def test_Identity_doit():
Inn = Identity(Add(n, n, evaluate=False))
assert isinstance(Inn.rows, Add)
assert Inn.doit() == Identity(2 * n)
assert isinstance(Inn.doit().rows, Mul)
def rule_gamma(expr, level=0):
""" Simplify products of gamma functions further. """
if expr.is_Atom:
return expr
def gamma_rat(x):
# helper to simplify ratios of gammas
was = x.count(gamma)
xx = x.replace(
gamma, lambda n: _rf(1, (n - 1).expand()).replace(
_rf, lambda a, b: gamma(a + b) / gamma(a)))
if xx.count(gamma) < was:
x = xx
return x
def gamma_factor(x):
# return True if there is a gamma factor in shallow args
if isinstance(x, gamma):
return True
if x.is_Add or x.is_Mul:
return any(gamma_factor(xi) for xi in x.args)
if x.is_Pow and (x.exp.is_integer or x.base.is_positive):
return gamma_factor(x.base)
return False
# recursion step
if level == 0:
expr = expr.func(*[rule_gamma(x, level + 1) for x in expr.args])
level += 1
if not expr.is_Mul:
return expr
# non-commutative step
if level == 1:
args, nc = expr.args_cnc()
if not args:
return expr
if nc:
return rule_gamma(Mul._from_args(args),
level + 1) * Mul._from_args(nc)
level += 1
# pure gamma handling, not factor absorption
if level == 2:
T, F = sift(expr.args, gamma_factor, binary=True)
gamma_ind = Mul(*F)
d = Mul(*T)
nd, dd = d.as_numer_denom()
for ipass in range(2):
args = list(ordered(Mul.make_args(nd)))
for i, ni in enumerate(args):
if ni.is_Add:
ni, dd = Add(*[
rule_gamma(gamma_rat(a / dd), level + 1)
for a in ni.args
]).as_numer_denom()
args[i] = ni
if not dd.has(gamma):
break
nd = Mul(*args)
if ipass == 0 and not gamma_factor(nd):
break
nd, dd = dd, nd # now process in reversed order
expr = gamma_ind * nd / dd
if not (expr.is_Mul and (gamma_factor(dd) or gamma_factor(nd))):
return expr
level += 1
# iteration until constant
if level == 3:
while True:
was = expr
expr = rule_gamma(expr, 4)
if expr == was:
return expr
numer_gammas = []
denom_gammas = []
numer_others = []
denom_others = []
def explicate(p):
if p is S.One:
return None, []
b, e = p.as_base_exp()
if e.is_Integer:
if isinstance(b, gamma):
return True, [b.args[0]] * e
else:
return False, [b] * e
else:
return False, [p]
newargs = list(ordered(expr.args))
while newargs:
n, d = newargs.pop().as_numer_denom()
isg, l = explicate(n)
if isg:
numer_gammas.extend(l)
elif isg is False:
numer_others.extend(l)
isg, l = explicate(d)
if isg:
denom_gammas.extend(l)
elif isg is False:
denom_others.extend(l)
# =========== level 2 work: pure gamma manipulation =========
if not as_comb:
# Try to reduce the number of gamma factors by applying the
# reflection formula gamma(x)*gamma(1-x) = pi/sin(pi*x)
for gammas, numer, denom in [
(numer_gammas, numer_others, denom_others),
(denom_gammas, denom_others, numer_others)
]:
new = []
while gammas:
g1 = gammas.pop()
if g1.is_integer:
new.append(g1)
continue
for i, g2 in enumerate(gammas):
n = g1 + g2 - 1
if not n.is_Integer:
continue
numer.append(S.Pi)
denom.append(sin(S.Pi * g1))
gammas.pop(i)
if n > 0:
for k in range(n):
numer.append(1 - g1 + k)
elif n < 0:
for k in range(-n):
denom.append(-g1 - k)
break
else:
new.append(g1)
# /!\ updating IN PLACE
gammas[:] = new
# Try to reduce the number of gammas by using the duplication
# theorem to cancel an upper and lower: gamma(2*s)/gamma(s) =
# 2**(2*s + 1)/(4*sqrt(pi))*gamma(s + 1/2). Although this could
# be done with higher argument ratios like gamma(3*x)/gamma(x),
# this would not reduce the number of gammas as in this case.
for ng, dg, no, do in [
(numer_gammas, denom_gammas, numer_others, denom_others),
(denom_gammas, numer_gammas, denom_others, numer_others)
]:
while True:
for x in ng:
for y in dg:
n = x - 2 * y
if n.is_Integer:
break
else:
continue
break
else:
break
ng.remove(x)
dg.remove(y)
if n > 0:
for k in range(n):
no.append(2 * y + k)
elif n < 0:
for k in range(-n):
do.append(2 * y - 1 - k)
ng.append(y + S(1) / 2)
no.append(2**(2 * y - 1))
do.append(sqrt(S.Pi))
