def _matches(self, expr, repl_dict={}):
# weed out negative one prefixes
sign = 1
a, b = self.as_two_terms()
if a is S.NegativeOne:
if b.is_Mul:
sign = -sign
else:
# the remainder, b, is not a Mul anymore
return b.matches(-expr, repl_dict)
expr = sympify(expr)
if expr.is_Mul and expr.args[0] is S.NegativeOne:
expr = -expr
sign = -sign
if not expr.is_Mul:
# expr can only match if it matches b and a matches +/- 1
if len(self.args) == 2:
# quickly test for equality
if b == expr:
return a.matches(Rational(sign), repl_dict)
# do more expensive match
dd = b.matches(expr, repl_dict)
if dd is None:
return None
dd = a.matches(Rational(sign), dd)
return dd
return None
d = repl_dict.copy()
# weed out identical terms
pp = list(self.args)
ee = list(expr.args)
for p in self.args:
if p in expr.args:
ee.remove(p)
pp.remove(p)
# only one symbol left in pattern -> match the remaining expression
if len(pp) == 1 and isinstance(pp[0], C.Wild):
if len(ee) == 1:
d[pp[0]] = sign * ee[0]
else:
d[pp[0]] = sign * expr.func(*ee)
return d
if len(ee) != len(pp):
return None
for p, e in zip(pp, ee):
d = p.xreplace(d).matches(e, d)
if d is None:
return None
return d
def test_rational_is_integer(self):
self.assertTrue(Rational.is_integer(F(-1, 1)))
self.assertTrue(Rational.is_integer(F(0, 1)))
self.assertTrue(Rational.is_integer(F(1, 1)))
self.assertTrue(Rational.is_integer(F(42, 1)))
self.assertTrue(Rational.is_integer(F(2, 2)))
self.assertTrue(Rational.is_integer(F(8, 4)))
self.assertFalse(Rational.is_integer(F(1, 2)))
self.assertFalse(Rational.is_integer(F(1, 3)))
self.assertFalse(Rational.is_integer(F(2, 3)))
def _eval_nseries(self, x, x0, n):
assert len(self.args) == 1
arg = self.args[0]
arg0 = arg.limit(x, 0)
from sympy import oo
if arg0 in [-oo, oo]:
raise PoleError("Cannot expand around %s" % (arg))
if arg0 is not S.Zero:
e = self.func(arg)
e1 = e.expand()
if e == e1:
#for example when e = sin(x+1) or e = sin(cos(x))
#let's try the general algorithm
term = e.subs(x, S.Zero)
series = term
fact = S.One
for i in range(n - 1):
i += 1
fact *= Rational(i)
e = e.diff(x)
term = e.subs(x, S.Zero) * (x**i) / fact
term = term.expand()
series += term
return series + C.Order(x**n, x)
return self.nseries(x, x0, n)
l = []
g = None
for i in xrange(n + 2):
g = self.taylor_term(i, arg, g)
g = g.nseries(x, x0, n)
l.append(g)
return Add(*l) + C.Order(x**n, x)
def _eval_nseries(self, x, n):
"""
This function does compute series for multivariate functions,
but the expansion is always in terms of *one* variable.
Examples:
>>> from sympy import atan2, O
>>> from sympy.abc import x, y
>>> atan2(x, y).series(x, n=2)
atan2(0, y) + x/y + O(x**2)
>>> atan2(x, y).series(y, n=2)
atan2(x, 0) - y/x + O(y**2)
"""
if self.func.nargs is None:
raise NotImplementedError('series for user-defined \
functions are not supported.')
args = self.args
args0 = [t.limit(x, 0) for t in args]
if any([t is S.NaN or t.is_bounded is False for t in args0]):
raise PoleError("Cannot expand %s around 0" % (args))
if (self.func.nargs == 1 and args0[0]) or self.func.nargs > 1:
e = self
e1 = e.expand()
if e == e1:
#for example when e = sin(x+1) or e = sin(cos(x))
#let's try the general algorithm
term = e.subs(x, S.Zero)
if term.is_bounded is False or term is S.NaN:
raise PoleError("Cannot expand %s around 0" % (self))
series = term
fact = S.One
for i in range(n - 1):
i += 1
fact *= Rational(i)
e = e.diff(x)
subs = e.subs(x, S.Zero)
if subs is S.NaN:
# try to evaluate a limit if we have to
subs = e.limit(x, S.Zero)
if subs.is_bounded is False:
raise PoleError("Cannot expand %s around 0" % (self))
term = subs * (x**i) / fact
term = term.expand()
series += term
return series + C.Order(x**n, x)
return e1.nseries(x, n=n)
arg = self.args[0]
l = []
g = None
for i in xrange(n + 2):
g = self.taylor_term(i, arg, g)
g = g.nseries(x, n=n)
l.append(g)
return Add(*l) + C.Order(x**n, x)
def _eval_nseries(self, x, n):
if self.func.nargs != 1:
raise NotImplementedError('series for user-defined and \
multi-arg functions are not supported.')
