)
(old_c, old_nc) = breakup(old)
# update the coefficients if we had an extraction
if getattr(co_xmul, 'is_Rational', False):
c.pop(co_self)
c[co_xmul] = S.One
old_c.pop(co_old)
# do quick tests to see if we can't succeed
ok = True
if (
# more non-commutative terms
len(old_nc) > len(nc)):
ok = False
elif (
# more commutative terms
len(old_c) > len(c)):
ok = False
elif (
# unmatched non-commutative bases
set(_[0] for _ in old_nc).difference(set(_[0] for _ in nc))):
ok = False
elif (
# unmatched commutative terms
set(old_c).difference(set(c))):
ok = False
elif (
# differences in sign
any(sign(c[b]) != sign(old_c[b]) for b in old_c)):
ok = False
if not ok:
return fallback()
if not old_c:
cdid = None
else:
rat = []
for (b, old_e) in old_c.items():
c_e = c[b]
rat.append(ndiv(c_e, old_e))
if not rat[-1]:
return fallback()
cdid = min(rat)
if not old_nc:
ncdid = None
for i in range(len(nc)):
nc[i] = rejoin(*nc[i])
else:
ncdid = 0 # number of nc replacements we did
take = len(old_nc) # how much to look at each time
limit = cdid or S.Infinity # max number that we can take
failed = [] # failed terms will need subs if other terms pass
i = 0
while limit and i + take <= len(nc):
hit = False
# the bases must be equivalent in succession, and
# the powers must be extractively compatible on the
# first and last factor but equal inbetween.
rat = []
for j in range(take):
if nc[i + j][0] != old_nc[j][0]:
break
elif j == 0:
rat.append(ndiv(nc[i + j][1], old_nc[j][1]))
elif j == take - 1:
rat.append(ndiv(nc[i + j][1], old_nc[j][1]))
elif nc[i + j][1] != old_nc[j][1]:
break
else:
rat.append(1)
j += 1
else:
ndo = min(rat)
if ndo:
if take == 1:
if cdid:
ndo = min(cdid, ndo)
nc[i] = Pow(new, ndo) * rejoin(
nc[i][0], nc[i][1] - ndo * old_nc[0][1])
else:
ndo = 1
# the left residual
l = rejoin(nc[i][0], nc[i][1] - ndo * old_nc[0][1])
# eliminate all middle terms
mid = new
# the right residual (which may be the same as the middle if take == 2)
ir = i + take - 1
r = (nc[ir][0], nc[ir][1] - ndo * old_nc[-1][1])
if r[1]:
if i + take < len(nc):
nc[i:i + take] = [l * mid, r]
else:
r = rejoin(*r)
nc[i:i + take] = [l * mid * r]
else:
# there was nothing left on the right
nc[i:i + take] = [l * mid]
limit -= ndo
ncdid += ndo
hit = True
if not hit:
# do the subs on this failing factor
failed.append(i)
i += 1
else:
if not ncdid:
return fallback()
# although we didn't fail, certain nc terms may have
# failed so we rebuild them after attempting a partial
# subs on them
failed.extend(range(i, len(nc)))
for i in failed:
nc[i] = rejoin(*nc[i]).subs(old, new)
# rebuild the expression
if cdid is None:
do = ncdid
elif ncdid is None:
do = cdid
else:
do = min(ncdid, cdid)
margs = []
for b in c:
if b in old_c:
# calculate the new exponent
e = c[b] - old_c[b] * do
margs.append(rejoin(b, e))
else:
margs.append(rejoin(b.subs(old, new), c[b]))
if cdid and not ncdid:
# in case we are replacing commutative with non-commutative,
# we want the new term to come at the front just like the
# rest of this routine
margs = [Pow(new, cdid)] + margs
return Mul(*margs) * Mul(*nc)
def _eval_subs(self, old, new):
from sympy import sign
from sympy.simplify.simplify import powdenest
if self == old:
return new
def fallback():
"""Return this value when partial subs has failed."""
return self.__class__(*[s._eval_subs(old, new) for s in
self.args])
def breakup(eq):
"""break up powers assuming (not checking) that eq is a Mul:
b**(Rational*e) -> b**e, Rational
commutatives come back as a dictionary {b**e: Rational}
noncommutatives come back as a list [(b**e, Rational)]
"""
(c, nc) = (dict(), list())
for (i, a) in enumerate(Mul.make_args(eq) or [eq]): # remove or [eq] after 2114 accepted
a = powdenest(a)
(b, e) = a.as_base_exp()
if not e is S.One:
(co, _) = e.as_coeff_mul()
b = Pow(b, e/co)
e = co
if a.is_commutative:
if b in c: # handle I and -1 like things where b, e for I is -1, 1/2
c[b] += e
else:
c[b] = e
else:
nc.append([b, e])
return (c, nc)
def rejoin(b, co):
"""
Put rational back with exponent; in general this is not ok, but
since we took it from the exponent for analysis, it's ok to put
it back.
"""
(b, e) = b.as_base_exp()
return Pow(b, e*co)
def ndiv(a, b):
"""if b divides a in an extractive way (like 1/4 divides 1/2
but not vice versa, and 2/5 does not divide 1/3) then return
the integer number of times it divides, else return 0.