# Try to reduce the number of gamma factors by applying the
# multiplication theorem (used when n gammas with args differing
# by 1/n mod 1 are encountered).
#
# run of 2 with args differing by 1/2
#
# >>> gammasimp(gamma(x)*gamma(x+S.Half))
# 2*sqrt(2)*2**(-2*x - 1/2)*sqrt(pi)*gamma(2*x)
#
# run of 3 args differing by 1/3 (mod 1)
#
# >>> gammasimp(gamma(x)*gamma(x+S(1)/3)*gamma(x+S(2)/3))
# 6*3**(-3*x - 1/2)*pi*gamma(3*x)
# >>> gammasimp(gamma(x)*gamma(x+S(1)/3)*gamma(x+S(5)/3))
# 2*3**(-3*x - 1/2)*pi*(3*x + 2)*gamma(3*x)
#
def _run(coeffs):
# find runs in coeffs such that the difference in terms (mod 1)
# of t1, t2, ..., tn is 1/n
u = list(uniq(coeffs))
for i in range(len(u)):
dj = ([((u[j] - u[i]) % 1, j)
for j in range(i + 1, len(u))])
for one, j in dj:
if one.p == 1 and one.q != 1:
n = one.q
got = [i]
get = list(range(1, n))
for d, j in dj:
m = n * d
if m.is_Integer and m in get:
get.remove(m)
got.append(j)
if not get:
break
else:
continue
for i, j in enumerate(got):
c = u[j]
coeffs.remove(c)
got[i] = c
return one.q, got[0], got[1:]
def _mult_thm(gammas, numer, denom):
# pull off and analyze the leading coefficient from each gamma arg
# looking for runs in those Rationals
# expr -> coeff + resid -> rats[resid] = coeff
rats = {}
for g in gammas:
c, resid = g.as_coeff_Add()
rats.setdefault(resid, []).append(c)
# look for runs in Rationals for each resid
keys = sorted(rats, key=default_sort_key)
for resid in keys:
coeffs = list(sorted(rats[resid]))
new = []
while True:
run = _run(coeffs)
if run is None:
break
# process the sequence that was found:
# 1) convert all the gamma functions to have the right
# argument (could be off by an integer)
# 2) append the factors corresponding to the theorem
# 3) append the new gamma function
n, ui, other = run
# (1)
for u in other:
con = resid + u - 1
for k in range(int(u - ui)):
numer.append(con - k)
con = n * (resid + ui) # for (2) and (3)
# (2)
numer.append(
(2 * S.Pi)**(S(n - 1) / 2) * n**(S(1) / 2 - con))
# (3)
new.append(con)
# restore resid to coeffs
rats[resid] = [resid + c for c in coeffs] + new
# rebuild the gamma arguments
g = []
for resid in keys:
g += rats[resid]
# /!\ updating IN PLACE
gammas[:] = g
for l, numer, denom in [(numer_gammas, numer_others, denom_others),
(denom_gammas, denom_others, numer_others)
]:
_mult_thm(l, numer, denom)
# =========== level >= 2 work: factor absorption =========
if level >= 2:
# Try to absorb factors into the gammas: x*gamma(x) -> gamma(x + 1)
# and gamma(x)/(x - 1) -> gamma(x - 1)
# This code (in particular repeated calls to find_fuzzy) can be very
# slow.
def find_fuzzy(l, x):
if not l:
return
S1, T1 = compute_ST(x)
for y in l:
S2, T2 = inv[y]
if T1 != T2 or (not S1.intersection(S2) and
(S1 != set() or S2 != set())):
continue
# XXX we want some simplification (e.g. cancel or
# simplify) but no matter what it's slow.
a = len(cancel(x / y).free_symbols)
b = len(x.free_symbols)
c = len(y.free_symbols)
# TODO is there a better heuristic?
if a == 0 and (b > 0 or c > 0):
return y
# We thus try to avoid expensive calls by building the following
# "invariants": For every factor or gamma function argument
# - the set of free symbols S
# - the set of functional components T
# We will only try to absorb if T1==T2 and (S1 intersect S2 != emptyset
# or S1 == S2 == emptyset)
inv = {}
def compute_ST(expr):
if expr in inv:
return inv[expr]
return (expr.free_symbols, expr.atoms(Function).union(
set(e.exp for e in expr.atoms(Pow))))
def update_ST(expr):
inv[expr] = compute_ST(expr)
for expr in numer_gammas + denom_gammas + numer_others + denom_others:
update_ST(expr)
for gammas, numer, denom in [
(numer_gammas, numer_others, denom_others),
(denom_gammas, denom_others, numer_others)
]:
new = []
while gammas:
g = gammas.pop()
cont = True
while cont:
cont = False
y = find_fuzzy(numer, g)
if y is not None:
numer.remove(y)
if y != g:
numer.append(y / g)
update_ST(y / g)
g += 1
cont = True
y = find_fuzzy(denom, g - 1)
if y is not None:
denom.remove(y)
if y != g - 1:
numer.append((g - 1) / y)
update_ST((g - 1) / y)
g -= 1
cont = True
new.append(g)
# /!\ updating IN PLACE
gammas[:] = new
# =========== rebuild expr ==================================
return Mul(*[gamma(g) for g in numer_gammas]) \
/ Mul(*[gamma(g) for g in denom_gammas]) \
* Mul(*numer_others) / Mul(*denom_others)
def _eval_trace(self):
if self.rowblocksizes == self.colblocksizes:
return Add(
*[Trace(self.blocks[i, i]) for i in range(self.blockshape[0])])
raise NotImplementedError(
"Can't perform trace of irregular blockshape")
def __new__(cls, expr, *symbols):
expr = sympify(expr)
if expr is S.NaN:
return S.NaN
point = S.Zero
if symbols:
symbols = map(sympify, symbols)
if symbols[-1] in (S.Infinity, S.Zero):
point = symbols[-1]
symbols = symbols[:-1]
if not all(isinstance(s, Symbol) for s in symbols):
raise NotImplementedError(
'Order at points other than 0 or oo not supported.')
if not symbols:
symbols = list(expr.free_symbols)
if expr.is_Order:
v = set(expr.variables)
symbols = v | set(symbols)
if symbols == v:
return expr
symbols = list(symbols)
elif symbols:
symbols = list(set(symbols))
args = tuple(symbols) + (point, )
if len(symbols) > 1:
# XXX: better way? We need this expand() to
# workaround e.g: expr = x*(x + y).
# (x*(x + y)).as_leading_term(x, y) currently returns
# x*y (wrong order term!). That's why we want to deal with
# expand()'ed expr (handled in "if expr.is_Add" branch below).
expr = expr.expand()
if expr.is_Add:
lst = expr.extract_leading_order(*args)
expr = Add(*[f.expr for (e, f) in lst])
elif expr:
expr = expr.as_leading_term(*symbols)
expr = expr.as_independent(*symbols, as_Add=False)[1]
expr = expand_power_base(expr)
expr = expand_log(expr)
if len(symbols) == 1:
# The definition of O(f(x)) symbol explicitly stated that
# the argument of f(x) is irrelevant. That's why we can
# combine some power exponents (only "on top" of the
# expression tree for f(x)), e.g.:
# x**p * (-x)**q -> x**(p+q) for real p, q.
x = symbols[0]
margs = list(
Mul.make_args(expr.as_independent(x, as_Add=False)[1]))
for i, t in enumerate(margs):
if t.is_Pow:
b, q = t.args
if b in (x, -x) and q.is_real and not q.has(x):
margs[i] = x**q
elif b.is_Pow and not b.exp.has(x):
b, r = b.args
if b in (x, -x) and r.is_real:
margs[i] = x**(r * q)
elif b.is_Mul and b.args[0] is S.NegativeOne:
b = -b
if b.is_Pow and not b.exp.has(x):
b, r = b.args
if b in (x, -x) and r.is_real:
margs[i] = x**(r * q)
expr = Mul(*margs)
if expr is S.Zero:
return expr
if not expr.has(*symbols):
expr = S.One
# create Order instance:
symbols.sort(key=default_sort_key)
args = (expr, ) + tuple(symbols) + (point, )
obj = Expr.__new__(cls, *args)
return obj
def is_log_deriv_k_t_radical(fa, fd, DE, Df=True):
"""
Checks if Df is the logarithmic derivative of a k(t)-radical.
b in k(t) can be written as the logarithmic derivative of a k(t) radical if
there exist n in ZZ and u in k(t) with n, u != 0 such that n*b == Du/u.