arg = self.args[0]
arg0 = arg.limit(x, 0)
from sympy import oo
if arg0 == S.NaN or arg0.is_bounded == False:
raise PoleError("Cannot expand %s around 0" % (arg))
if arg0:
e = self
e1 = e.expand()
if e == e1:
#for example when e = sin(x+1) or e = sin(cos(x))
#let's try the general algorithm
term = e.subs(x, S.Zero)
if arg0 == S.NaN or term.is_bounded == False:
raise PoleError("Cannot expand %s around 0" % (self))
series = term
fact = S.One
for i in range(n - 1):
i += 1
fact *= Rational(i)
e = e.diff(x)
subs = e.subs(x, S.Zero)
if subs == S.NaN:
# try to evaluate a limit if we have to
subs = e.limit(x, S.Zero)
if subs.is_bounded == False:
raise PoleError("Cannot expand %s around 0" %
(self))
term = subs * (x**i) / fact
term = term.expand()
series += term
return series + C.Order(x**n, x)
return e1.nseries(x, n=n)
l = []
g = None
for i in xrange(n + 2):
g = self.taylor_term(i, arg, g)
g = g.nseries(x, n=n)
l.append(g)
return Add(*l) + C.Order(x**n, x)
def _matches(self, expr, repl_dict={}, evaluate=False):
if evaluate:
return self.subs(repl_dict).matches(expr, repl_dict)
# weed out negative one prefixes
sign = 1
a, b = self.as_two_terms()
if a is S.NegativeOne:
if b.is_Mul:
sign = -sign
else:
# the remainder, b, is not a Mul anymore
return b.matches(-expr, repl_dict, evaluate)
expr = sympify(expr)
if expr.is_Mul and expr.args[0] is S.NegativeOne:
expr = -expr; sign = -sign
if not expr.is_Mul:
# expr can only match if it matches b and a matches +/- 1
if len(self.args) == 2:
# quickly test for equality
if b == expr:
return a.matches(Rational(sign), repl_dict, evaluate)
# do more expensive match
dd = b.matches(expr, repl_dict, evaluate)
if dd == None:
return None
dd = a.matches(Rational(sign), dd, evaluate)
return dd
return None
d = repl_dict.copy()
# weed out identical terms
pp = list(self.args)
ee = list(expr.args)
for p in self.args:
if p in expr.args:
ee.remove(p)
pp.remove(p)
# only one symbol left in pattern -> match the remaining expression
if len(pp) == 1 and isinstance(pp[0], C.Wild):
if len(ee) == 1:
d[pp[0]] = sign * ee[0]
else:
d[pp[0]] = sign * (type(expr)(*ee))
return d
if len(ee) != len(pp):
return None
i = 0
for p, e in zip(pp, ee):
if i == 0 and sign != 1:
try:
e = sign * e
except TypeError:
return None
d = p.matches(e, d, evaluate=not i)
i += 1
if d is None:
return None
return d
def flatten(cls, seq):
"""Return commutative, noncommutative and order arguments by
combining related terms.
** Developer Notes **
* In an expression like ``a*b*c``, python process this through sympy
as ``Mul(Mul(a, b), c)``. This can have undesirable consequences.
- Sometimes terms are not combined as one would like:
{c.f. http://code.google.com/p/sympy/issues/detail?id=1497}
>>> from sympy import Mul, sqrt
>>> from sympy.abc import x, y, z
>>> 2*(x + 1) # this is the 2-arg Mul behavior
2*x + 2
>>> y*(x + 1)*2
2*y*(x + 1)
>>> 2*(x + 1)*y # 2-arg result will be obtained first
y*(2*x + 2)
>>> Mul(2, x + 1, y) # all 3 args simultaneously processed
2*y*(x + 1)
>>> 2*((x + 1)*y) # parentheses can control this behavior
2*y*(x + 1)
Powers with compound bases may not find a single base to
combine with unless all arguments are processed at once.
Post-processing may be necessary in such cases.