"""
if not b.q % a.q or not a.q % b.q:
return int(a/b)
return 0
if not old.is_Mul:
return fallback()
# handle the leading coefficient and use it to decide if anything
# should even be started; we always know where to find the Rational
# so it's a quick test
coeff = S.One
co_self = self.args[0]
co_old = old.args[0]
if co_old.is_Rational and co_self.is_Rational:
co_xmul = co_self.extract_multiplicatively(co_old)
elif co_old.is_Rational:
co_xmul = None
else:
co_xmul = True
if not co_xmul:
return fallback()
(c, nc) = breakup(self)
(old_c, old_nc) = breakup(old)
# update the coefficients if we had an extraction
if getattr(co_xmul, 'is_Rational', False):
c.pop(co_self)
c[co_xmul] = S.One
old_c.pop(co_old)
# do quick tests to see if we can't succeed
ok = True
if (
# more non-commutative terms
len(old_nc) > len(nc)):
ok = False
elif (
# more commutative terms
len(old_c) > len(c)):
ok = False
elif (
# unmatched non-commutative bases
set(_[0] for _ in old_nc).difference(set(_[0] for _ in nc))):
ok = False
elif (
# unmatched commutative terms
set(old_c).difference(set(c))):
ok = False
elif (
# differences in sign
any(sign(c[b]) != sign(old_c[b]) for b in old_c)):
ok = False
if not ok:
return fallback()
if not old_c:
cdid = None
else:
rat = []
for (b, old_e) in old_c.items():
c_e = c[b]
rat.append(ndiv(c_e, old_e))
if not rat[-1]:
return fallback()
cdid = min(rat)
if not old_nc:
ncdid = None
for i in range(len(nc)):
nc[i] = rejoin(*nc[i])
else:
ncdid = 0 # number of nc replacements we did
take = len(old_nc) # how much to look at each time
limit = cdid or S.Infinity # max number that we can take
failed = [] # failed terms will need subs if other terms pass
i = 0
while limit and i + take <= len(nc):
hit = False
# the bases must be equivalent in succession, and
# the powers must be extractively compatible on the
# first and last factor but equal inbetween.
rat = []
for j in range(take):
if nc[i + j][0] != old_nc[j][0]:
break
elif j == 0:
rat.append(ndiv(nc[i + j][1], old_nc[j][1]))
elif j == take - 1:
rat.append(ndiv(nc[i + j][1], old_nc[j][1]))
elif nc[i + j][1] != old_nc[j][1]:
break
else:
rat.append(1)
j += 1
else:
ndo = min(rat)
if ndo:
if take == 1:
if cdid:
ndo = min(cdid, ndo)
nc[i] = Pow(new, ndo)*rejoin(nc[i][0],
nc[i][1] - ndo*old_nc[0][1])
else:
ndo = 1
# the left residual
l = rejoin(nc[i][0], nc[i][1] - ndo*
old_nc[0][1])
# eliminate all middle terms
mid = new
# the right residual (which may be the same as the middle if take == 2)
ir = i + take - 1
r = (nc[ir][0], nc[ir][1] - ndo*
old_nc[-1][1])
if r[1]:
if i + take < len(nc):
nc[i:i + take] = [l*mid, r]
else:
r = rejoin(*r)
nc[i:i + take] = [l*mid*r]
else:
# there was nothing left on the right
nc[i:i + take] = [l*mid]
limit -= ndo
ncdid += ndo
hit = True
if not hit:
# do the subs on this failing factor
failed.append(i)
i += 1
else:
if not ncdid:
return fallback()
# although we didn't fail, certain nc terms may have
# failed so we rebuild them after attempting a partial
# subs on them
failed.extend(range(i, len(nc)))
for i in failed:
nc[i] = rejoin(*nc[i]).subs(old, new)
# rebuild the expression
if cdid is None:
do = ncdid
elif ncdid is None:
do = cdid
else:
do = min(ncdid, cdid)
margs = []
for b in c:
if b in old_c:
# calculate the new exponent
e = c[b] - old_c[b]*do
margs.append(rejoin(b, e))
else:
margs.append(rejoin(b.subs(old, new), c[b]))
if cdid and not ncdid:
# in case we are replacing commutative with non-commutative,
# we want the new term to come at the front just like the
# rest of this routine
margs = [Pow(new, cdid)] + margs
return Mul(*margs)*Mul(*nc)
def has(self, *patterns, **flags):
"""Return True if self has any of the patterns. If the `all` flag
is True then return True if all of the patterns are present.
>>> from sympy import sin, S
>>> from sympy.abc import x, y, z
>>> (x**2 + sin(x*y)).has(z)
False
>>> (x**2 + sin(x*y)).has(x, y, z)
True
When `all` is True then True is returned only if all of the
patterns are present:
>>> (x**2 + sin(x*y)).has(x, y, z, all=True)
False
If there are no patterns, False is always returned:
"something doesn't have nothing"
>>> (x).has()
False
>>> (S.One).has()
False
"""
from sympy.core.symbol import Wild
def search(expr, target, hit):
if hasattr(expr, '__iter__') and hasattr(expr, '__len__'):
# this 'if' clause is needed until all objects use
# sympy containers
for i in expr:
if search(i, target, hit):
return True
elif not isinstance(expr, Basic):
pass
elif target(expr) and hit(expr):
return True
else:
for term in expr.iter_basic_args():
if search(term, target, hit):
return True
return False
def _has(p):
p = sympify(p)
if isinstance(p, BasicType):
return search(self, lambda w: isinstance(w, p), lambda w: True)
if p.is_Atom and not isinstance(p, Wild):
return search(self, lambda w: isinstance(w, p.func), lambda w: w in [p])
return search(self, lambda w: p.matches(w) is not None, lambda w: True)
if not patterns:
return False # something doesn't have nothing
patterns = set(patterns)
if flags.get('all', False):
return all(_has(p) for p in patterns)
else:
return any(_has(p) for p in patterns)