Either returns (ans, u, n, const) or None, which means that Df cannot be
written as the logarithmic derivative of a k(t)-radical. ans is a list of
tuples such that Mul(*[i**j for i, j in ans]) == u. This is useful for
seeing exactly what elements of k(t) produce u.
This function uses the structure theorem approach, which says that for any
f in K, Df is the logarithmic derivative of a K-radical if and only if there
are ri in QQ such that::
--- --- Dt
\ r * Dt + \ r * i
/ i i / i --- = Df.
--- --- t
i in L i in E i
K/C(x) K/C(x)
Where C = Const(K), L_K/C(x) = { i in {1, ..., n} such that t_i is
transcendental over C(x)(t_1, ..., t_i-1) and Dt_i = Da_i/a_i, for some a_i
in C(x)(t_1, ..., t_i-1)* } (i.e., the set of all indices of logarithmic
monomials of K over C(x)), and E_K/C(x) = { i in {1, ..., n} such that t_i
is transcendental over C(x)(t_1, ..., t_i-1) and Dt_i/t_i = Da_i, for some
a_i in C(x)(t_1, ..., t_i-1) } (i.e., the set of all indices of
hyperexponential monomials of K over C(x)). If K is an elementary extension
over C(x), then the cardinality of L_K/C(x) U E_K/C(x) is exactly the
transcendence degree of K over C(x). Furthermore, because Const_D(K) ==
Const_D(C(x)) == C, deg(Dt_i) == 1 when t_i is in E_K/C(x) and
deg(Dt_i) == 0 when t_i is in L_K/C(x), implying in particular that E_K/C(x)
and L_K/C(x) are disjoint.
The sets L_K/C(x) and E_K/C(x) must, by their nature, be computed
recursively using this same function. Therefore, it is required to pass
them as indices to D (or T). L_args are the arguments of the logarithms
indexed by L_K (i.e., if i is in L_K, then T[i] == log(L_args[i])). This is
needed to compute the final answer u such that n*f == Du/u.
exp(f) will be the same as u up to a multiplicative constant. This is
because they will both behave the same as monomials. For example, both
exp(x) and exp(x + 1) == E*exp(x) satisfy Dt == t. Therefore, the term const
is returned. const is such that exp(const)*f == u. This is calculated by
subtracting the arguments of one exponential from the other. Therefore, it
is necessary to pass the arguments of the exponential terms in E_args.
To handle the case where we are given Df, not f, use
is_log_deriv_k_t_radical_in_field().
"""
H = []
if Df:
dfa, dfd = (fd * derivation(fa, DE) - fa * derivation(fd, DE)).cancel(
fd**2, include=True)
else:
dfa, dfd = fa, fd
# Our assumption here is that each monomial is recursively transcendental
if len(DE.L_K) + len(DE.E_K) != len(DE.D) - 1:
if [i for i in DE.cases if i == 'tan'] or \
set([i for i in DE.cases if i == 'primitive']) - set(DE.L_K):
raise NotImplementedError(
"Real version of the structure "
"theorems with hypertangent support is not yet implemented.")
# TODO: What should really be done in this case?
raise NotImplementedError("Nonelementary extensions not supported "
"in the structure theorems.")
E_part = [DE.D[i].quo(Poly(DE.T[i], DE.T[i])).as_expr() for i in DE.E_K]
L_part = [DE.D[i].as_expr() for i in DE.L_K]
lhs = Matrix([E_part + L_part])
rhs = Matrix([dfa.as_expr() / dfd.as_expr()])
A, u = constant_system(lhs, rhs, DE)
if not all(derivation(i, DE, basic=True).is_zero for i in u) or not A:
# If the elements of u are not all constant
# Note: See comment in constant_system
# Also note: derivation(basic=True) calls cancel()
return None
else:
if not all(i.is_Rational for i in u):
# TODO: But maybe we can tell if they're not rational, like
# log(2)/log(3). Also, there should be an option to continue
# anyway, even if the result might potentially be wrong.
raise NotImplementedError("Cannot work with non-rational "
"coefficients in this case.")
else:
n = reduce(ilcm, [i.as_numer_denom()[1] for i in u])
u *= n
terms = [DE.T[i] for i in DE.E_K] + DE.L_args
ans = list(zip(terms, u))
result = Mul(*[Pow(i, j) for i, j in ans])
# exp(f) will be the same as result up to a multiplicative
# constant. We now find the log of that constant.
argterms = DE.E_args + [DE.T[i] for i in DE.L_K]
const = cancel(fa.as_expr() / fd.as_expr() -
Add(*[Mul(i, j / n) for i, j in zip(argterms, u)]))
return (ans, result, n, const)
def trigsimp_groebner(expr,
hints=[],
quick=False,
order="grlex",
polynomial=False):
"""
Simplify trigonometric expressions using a groebner basis algorithm.
This routine takes a fraction involving trigonometric or hyperbolic
expressions, and tries to simplify it. The primary metric is the
total degree. Some attempts are made to choose the simplest possible
expression of the minimal degree, but this is non-rigorous, and also
very slow (see the ``quick=True`` option).
If ``polynomial`` is set to True, instead of simplifying numerator and
denominator together, this function just brings numerator and denominator
into a canonical form. This is much faster, but has potentially worse
results. However, if the input is a polynomial, then the result is
guaranteed to be an equivalent polynomial of minimal degree.
The most important option is hints. Its entries can be any of the
following:
- a natural number
- a function
- an iterable of the form (func, var1, var2, ...)
- anything else, interpreted as a generator
A number is used to indicate that the search space should be increased.
A function is used to indicate that said function is likely to occur in a
simplified expression.
An iterable is used indicate that func(var1 + var2 + ...) is likely to
occur in a simplified .
An additional generator also indicates that it is likely to occur.
(See examples below).
This routine carries out various computationally intensive algorithms.
The option ``quick=True`` can be used to suppress one particularly slow
step (at the expense of potentially more complicated results, but never at
the expense of increased total degree).