{c.f. http://code.google.com/p/sympy/issues/detail?id=2629}
>>> a = sqrt(x*sqrt(y))
>>> a**3
(x*sqrt(y))**(3/2)
>>> Mul(a,a,a)
(x*sqrt(y))**(3/2)
>>> a*a*a
x*sqrt(y)*sqrt(x*sqrt(y))
>>> _.subs(a.base, z).subs(z, a.base)
(x*sqrt(y))**(3/2)
- If more than two terms are being multiplied then all the
previous terms will be re-processed for each new argument.
So if each of ``a``, ``b`` and ``c`` were :class:`Mul`
expression, then ``a*b*c`` (or building up the product
with ``*=``) will process all the arguments of ``a`` and
``b`` twice: once when ``a*b`` is computed and again when
``c`` is multiplied.
Using ``Mul(a, b, c)`` will process all arguments once.
* The results of Mul are cached according to arguments, so flatten
will only be called once for ``Mul(a, b, c)``. If you can
structure a calculation so the arguments are most likely to be
repeats then this can save time in computing the answer. For
example, say you had a Mul, M, that you wished to divide by ``d[i]``
and multiply by ``n[i]`` and you suspect there are many repeats
in ``n``. It would be better to compute ``M*n[i]/d[i]`` rather
than ``M/d[i]*n[i]`` since every time n[i] is a repeat, the
product, ``M*n[i]`` will be returned without flattening -- the
cached value will be returned. If you divide by the ``d[i]``
first (and those are more unique than the ``n[i]``) then that will
create a new Mul, ``M/d[i]`` the args of which will be traversed
again when it is multiplied by ``n[i]``.
{c.f. http://code.google.com/p/sympy/issues/detail?id=2607}
This consideration is moot if the cache is turned off.
NB
The validity of the above notes depends on the implementation
details of Mul and flatten which may change at any time. Therefore,
you should only consider them when your code is highly performance
sensitive.
"""
# apply associativity, separate commutative part of seq
c_part = [] # out: commutative factors
nc_part = [] # out: non-commutative factors
nc_seq = []
coeff = S.One # standalone term
# e.g. 3 * ...
c_powers = [] # (base,exp) n
# e.g. (x,n) for x
num_exp = [] # (num-base, exp) y
# e.g. (3, y) for ... * 3 * ...
neg1e = 0 # exponent on -1 extracted from Number-based Pow
pnum_rat = {} # (num-base, Rat-exp) 1/2
# e.g. (3, 1/2) for ... * 3 * ...
order_symbols = None
# --- PART 1 ---
#
# "collect powers and coeff":
#
# o coeff
# o c_powers
# o num_exp
# o neg1e
# o pnum_rat
#
# NOTE: this is optimized for all-objects-are-commutative case
for o in seq:
# O(x)
if o.is_Order:
o, order_symbols = o.as_expr_variables(order_symbols)
# Mul([...])
if o.is_Mul:
if o.is_commutative:
seq.extend(o.args) # XXX zerocopy?
else:
# NCMul can have commutative parts as well
for q in o.args:
if q.is_commutative:
seq.append(q)
else:
nc_seq.append(q)
# append non-commutative marker, so we don't forget to
# process scheduled non-commutative objects
seq.append(NC_Marker)
continue
# 3
elif o.is_Number:
if o is S.NaN or coeff is S.ComplexInfinity and o is S.Zero:
# we know for sure the result will be nan
return [S.NaN], [], None
elif coeff.is_Number: # it could be zoo
coeff *= o
if coeff is S.NaN:
# we know for sure the result will be nan
return [S.NaN], [], None
continue
elif o is S.ComplexInfinity:
if not coeff or coeff is S.ComplexInfinity:
# we know for sure the result will be nan
return [S.NaN], [], None
coeff = S.ComplexInfinity
continue
elif o.is_commutative:
# e
# o = b
b, e = o.as_base_exp()
# y
# 3
if o.is_Pow and b.is_Number:
# get all the factors with numeric base so they can be
# combined below, but don't combine negatives unless
# the exponent is an integer
if e.is_Rational:
if e.is_Integer:
coeff *= Pow(b, e) # it is an unevaluated power
continue
elif e.is_negative: # also a sign of an unevaluated power
seq.append(Pow(b, e))
continue
elif b.is_negative:
neg1e += e
b = -b
if b is not S.One:
pnum_rat.setdefault(b, []).append(e)
continue
elif b.is_positive or e.is_integer:
num_exp.append((b, e))
continue
c_powers.append((b,e))
# NON-COMMUTATIVE
# TODO: Make non-commutative exponents not combine automatically
else:
if o is not NC_Marker:
nc_seq.append(o)
# process nc_seq (if any)
while nc_seq:
o = nc_seq.pop(0)
if not nc_part:
nc_part.append(o)
continue
# b c b+c
# try to combine last terms: a * a -> a
o1 = nc_part.pop()
b1,e1 = o1.as_base_exp()
b2,e2 = o.as_base_exp()