Examples
========
>>> from sympy.abc import x, y
>>> from sympy import sin, tan, cos, sinh, cosh, tanh
>>> from sympy.simplify.trigsimp import trigsimp_groebner
Suppose you want to simplify ``sin(x)*cos(x)``. Naively, nothing happens:
>>> ex = sin(x)*cos(x)
>>> trigsimp_groebner(ex)
sin(x)*cos(x)
This is because ``trigsimp_groebner`` only looks for a simplification
involving just ``sin(x)`` and ``cos(x)``. You can tell it to also try
``2*x`` by passing ``hints=[2]``:
>>> trigsimp_groebner(ex, hints=[2])
sin(2*x)/2
>>> trigsimp_groebner(sin(x)**2 - cos(x)**2, hints=[2])
-cos(2*x)
Increasing the search space this way can quickly become expensive. A much
faster way is to give a specific expression that is likely to occur:
>>> trigsimp_groebner(ex, hints=[sin(2*x)])
sin(2*x)/2
Hyperbolic expressions are similarly supported:
>>> trigsimp_groebner(sinh(2*x)/sinh(x))
2*cosh(x)
Note how no hints had to be passed, since the expression already involved
``2*x``.
The tangent function is also supported. You can either pass ``tan`` in the
hints, to indicate that tan should be tried whenever cosine or sine are,
or you can pass a specific generator:
>>> trigsimp_groebner(sin(x)/cos(x), hints=[tan])
tan(x)
>>> trigsimp_groebner(sinh(x)/cosh(x), hints=[tanh(x)])
tanh(x)
Finally, you can use the iterable form to suggest that angle sum formulae
should be tried:
>>> ex = (tan(x) + tan(y))/(1 - tan(x)*tan(y))
>>> trigsimp_groebner(ex, hints=[(tan, x, y)])
tan(x + y)
"""
# TODO
# - preprocess by replacing everything by funcs we can handle
# - optionally use cot instead of tan
# - more intelligent hinting.
# For example, if the ideal is small, and we have sin(x), sin(y),
# add sin(x + y) automatically... ?
# - algebraic numbers ...
# - expressions of lowest degree are not distinguished properly
# e.g. 1 - sin(x)**2
# - we could try to order the generators intelligently, so as to influence
# which monomials appear in the quotient basis
# THEORY
# ------
# Ratsimpmodprime above can be used to "simplify" a rational function
# modulo a prime ideal. "Simplify" mainly means finding an equivalent
# expression of lower total degree.
#
# We intend to use this to simplify trigonometric functions. To do that,
# we need to decide (a) which ring to use, and (b) modulo which ideal to
# simplify. In practice, (a) means settling on a list of "generators"
# a, b, c, ..., such that the fraction we want to simplify is a rational
# function in a, b, c, ..., with coefficients in ZZ (integers).
# (2) means that we have to decide what relations to impose on the
# generators. There are two practical problems:
# (1) The ideal has to be *prime* (a technical term).
# (2) The relations have to be polynomials in the generators.
#
# We typically have two kinds of generators:
# - trigonometric expressions, like sin(x), cos(5*x), etc
# - "everything else", like gamma(x), pi, etc.
#
# Since this function is trigsimp, we will concentrate on what to do with
# trigonometric expressions. We can also simplify hyperbolic expressions,
# but the extensions should be clear.
#
# One crucial point is that all *other* generators really should behave
# like indeterminates. In particular if (say) "I" is one of them, then
# in fact I**2 + 1 = 0 and we may and will compute non-sensical
# expressions. However, we can work with a dummy and add the relation
# I**2 + 1 = 0 to our ideal, then substitute back in the end.
#
# Now regarding trigonometric generators. We split them into groups,
# according to the argument of the trigonometric functions. We want to
# organise this in such a way that most trigonometric identities apply in
# the same group. For example, given sin(x), cos(2*x) and cos(y), we would
# group as [sin(x), cos(2*x)] and [cos(y)].
#
# Our prime ideal will be built in three steps:
# (1) For each group, compute a "geometrically prime" ideal of relations.
# Geometrically prime means that it generates a prime ideal in
# CC[gens], not just ZZ[gens].
# (2) Take the union of all the generators of the ideals for all groups.
# By the geometric primality condition, this is still prime.
# (3) Add further inter-group relations which preserve primality.
#
# Step (1) works as follows. We will isolate common factors in the
# argument, so that all our generators are of the form sin(n*x), cos(n*x)
# or tan(n*x), with n an integer. Suppose first there are no tan terms.
# The ideal [sin(x)**2 + cos(x)**2 - 1] is geometrically prime, since
# X**2 + Y**2 - 1 is irreducible over CC.
# Now, if we have a generator sin(n*x), than we can, using trig identities,
# express sin(n*x) as a polynomial in sin(x) and cos(x). We can add this
# relation to the ideal, preserving geometric primality, since the quotient
# ring is unchanged.
# Thus we have treated all sin and cos terms.
# For tan(n*x), we add a relation tan(n*x)*cos(n*x) - sin(n*x) = 0.
# (This requires of course that we already have relations for cos(n*x) and
# sin(n*x).) It is not obvious, but it seems that this preserves geometric
# primality.
# XXX A real proof would be nice. HELP!
# Sketch that <S**2 + C**2 - 1, C*T - S> is a prime ideal of
# CC[S, C, T]:
# - it suffices to show that the projective closure in CP**3 is
# irreducible
# - using the half-angle substitutions, we can express sin(x), tan(x),
# cos(x) as rational functions in tan(x/2)
# - from this, we get a rational map from CP**1 to our curve
# - this is a morphism, hence the curve is prime
#
# Step (2) is trivial.
#
# Step (3) works by adding selected relations of the form
# sin(x + y) - sin(x)*cos(y) - sin(y)*cos(x), etc. Geometric primality is
# preserved by the same argument as before.
def parse_hints(hints):
"""Split hints into (n, funcs, iterables, gens)."""
n = 1
funcs, iterables, gens = [], [], []
for e in hints:
if isinstance(e, (SYMPY_INTS, Integer)):
n = e
elif isinstance(e, FunctionClass):
funcs.append(e)
elif iterable(e):
iterables.append((e[0], e[1:]))
# XXX sin(x+2y)?
# Note: we go through polys so e.g.
# sin(-x) -> -sin(x) -> sin(x)
gens.extend(
parallel_poly_from_expr([e[0](x) for x in e[1:]] +
[e[0](Add(*e[1:]))])[1].gens)
else:
gens.append(e)
return n, funcs, iterables, gens
def build_ideal(x, terms):
"""
Build generators for our ideal. Terms is an iterable with elements of
the form (fn, coeff), indicating that we have a generator fn(coeff*x).
If any of the terms is trigonometric, sin(x) and cos(x) are guaranteed
to appear in terms. Similarly for hyperbolic functions. For tan(n*x),
sin(n*x) and cos(n*x) are guaranteed.
"""
I = []
y = Dummy('y')
for fn, coeff in terms:
for c, s, t, rel in ([cos, sin, tan,
cos(x)**2 + sin(x)**2 - 1], [
cosh, sinh, tanh,
cosh(x)**2 - sinh(x)**2 - 1
]):
if coeff == 1 and fn in [c, s]:
I.append(rel)
elif fn == t:
I.append(t(coeff * x) * c(coeff * x) - s(coeff * x))
elif fn in [c, s]:
cn = fn(coeff * y).expand(trig=True).subs(y, x)
I.append(fn(coeff * x) - cn)
return list(set(I))
def analyse_gens(gens, hints):
"""
Analyse the generators ``gens``, using the hints ``hints``.
The meaning of ``hints`` is described in the main docstring.
Return a new list of generators, and also the ideal we should
work with.
"""
# First parse the hints
n, funcs, iterables, extragens = parse_hints(hints)
debug('n=%s' % n, 'funcs:', funcs, 'iterables:', iterables,
'extragens:', extragens)
# We just add the extragens to gens and analyse them as before
gens = list(gens)
gens.extend(extragens)
# remove duplicates
funcs = list(set(funcs))
iterables = list(set(iterables))
gens = list(set(gens))
# all the functions we can do anything with
allfuncs = {sin, cos, tan, sinh, cosh, tanh}
# sin(3*x) -> ((3, x), sin)
trigterms = [(g.args[0].as_coeff_mul(), g.func) for g in gens
if g.func in allfuncs]
# Our list of new generators - start with anything that we cannot
# work with (i.e. is not a trigonometric term)
freegens = [g for g in gens if g.func not in allfuncs]
newgens = []
trigdict = {}
for (coeff, var), fn in trigterms:
trigdict.setdefault(var, []).append((coeff, fn))
res = [] # the ideal
for key, val in trigdict.items():
# We have now assembeled a dictionary. Its keys are common
# arguments in trigonometric expressions, and values are lists of
# pairs (fn, coeff). x0, (fn, coeff) in trigdict means that we
# need to deal with fn(coeff*x0). We take the rational gcd of the
# coeffs, call it ``gcd``. We then use x = x0/gcd as "base symbol",
# all other arguments are integral multiples thereof.
# We will build an ideal which works with sin(x), cos(x).
# If hint tan is provided, also work with tan(x). Moreover, if
# n > 1, also work with sin(k*x) for k <= n, and similarly for cos
# (and tan if the hint is provided). Finally, any generators which
# the ideal does not work with but we need to accommodate (either
# because it was in expr or because it was provided as a hint)
# we also build into the ideal.
# This selection process is expressed in the list ``terms``.
# build_ideal then generates the actual relations in our ideal,
# from this list.
fns = [x[1] for x in val]
val = [x[0] for x in val]
gcd = reduce(igcd, val)
terms = [(fn, v / gcd) for (fn, v) in zip(fns, val)]
fs = set(funcs + fns)
for c, s, t in ([cos, sin, tan], [cosh, sinh, tanh]):
if any(x in fs for x in (c, s, t)):
fs.add(c)
fs.add(s)
for fn in fs:
for k in range(1, n + 1):
terms.append((fn, k))
extra = []
for fn, v in terms:
if fn == tan:
extra.append((sin, v))
extra.append((cos, v))
if fn in [sin, cos] and tan in fs:
extra.append((tan, v))
if fn == tanh:
extra.append((sinh, v))
extra.append((cosh, v))
if fn in [sinh, cosh] and tanh in fs:
extra.append((tanh, v))
terms.extend(extra)
x = gcd * Mul(*key)
r = build_ideal(x, terms)
res.extend(r)
newgens.extend(set(fn(v * x) for fn, v in terms))
# Add generators for compound expressions from iterables
for fn, args in iterables:
if fn == tan:
# Tan expressions are recovered from sin and cos.
iterables.extend([(sin, args), (cos, args)])
elif fn == tanh:
# Tanh expressions are recovered from sihn and cosh.
iterables.extend([(sinh, args), (cosh, args)])
else:
dummys = symbols('d:%i' % len(args), cls=Dummy)
expr = fn(Add(*dummys)).expand(trig=True).subs(
list(zip(dummys, args)))
res.append(fn(Add(*args)) - expr)
if myI in gens:
res.append(myI**2 + 1)
freegens.remove(myI)
newgens.append(myI)
return res, freegens, newgens
myI = Dummy('I')
expr = expr.subs(S.ImaginaryUnit, myI)
subs = [(myI, S.ImaginaryUnit)]
num, denom = cancel(expr).as_numer_denom()
try:
(pnum, pdenom), opt = parallel_poly_from_expr([num, denom])
except PolificationFailed:
return expr
debug('initial gens:', opt.gens)
ideal, freegens, gens = analyse_gens(opt.gens, hints)
debug('ideal:', ideal)
debug('new gens:', gens, " -- len", len(gens))
debug('free gens:', freegens, " -- len", len(gens))
# NOTE we force the domain to be ZZ to stop polys from injecting generators
# (which is usually a sign of a bug in the way we build the ideal)
if not gens:
return expr
G = groebner(ideal, order=order, gens=gens, domain=ZZ)
debug('groebner basis:', list(G), " -- len", len(G))
# If our fraction is a polynomial in the free generators, simplify all
# coefficients separately:
from sympy.simplify.ratsimp import ratsimpmodprime
if freegens and pdenom.has_only_gens(*set(gens).intersection(pdenom.gens)):
num = Poly(num, gens=gens + freegens).eject(*gens)
res = []
for monom, coeff in num.terms():
ourgens = set(parallel_poly_from_expr([coeff, denom])[1].gens)
# We compute the transitive closure of all generators that can
# be reached from our generators through relations in the ideal.
changed = True
while changed:
changed = False
for p in ideal:
p = Poly(p)
if not ourgens.issuperset(p.gens) and \
not p.has_only_gens(*set(p.gens).difference(ourgens)):
changed = True
ourgens.update(p.exclude().gens)
# NOTE preserve order!
realgens = [x for x in gens if x in ourgens]