new_exp = e1 + e2
# Only allow powers to combine if the new exponent is
# not an Add. This allow things like a**2*b**3 == a**5
# if a.is_commutative == False, but prohibits
# a**x*a**y and x**a*x**b from combining (x,y commute).
if b1==b2 and (not new_exp.is_Add):
o12 = b1 ** new_exp
# now o12 could be a commutative object
if o12.is_commutative:
seq.append(o12)
continue
else:
nc_seq.insert(0, o12)
else:
nc_part.append(o1)
nc_part.append(o)
# We do want a combined exponent if it would not be an Add, such as
# y 2y 3y
# x * x -> x
# We determine this if two exponents have the same term in as_coeff_mul
#
# Unfortunately, this isn't smart enough to consider combining into
# exponents that might already be adds, so things like:
# z - y y
# x * x will be left alone. This is because checking every possible
# combination can slow things down.
# gather exponents of common bases...
# in c_powers
new_c_powers = []
common_b = {} # b:e
for b, e in c_powers:
co = e.as_coeff_mul()
common_b.setdefault(b, {}).setdefault(co[1], []).append(co[0])
for b, d in common_b.items():
for di, li in d.items():
d[di] = Add(*li)
for b, e in common_b.items():
for t, c in e.items():
new_c_powers.append((b, c*Mul(*t)))
c_powers = new_c_powers
# and in num_exp
new_num_exp = []
common_b = {} # b:e
for b, e in num_exp:
co = e.as_coeff_mul()
common_b.setdefault(b, {}).setdefault(co[1], []).append(co[0])
for b, d in common_b.items():
for di, li in d.items():
d[di] = Add(*li)
for b, e in common_b.items():
for t, c in e.items():
new_num_exp.append((b,c*Mul(*t)))
num_exp = new_num_exp
# --- PART 2 ---
#
# o process collected powers (x**0 -> 1; x**1 -> x; otherwise Pow)
# o combine collected powers (2**x * 3**x -> 6**x)
# with numeric base
# ................................
# now we have:
# - coeff:
# - c_powers: (b, e)
# - num_exp: (2, e)
# - pnum_rat: {(1/3, [1/3, 2/3, 1/4])}
# 0 1
# x -> 1 x -> x
for b, e in c_powers:
if e is S.One:
if b.is_Number:
coeff *= b
else:
c_part.append(b)
elif not e is S.Zero:
c_part.append(Pow(b, e))
# x x x
# 2 * 3 -> 6
inv_exp_dict = {} # exp:Mul(num-bases) x x
# e.g. x:6 for ... * 2 * 3 * ...
for b, e in num_exp:
inv_exp_dict.setdefault(e, []).append(b)
for e, b in inv_exp_dict.items():
inv_exp_dict[e] = Mul(*b)
c_part.extend([Pow(b, e) for e, b in inv_exp_dict.iteritems() if e])
# b, e -> e, b
# {(1/5, [1/3]), (1/2, [1/12, 1/4]} -> {(1/3, [1/5, 1/2])}
comb_e = {}
for b, e in pnum_rat.iteritems():
comb_e.setdefault(Add(*e), []).append(b)
del pnum_rat
# process them, reducing exponents to values less than 1
# and updating coeff if necessary else adding them to
# num_rat for further processing
num_rat = []
for e, b in comb_e.iteritems():
b = Mul(*b)
if e.q == 1:
coeff *= Pow(b, e)
continue
if e.p > e.q:
e_i, ep = divmod(e.p, e.q)
coeff *= Pow(b, e_i)
e = Rational(ep, e.q)
num_rat.append((b, e))
del comb_e
# extract gcd of bases in num_rat
# 2**(1/3)*6**(1/4) -> 2**(1/3+1/4)*3**(1/4)
pnew = {}
i = 0 # steps through num_rat which may grow
while i < len(num_rat):
bi, ei = num_rat[i]
grow = []
for j in range(i + 1, len(num_rat)):
bj, ej = num_rat[j]
g = igcd(bi, bj)
if g != 1:
# 4**r1*6**r2 -> 2**(r1+r2) * 2**r1 * 3**r2
# this might have a gcd with something else
e = ei + ej
if e.q == 1:
coeff *= Pow(g, e)
else:
if e.p > e.q:
e_i, ep = divmod(e.p, e.q) # change e in place
coeff *= Pow(g, e_i)
e = Rational(ep, e.q)
grow.append((g, e))
# update the jth item
num_rat[j] = (bj//g, ej)
# update bi that we are checking with
bi = bi//g
if bi is S.One:
break
if bi is not S.One:
obj = Pow(bi, ei)
if obj.is_Number:
coeff *= obj
else:
if obj.is_Mul: # sqrt(12) -> 2*sqrt(3)
c, obj = obj.args # expecting only 2 args
coeff *= c
assert obj.is_Pow
bi, ei = obj.args
pnew.setdefault(ei, []).append(bi)
num_rat.extend(grow)
i += 1
# combine bases of the new powers
for e, b in pnew.iteritems():
pnew[e] = Mul(*b)
# see if there is a base with matching coefficient
# that the -1 can be joined with
if neg1e:
p = Pow(S.NegativeOne, neg1e)
if p.is_Number:
coeff *= p
else:
if p.is_Mul:
c, p = p.args
coeff *= c
assert p.is_Pow and p.base is S.NegativeOne
neg1e = p.args[1]
for e, b in pnew.iteritems():
if e == neg1e and b.is_positive:
pnew[e] = -b
break
else:
c_part.append(p)
# add all the pnew powers
c_part.extend([Pow(b, e) for e, b in pnew.iteritems()])
# oo, -oo
if (coeff is S.Infinity) or (coeff is S.NegativeInfinity):
new_c_part = []
coeff_sign = 1
for t in c_part:
if t.is_positive:
continue
if t.is_negative:
coeff_sign *= -1
continue
new_c_part.append(t)
c_part = new_c_part
new_nc_part = []
for t in nc_part:
if t.is_positive:
continue
if t.is_negative:
coeff_sign *= -1
continue
new_nc_part.append(t)
nc_part = new_nc_part
coeff *= coeff_sign
# zoo
if coeff is S.ComplexInfinity:
# zoo might be
# unbounded_real + bounded_im
# bounded_real + unbounded_im
# unbounded_real + unbounded_im
# and non-zero real or imaginary will not change that status.
c_part = [c for c in c_part if not (c.is_nonzero and
c.is_real is not None)]
nc_part = [c for c in nc_part if not (c.is_nonzero and
c.is_real is not None)]
# 0
elif coeff is S.Zero:
# we know for sure the result will be 0
return [coeff], [], order_symbols
# order commutative part canonically
c_part.sort(key=cmp_to_key(Basic.compare))
# current code expects coeff to be always in slot-0
if coeff is not S.One:
c_part.insert(0, coeff)
# we are done
if len(c_part)==2 and c_part[0].is_Number and c_part[1].is_Add:
# 2*(1+a) -> 2 + 2 * a
coeff = c_part[0]
c_part = [Add(*[coeff*f for f in c_part[1].args])]
return c_part, nc_part, order_symbols
def _rgcd(a, b):
return Rational(Integer(igcd(a.p, b.p)), Integer(ilcm(a.q, b.q)))
def _eval_nseries(self, x, n, logx):
"""
This function does compute series for multivariate functions,
but the expansion is always in terms of *one* variable.
Examples:
>>> from sympy import atan2, O
>>> from sympy.abc import x, y
>>> atan2(x, y).series(x, n=2)
atan2(0, y) + x/y + O(x**2)
>>> atan2(x, y).series(y, n=2)
atan2(x, 0) - y/x + O(y**2)
This function also computes asymptotic expansions, if necessary
and possible:
>>> from sympy import loggamma
>>> loggamma(1/x)._eval_nseries(x,0,None)
log(x)/2 - log(x)/x - 1/x + O(1)
"""
if self.func.nargs is None:
raise NotImplementedError('series for user-defined \
functions are not supported.')
args = self.args
args0 = [t.limit(x, 0) for t in args]
if any([t.is_bounded == False for t in args0]):
from sympy import Dummy, oo, zoo, nan
a = [t.compute_leading_term(x, logx=logx) for t in args]
a0 = [t.limit(x, 0) for t in a]
if any([t.has(oo, -oo, zoo, nan) for t in a0]):
return self._eval_aseries(n, args0, x,
logx)._eval_nseries(x, n, logx)