# The generators of the ideal have now been (implicitly) split
# into two groups: those involving ourgens and those that don't.
# Since we took the transitive closure above, these two groups
# live in subgrings generated by a *disjoint* set of variables.
# Any sensible groebner basis algorithm will preserve this disjoint
# structure (i.e. the elements of the groebner basis can be split
# similarly), and and the two subsets of the groebner basis then
# form groebner bases by themselves. (For the smaller generating
# sets, of course.)
ourG = [
g.as_expr() for g in G.polys
if g.has_only_gens(*ourgens.intersection(g.gens))
]
res.append(Mul(*[a**b for a, b in zip(freegens, monom)]) * \
ratsimpmodprime(coeff/denom, ourG, order=order,
gens=realgens, quick=quick, domain=ZZ,
polynomial=polynomial).subs(subs))
return Add(*res)
# NOTE The following is simpler and has less assumptions on the
# groebner basis algorithm. If the above turns out to be broken,
# use this.
return Add(*[Mul(*[a**b for a, b in zip(freegens, monom)]) * \
ratsimpmodprime(coeff/denom, list(G), order=order,
gens=gens, quick=quick, domain=ZZ)
for monom, coeff in num.terms()])
else:
return ratsimpmodprime(expr,
list(G),
order=order,
gens=freegens + gens,
quick=quick,
domain=ZZ,
polynomial=polynomial).subs(subs)
def __trigsimp(expr, deep=False):
"""recursive helper for trigsimp"""
from sympy.simplify.fu import TR10i
if _trigpat is None:
_trigpats()
a, b, c, d, matchers_division, matchers_add, \
matchers_identity, artifacts = _trigpat
if expr.is_Mul:
# do some simplifications like sin/cos -> tan:
if not expr.is_commutative:
com, nc = expr.args_cnc()
expr = _trigsimp(Mul._from_args(com), deep) * Mul._from_args(nc)
else:
for i, (pattern, simp, ok1, ok2) in enumerate(matchers_division):
if not _dotrig(expr, pattern):
continue
newexpr = _match_div_rewrite(expr, i)
if newexpr is not None:
if newexpr != expr:
expr = newexpr
break
else:
continue
# use SymPy matching instead
res = expr.match(pattern)
if res and res.get(c, 0):
if not res[c].is_integer:
ok = ok1.subs(res)
if not ok.is_positive:
continue
ok = ok2.subs(res)
if not ok.is_positive:
continue
# if "a" contains any of trig or hyperbolic funcs with
# argument "b" then skip the simplification
if any(w.args[0] == res[b] for w in res[a].atoms(
TrigonometricFunction, HyperbolicFunction)):
continue
# simplify and finish:
expr = simp.subs(res)
break # process below
if expr.is_Add:
args = []
for term in expr.args:
if not term.is_commutative:
com, nc = term.args_cnc()
nc = Mul._from_args(nc)
term = Mul._from_args(com)
else:
nc = S.One
term = _trigsimp(term, deep)
for pattern, result in matchers_identity:
res = term.match(pattern)
if res is not None:
term = result.subs(res)
break
args.append(term * nc)
if args != expr.args:
expr = Add(*args)
expr = min(expr, expand(expr), key=count_ops)
if expr.is_Add:
for pattern, result in matchers_add:
if not _dotrig(expr, pattern):
continue
expr = TR10i(expr)
if expr.has(HyperbolicFunction):
res = expr.match(pattern)
# if "d" contains any trig or hyperbolic funcs with
# argument "a" or "b" then skip the simplification;
# this isn't perfect -- see tests
if res is None or not (a in res and b in res) or any(
w.args[0] in (res[a], res[b])
for w in res[d].atoms(TrigonometricFunction,
HyperbolicFunction)):
continue
expr = result.subs(res)
break
# Reduce any lingering artifacts, such as sin(x)**2 changing
# to 1 - cos(x)**2 when sin(x)**2 was "simpler"
for pattern, result, ex in artifacts:
if not _dotrig(expr, pattern):
continue
# Substitute a new wild that excludes some function(s)
# to help influence a better match. This is because
# sometimes, for example, 'a' would match sec(x)**2
a_t = Wild('a', exclude=[ex])
pattern = pattern.subs(a, a_t)
result = result.subs(a, a_t)
m = expr.match(pattern)
was = None
while m and was != expr:
was = expr
if m[a_t] == 0 or \
-m[a_t] in m[c].args or m[a_t] + m[c] == 0:
break
if d in m and m[a_t] * m[d] + m[c] == 0:
break
expr = result.subs(m)
m = expr.match(pattern)
m.setdefault(c, S.Zero)
elif expr.is_Mul or expr.is_Pow or deep and expr.args:
expr = expr.func(*[_trigsimp(a, deep) for a in expr.args])
try:
if not expr.has(*_trigs):
raise TypeError
e = expr.atoms(exp)
new = expr.rewrite(exp, deep=deep)
if new == e:
raise TypeError
fnew = factor(new)
if fnew != new:
new = sorted([new, factor(new)], key=count_ops)[0]
# if all exp that were introduced disappeared then accept it
if not (new.atoms(exp) - e):
expr = new
except TypeError:
pass
return expr
def _eval_trace(self):
from .trace import trace
return Add(*[trace(arg) for arg in self.args]).doit()
def _entry(self, i, j):
return Add(*[arg._entry(i, j) for arg in self.args])
def handle(expr):
# Handle first reduces to the case
# expr = 1/d, where d is an add, or d is base**p/2.
# We do this by recursively calling handle on each piece.
from sympy.simplify.simplify import nsimplify
n, d = fraction(expr)
if expr.is_Atom or (d.is_Atom and n.is_Atom):
return expr
elif not n.is_Atom:
n = n.func(*[handle(a) for a in n.args])
return _unevaluated_Mul(n, handle(1 / d))
elif n is not S.One:
return _unevaluated_Mul(n, handle(1 / d))
elif d.is_Mul:
return _unevaluated_Mul(*[handle(1 / d) for d in d.args])
# By this step, expr is 1/d, and d is not a mul.
if not symbolic and d.free_symbols:
return expr
if ispow2(d):
d2 = sqrtdenest(sqrt(d.base))**fraction(d.exp)[0]
if d2 != d:
return handle(1 / d2)
elif d.is_Pow and (d.exp.is_integer or d.base.is_positive):
# (1/d**i) = (1/d)**i
return handle(1 / d.base)**d.exp
if not (d.is_Add or ispow2(d)):
return 1 / d.func(*[handle(a) for a in d.args])
# handle 1/d treating d as an Add (though it may not be)
keep = True # keep changes that are made
# flatten it and collect radicals after checking for special
# conditions
d = _mexpand(d)
# did it change?
if d.is_Atom:
return 1 / d
# is it a number that might be handled easily?
if d.is_number:
_d = nsimplify(d)
if _d.is_Number and _d.equals(d):
return 1 / _d
while True:
# collect similar terms
collected = defaultdict(list)
for m in Add.make_args(d): # d might have become non-Add
p2 = []
other = []
for i in Mul.make_args(m):
if ispow2(i, log2=True):
p2.append(i.base if i.exp is S.Half else i.base**(
2 * i.exp))
elif i is S.ImaginaryUnit:
p2.append(S.NegativeOne)
else:
other.append(i)
collected[tuple(ordered(p2))].append(Mul(*other))
rterms = list(ordered(list(collected.items())))
rterms = [(Mul(*i), Add(*j)) for i, j in rterms]
nrad = len(rterms) - (1 if rterms[0][0] is S.One else 0)
if nrad < 1:
break
elif nrad > max_terms:
# there may have been invalid operations leading to this point
# so don't keep changes, e.g. this expression is troublesome
# in collecting terms so as not to raise the issue of 2834:
# r = sqrt(sqrt(5) + 5)
# eq = 1/(sqrt(5)*r + 2*sqrt(5)*sqrt(-sqrt(5) + 5) + 5*r)
keep = False
break
if len(rterms) > 4:
# in general, only 4 terms can be removed with repeated squaring
# but other considerations can guide selection of radical terms
# so that radicals are removed
if all(
[x.is_Integer and (y**2).is_Rational for x, y in rterms]):
nd, d = rad_rationalize(
S.One,
Add._from_args([sqrt(x) * y for x, y in rterms]))
n *= nd
else:
# is there anything else that might be attempted?
keep = False
break
from sympy.simplify.powsimp import powsimp, powdenest
num = powsimp(_num(rterms))
n *= num
d *= num
d = powdenest(_mexpand(d), force=symbolic)
if d.is_Atom:
break
if not keep:
return expr
return _unevaluated_Mul(n, 1 / d)
def _entry(self, i, j, expand=None):
return Add(*[arg._entry(i, j) for arg in self.args])
def collect_const(expr, *vars, **kwargs):
"""A non-greedy collection of terms with similar number coefficients in
an Add expr. If ``vars`` is given then only those constants will be
targeted. Although any Number can also be targeted, if this is not
desired set ``Numbers=False`` and no Float or Rational will be collected.
Examples
========
>>> from sympy import sqrt
>>> from sympy.abc import a, s, x, y, z
>>> from sympy.simplify.radsimp import collect_const
>>> collect_const(sqrt(3) + sqrt(3)*(1 + sqrt(2)))
sqrt(3)*(sqrt(2) + 2)
>>> collect_const(sqrt(3)*s + sqrt(7)*s + sqrt(3) + sqrt(7))
(sqrt(3) + sqrt(7))*(s + 1)
>>> s = sqrt(2) + 2
>>> collect_const(sqrt(3)*s + sqrt(3) + sqrt(7)*s + sqrt(7))
(sqrt(2) + 3)*(sqrt(3) + sqrt(7))
>>> collect_const(sqrt(3)*s + sqrt(3) + sqrt(7)*s + sqrt(7), sqrt(3))
sqrt(7) + sqrt(3)*(sqrt(2) + 3) + sqrt(7)*(sqrt(2) + 2)
The collection is sign-sensitive, giving higher precedence to the
unsigned values:
>>> collect_const(x - y - z)
x - (y + z)
>>> collect_const(-y - z)
-(y + z)
>>> collect_const(2*x - 2*y - 2*z, 2)
2*(x - y - z)
>>> collect_const(2*x - 2*y - 2*z, -2)
2*x - 2*(y + z)
See Also
========
collect, collect_sqrt, rcollect
"""
if not expr.is_Add:
return expr
recurse = False
Numbers = kwargs.get('Numbers', True)
if not vars:
recurse = True
vars = set()
for a in expr.args:
for m in Mul.make_args(a):
if m.is_number:
vars.add(m)
else:
vars = sympify(vars)
if not Numbers:
vars = [v for v in vars if not v.is_Number]
vars = list(ordered(vars))
for v in vars:
terms = defaultdict(list)
Fv = Factors(v)
for m in Add.make_args(expr):
f = Factors(m)
q, r = f.div(Fv)
if r.is_one:
# only accept this as a true factor if
# it didn't change an exponent from an Integer
# to a non-Integer, e.g. 2/sqrt(2) -> sqrt(2)
# -- we aren't looking for this sort of change
fwas = f.factors.copy()
fnow = q.factors
if not any(k in fwas and fwas[k].is_Integer
and not fnow[k].is_Integer for k in fnow):
terms[v].append(q.as_expr())
continue
terms[S.One].append(m)
args = []
hit = False
uneval = False
for k in ordered(terms):
v = terms[k]
if k is S.One:
args.extend(v)
continue
if len(v) > 1:
v = Add(*v)
hit = True
if recurse and v != expr:
vars.append(v)
else:
v = v[0]
# be careful not to let uneval become True unless
# it must be because it's going to be more expensive
# to rebuild the expression as an unevaluated one
if Numbers and k.is_Number and v.is_Add:
args.append(_keep_coeff(k, v, sign=True))
uneval = True
else:
args.append(k * v)
if hit:
if uneval:
expr = _unevaluated_Add(*args)
else:
expr = Add(*args)
if not expr.is_Add:
break
return expr
def collect_sqrt(expr, evaluate=None):
"""Return expr with terms having common square roots collected together.
If ``evaluate`` is False a count indicating the number of sqrt-containing
terms will be returned and, if non-zero, the terms of the Add will be
returned, else the expression itself will be returned as a single term.
If ``evaluate`` is True, the expression with any collected terms will be
returned.
Note: since I = sqrt(-1), it is collected, too.