# Careful: the argument goes to oo, but only logarithmically so. We
# are supposed to do a power series expansion "around the
# logarithmic term". e.g.
# f(1+x+log(x))
# -> f(1+logx) + x*f'(1+logx) + O(x**2)
# where 'logx' is given in the argument
a = [t._eval_nseries(x, n, logx) for t in args]
z = [r - r0 for (r, r0) in zip(a, a0)]
p = [Dummy() for t in z]
q = []
v = None
w = None
for ai, zi, pi in zip(a0, z, p):
if zi.has(x):
if v is not None: raise NotImplementedError
q.append(ai + pi)
v = pi
w = zi
else:
q.append(ai)
e1 = self.func(*q)
if v is None:
return e1
s = e1._eval_nseries(v, n, logx)
o = s.getO()
s = s.removeO()
s = s.subs(v, zi).expand() + C.Order(o.expr.subs(v, zi), x)
return s
if (self.func.nargs == 1 and args0[0]) or self.func.nargs > 1:
e = self
e1 = e.expand()
if e == e1:
#for example when e = sin(x+1) or e = sin(cos(x))
#let's try the general algorithm
term = e.subs(x, S.Zero)
if term.is_bounded is False or term is S.NaN:
raise PoleError("Cannot expand %s around 0" % (self))
series = term
fact = S.One
for i in range(n - 1):
i += 1
fact *= Rational(i)
e = e.diff(x)
subs = e.subs(x, S.Zero)
if subs is S.NaN:
# try to evaluate a limit if we have to
subs = e.limit(x, S.Zero)
if subs.is_bounded is False:
raise PoleError("Cannot expand %s around 0" % (self))
term = subs * (x**i) / fact
term = term.expand()
series += term
return series + C.Order(x**n, x)
return e1.nseries(x, n=n, logx=logx)
arg = self.args[0]
l = []
g = None
for i in xrange(n + 2):
g = self.taylor_term(i, arg, g)
g = g.nseries(x, n=n, logx=logx)
l.append(g)
return Add(*l) + C.Order(x**n, x)
def flatten(cls, seq):
# apply associativity, separate commutative part of seq
c_part = [] # out: commutative factors
nc_part = [] # out: non-commutative factors
nc_seq = []
coeff = S.One # standalone term
# e.g. 3 * ...
c_powers = [] # (base,exp) n
# e.g. (x,n) for x
num_exp = [] # (num-base, exp) y
# e.g. (3, y) for ... * 3 * ...
neg1e = 0 # exponent on -1 extracted from Number-based Pow
pnum_rat = {} # (num-base, Rat-exp) 1/2
# e.g. (3, 1/2) for ... * 3 * ...
order_symbols = None
# --- PART 1 ---
#
# "collect powers and coeff":
#
# o coeff
# o c_powers
# o num_exp
# o neg1e
# o pnum_rat
#
# NOTE: this is optimized for all-objects-are-commutative case
for o in seq:
# O(x)
if o.is_Order:
o, order_symbols = o.as_expr_variables(order_symbols)
# Mul([...])
if o.is_Mul:
if o.is_commutative:
seq.extend(o.args) # XXX zerocopy?
else:
# NCMul can have commutative parts as well
for q in o.args:
if q.is_commutative:
seq.append(q)
else:
nc_seq.append(q)
# append non-commutative marker, so we don't forget to
# process scheduled non-commutative objects
seq.append(NC_Marker)
continue
# 3
elif o.is_Number:
if o is S.NaN or coeff is S.ComplexInfinity and o is S.Zero:
# we know for sure the result will be nan
return [S.NaN], [], None
elif coeff.is_Number: # it could be zoo
coeff *= o
if coeff is S.NaN:
# we know for sure the result will be nan
return [S.NaN], [], None
continue
elif o is S.ComplexInfinity:
if not coeff or coeff is S.ComplexInfinity:
# we know for sure the result will be nan
return [S.NaN], [], None
coeff = S.ComplexInfinity
continue
elif o.is_commutative:
# e
# o = b
b, e = o.as_base_exp()
# y
# 3
if o.is_Pow and b.is_Number:
# get all the factors with numeric base so they can be
# combined below, but don't combine negatives unless
# the exponent is an integer
if e.is_Rational:
if e.is_Integer:
coeff *= Pow(b, e) # it is an unevaluated power
continue
elif e.is_negative: # also a sign of an unevaluated power
seq.append(Pow(b, e))
continue
elif b.is_negative:
neg1e += e
b = -b
if b is not S.One:
pnum_rat.setdefault(b, []).append(e)
continue
elif b.is_positive or e.is_integer:
num_exp.append((b, e))
continue
c_powers.append((b, e))
# NON-COMMUTATIVE
# TODO: Make non-commutative exponents not combine automatically
else:
if o is not NC_Marker:
nc_seq.append(o)
# process nc_seq (if any)
while nc_seq:
o = nc_seq.pop(0)
if not nc_part:
nc_part.append(o)
continue
# b c b+c
# try to combine last terms: a * a -> a
o1 = nc_part.pop()
b1, e1 = o1.as_base_exp()
b2, e2 = o.as_base_exp()
new_exp = e1 + e2
# Only allow powers to combine if the new exponent is
# not an Add. This allow things like a**2*b**3 == a**5
# if a.is_commutative == False, but prohibits
# a**x*a**y and x**a*x**b from combining (x,y commute).
if b1 == b2 and (not new_exp.is_Add):
o12 = b1**new_exp
# now o12 could be a commutative object
if o12.is_commutative:
seq.append(o12)
continue
else:
nc_seq.insert(0, o12)
else:
nc_part.append(o1)
nc_part.append(o)
# We do want a combined exponent if it would not be an Add, such as
# y 2y 3y
# x * x -> x
# We determine this if two exponents have the same term in as_coeff_mul
#
# Unfortunately, this isn't smart enough to consider combining into
# exponents that might already be adds, so things like:
# z - y y
# x * x will be left alone. This is because checking every possible
# combination can slow things down.
# gather exponents of common bases...
# in c_powers
new_c_powers = []
common_b = {} # b:e
for b, e in c_powers:
co = e.as_coeff_mul()
common_b.setdefault(b, {}).setdefault(co[1], []).append(co[0])
for b, d in common_b.items():
for di, li in d.items():
d[di] = Add(*li)
for b, e in common_b.items():
for t, c in e.items():
new_c_powers.append((b, c * Mul(*t)))
c_powers = new_c_powers
# and in num_exp
new_num_exp = []
common_b = {} # b:e
for b, e in num_exp:
co = e.as_coeff_mul()
common_b.setdefault(b, {}).setdefault(co[1], []).append(co[0])
for b, d in common_b.items():
for di, li in d.items():
d[di] = Add(*li)
for b, e in common_b.items():
for t, c in e.items():
new_num_exp.append((b, c * Mul(*t)))
num_exp = new_num_exp
# --- PART 2 ---
#
# o process collected powers (x**0 -> 1; x**1 -> x; otherwise Pow)
# o combine collected powers (2**x * 3**x -> 6**x)
# with numeric base
# ................................
# now we have:
# - coeff:
# - c_powers: (b, e)
# - num_exp: (2, e)
# - pnum_rat: {(1/3, [1/3, 2/3, 1/4])}
# 0 1
# x -> 1 x -> x
for b, e in c_powers:
if e is S.One:
if b.is_Number:
coeff *= b
else:
c_part.append(b)
elif not e is S.Zero:
c_part.append(Pow(b, e))
# x x x
# 2 * 3 -> 6
inv_exp_dict = {} # exp:Mul(num-bases) x x
# e.g. x:6 for ... * 2 * 3 * ...
for b, e in num_exp:
inv_exp_dict.setdefault(e, []).append(b)
for e, b in inv_exp_dict.items():
inv_exp_dict[e] = Mul(*b)
c_part.extend([Pow(b, e) for e, b in inv_exp_dict.iteritems() if e])
# b, e -> e, b
# {(1/5, [1/3]), (1/2, [1/12, 1/4]} -> {(1/3, [1/5, 1/2])}
comb_e = {}
for b, e in pnum_rat.iteritems():
comb_e.setdefault(Add(*e), []).append(b)
del pnum_rat
# process them, reducing exponents to values less than 1
# and updating coeff if necessary else adding them to
# num_rat for further processing
num_rat = []
for e, b in comb_e.iteritems():
b = Mul(*b)
if e.q == 1:
coeff *= Pow(b, e)
continue
if e.p > e.q:
e_i, ep = divmod(e.p, e.q)
coeff *= Pow(b, e_i)
e = Rational(ep, e.q)
num_rat.append((b, e))
del comb_e
# extract gcd of bases in num_rat
# 2**(1/3)*6**(1/4) -> 2**(1/3+1/4)*3**(1/4)
pnew = {}
i = 0 # steps through num_rat which may grow
while i < len(num_rat):
bi, ei = num_rat[i]
grow = []
for j in range(i + 1, len(num_rat)):
bj, ej = num_rat[j]
g = igcd(bi, bj)
if g != 1:
# 4**r1*6**r2 -> 2**(r1+r2) * 2**r1 * 3**r2
# this might have a gcd with something else
e = ei + ej
if e.q == 1:
coeff *= Pow(g, e)
else:
if e.p > e.q:
e_i, ep = divmod(e.p, e.q) # change e in place
coeff *= Pow(g, e_i)
e = Rational(ep, e.q)
grow.append((g, e))
# update the jth item
num_rat[j] = (bj // g, ej)
# update bi that we are checking with
bi = bi // g
if bi is S.One:
break
if bi is not S.One:
obj = Pow(bi, ei)
if obj.is_Number:
coeff *= obj
else:
if obj.is_Mul: # 12**(1/2) -> 2*sqrt(3)
c, obj = obj.args # expecting only 2 args
coeff *= c
assert obj.is_Pow
bi, ei = obj.args
pnew.setdefault(ei, []).append(bi)
num_rat.extend(grow)
i += 1
# combine bases of the new powers
for e, b in pnew.iteritems():
pnew[e] = Mul(*b)
# see if there is a base with matching coefficient
# that the -1 can be joined with
if neg1e:
p = Pow(S.NegativeOne, neg1e)
if p.is_Number:
coeff *= p
else:
if p.is_Mul:
c, p = p.args
coeff *= c
assert p.is_Pow and p.base is S.NegativeOne
neg1e = p.args[1]
for e, b in pnew.iteritems():
if e == neg1e and b.is_positive:
pnew[e] = -b
break
else:
c_part.append(p)
# add all the pnew powers
c_part.extend([Pow(b, e) for e, b in pnew.iteritems()])
# oo, -oo
if (coeff is S.Infinity) or (coeff is S.NegativeInfinity):
new_c_part = []
coeff_sign = 1
for t in c_part:
if t.is_positive:
continue
if t.is_negative:
coeff_sign *= -1
continue
new_c_part.append(t)
c_part = new_c_part
new_nc_part = []
for t in nc_part:
if t.is_positive:
continue
if t.is_negative:
coeff_sign *= -1
continue
new_nc_part.append(t)
nc_part = new_nc_part
coeff *= coeff_sign
# zoo
if coeff is S.ComplexInfinity:
# zoo might be
# unbounded_real + bounded_im
# bounded_real + unbounded_im
# unbounded_real + unbounded_im
# and non-zero real or imaginary will not change that status.
c_part = [
c for c in c_part
if not (c.is_nonzero and c.is_real is not None)
]
nc_part = [
c for c in nc_part
if not (c.is_nonzero and c.is_real is not None)
]
# 0
elif coeff is S.Zero:
# we know for sure the result will be 0
return [coeff], [], order_symbols
# order commutative part canonically
c_part.sort(key=cmp_to_key(Basic.compare))
# current code expects coeff to be always in slot-0
if coeff is not S.One:
c_part.insert(0, coeff)
# we are done
if len(c_part) == 2 and c_part[0].is_Number and c_part[1].is_Add:
# 2*(1+a) -> 2 + 2 * a
coeff = c_part[0]
c_part = [Add(*[coeff * f for f in c_part[1].args])]
return c_part, nc_part, order_symbols
def _matches(self, expr, repl_dict={}, evaluate=False):
# weed out negative one prefixes
sign = 1
if self.args[0] == -1:
self = -self
sign = -sign
if expr.is_Mul and expr.args[0] == -1:
expr = -expr
sign = -sign
if evaluate:
return self.subs(repl_dict).matches(expr, repl_dict)
expr = sympify(expr)
if not isinstance(expr, self.__class__):
# if we can omit the first factor, we can match it to sign * one
if Mul(*self.args[1:]) == expr:
return self.args[0].matches(Rational(sign), repl_dict,
evaluate)
# two-factor product: if the 2nd factor matches, the first part must be sign * one
if len(self.args[:]) == 2:
dd = self.args[1].matches(expr, repl_dict, evaluate)
if dd == None:
return None
dd = self.args[0].matches(Rational(sign), dd, evaluate)
return dd
return None
if len(self.args[:]) == 0:
if self == expr:
return repl_dict
return None
d = repl_dict.copy()
# weed out identical terms
pp = list(self.args)
ee = list(expr.args)
for p in self.args:
if p in expr.args:
ee.remove(p)
pp.remove(p)
# only one symbol left in pattern -> match the remaining expression
from symbol import Wild
if len(pp) == 1 and isinstance(pp[0], Wild):
if len(ee) == 1:
d[pp[0]] = sign * ee[0]
else:
d[pp[0]] = sign * (type(expr)(*ee))
return d
if len(ee) != len(pp):
return None
i = 0
for p, e in zip(pp, ee):
if i == 0 and sign != 1:
try:
e = sign * e
except TypeError:
return None
d = p.matches(e, d, evaluate=not i)
i += 1
if d is None:
return None
return d