Examples
========
>>> from sympy import sqrt
>>> from sympy.simplify.radsimp import collect_sqrt
>>> from sympy.abc import a, b
>>> r2, r3, r5 = [sqrt(i) for i in [2, 3, 5]]
>>> collect_sqrt(a*r2 + b*r2)
sqrt(2)*(a + b)
>>> collect_sqrt(a*r2 + b*r2 + a*r3 + b*r3)
sqrt(2)*(a + b) + sqrt(3)*(a + b)
>>> collect_sqrt(a*r2 + b*r2 + a*r3 + b*r5)
sqrt(3)*a + sqrt(5)*b + sqrt(2)*(a + b)
If evaluate is False then the arguments will be sorted and
returned as a list and a count of the number of sqrt-containing
terms will be returned:
>>> collect_sqrt(a*r2 + b*r2 + a*r3 + b*r5, evaluate=False)
((sqrt(3)*a, sqrt(5)*b, sqrt(2)*(a + b)), 3)
>>> collect_sqrt(a*sqrt(2) + b, evaluate=False)
((b, sqrt(2)*a), 1)
>>> collect_sqrt(a + b, evaluate=False)
((a + b,), 0)
See Also
========
collect, collect_const, rcollect
"""
if evaluate is None:
evaluate = global_evaluate[0]
# this step will help to standardize any complex arguments
# of sqrts
coeff, expr = expr.as_content_primitive()
vars = set()
for a in Add.make_args(expr):
for m in a.args_cnc()[0]:
if m.is_number and (m.is_Pow and m.exp.is_Rational and m.exp.q == 2
or m is S.ImaginaryUnit):
vars.add(m)
# we only want radicals, so exclude Number handling; in this case
# d will be evaluated
d = collect_const(expr, *vars, Numbers=False)
hit = expr != d
if not evaluate:
nrad = 0
# make the evaluated args canonical
args = list(ordered(Add.make_args(d)))
for i, m in enumerate(args):
c, nc = m.args_cnc()
for ci in c:
# XXX should this be restricted to ci.is_number as above?
if ci.is_Pow and ci.exp.is_Rational and ci.exp.q == 2 or \
ci is S.ImaginaryUnit:
nrad += 1
break
args[i] *= coeff
if not (hit or nrad):
args = [Add(*args)]
return tuple(args), nrad
return coeff * d
def collect(expr,
syms,
func=None,
evaluate=None,
exact=False,
distribute_order_term=True):
"""
Collect additive terms of an expression.
This function collects additive terms of an expression with respect
to a list of expression up to powers with rational exponents. By the
term symbol here are meant arbitrary expressions, which can contain
powers, products, sums etc. In other words symbol is a pattern which
will be searched for in the expression's terms.
The input expression is not expanded by :func:`collect`, so user is
expected to provide an expression is an appropriate form. This makes
:func:`collect` more predictable as there is no magic happening behind the
scenes. However, it is important to note, that powers of products are
converted to products of powers using the :func:`expand_power_base`
function.
There are two possible types of output. First, if ``evaluate`` flag is
set, this function will return an expression with collected terms or
else it will return a dictionary with expressions up to rational powers
as keys and collected coefficients as values.
Examples
========
>>> from sympy import S, collect, expand, factor, Wild
>>> from sympy.abc import a, b, c, x, y, z
This function can collect symbolic coefficients in polynomials or
rational expressions. It will manage to find all integer or rational
powers of collection variable::
>>> collect(a*x**2 + b*x**2 + a*x - b*x + c, x)
c + x**2*(a + b) + x*(a - b)
The same result can be achieved in dictionary form::
>>> d = collect(a*x**2 + b*x**2 + a*x - b*x + c, x, evaluate=False)
>>> d[x**2]
a + b
>>> d[x]
a - b
>>> d[S.One]
c
You can also work with multivariate polynomials. However, remember that
this function is greedy so it will care only about a single symbol at time,
in specification order::
>>> collect(x**2 + y*x**2 + x*y + y + a*y, [x, y])
x**2*(y + 1) + x*y + y*(a + 1)
Also more complicated expressions can be used as patterns::
>>> from sympy import sin, log
>>> collect(a*sin(2*x) + b*sin(2*x), sin(2*x))
(a + b)*sin(2*x)
>>> collect(a*x*log(x) + b*(x*log(x)), x*log(x))
x*(a + b)*log(x)
You can use wildcards in the pattern::
>>> w = Wild('w1')
>>> collect(a*x**y - b*x**y, w**y)
x**y*(a - b)
It is also possible to work with symbolic powers, although it has more
complicated behavior, because in this case power's base and symbolic part
of the exponent are treated as a single symbol::
>>> collect(a*x**c + b*x**c, x)
a*x**c + b*x**c
>>> collect(a*x**c + b*x**c, x**c)
x**c*(a + b)
However if you incorporate rationals to the exponents, then you will get
well known behavior::
>>> collect(a*x**(2*c) + b*x**(2*c), x**c)
x**(2*c)*(a + b)
Note also that all previously stated facts about :func:`collect` function
apply to the exponential function, so you can get::
>>> from sympy import exp
>>> collect(a*exp(2*x) + b*exp(2*x), exp(x))
(a + b)*exp(2*x)
If you are interested only in collecting specific powers of some symbols
then set ``exact`` flag in arguments::
>>> collect(a*x**7 + b*x**7, x, exact=True)
a*x**7 + b*x**7
>>> collect(a*x**7 + b*x**7, x**7, exact=True)
x**7*(a + b)
You can also apply this function to differential equations, where
derivatives of arbitrary order can be collected. Note that if you
collect with respect to a function or a derivative of a function, all
derivatives of that function will also be collected. Use
``exact=True`` to prevent this from happening::
>>> from sympy import Derivative as D, collect, Function
>>> f = Function('f') (x)
>>> collect(a*D(f,x) + b*D(f,x), D(f,x))
(a + b)*Derivative(f(x), x)
>>> collect(a*D(D(f,x),x) + b*D(D(f,x),x), f)
(a + b)*Derivative(f(x), x, x)
>>> collect(a*D(D(f,x),x) + b*D(D(f,x),x), D(f,x), exact=True)
a*Derivative(f(x), x, x) + b*Derivative(f(x), x, x)
>>> collect(a*D(f,x) + b*D(f,x) + a*f + b*f, f)
(a + b)*f(x) + (a + b)*Derivative(f(x), x)
Or you can even match both derivative order and exponent at the same time::
>>> collect(a*D(D(f,x),x)**2 + b*D(D(f,x),x)**2, D(f,x))
(a + b)*Derivative(f(x), x, x)**2
Finally, you can apply a function to each of the collected coefficients.
For example you can factorize symbolic coefficients of polynomial::
>>> f = expand((x + a + 1)**3)
>>> collect(f, x, factor)
x**3 + 3*x**2*(a + 1) + 3*x*(a + 1)**2 + (a + 1)**3
.. note:: Arguments are expected to be in expanded form, so you might have
to call :func:`expand` prior to calling this function.
See Also
========
collect_const, collect_sqrt, rcollect
"""
if evaluate is None:
evaluate = global_evaluate[0]
def make_expression(terms):
product = []
for term, rat, sym, deriv in terms:
if deriv is not None:
var, order = deriv
while order > 0:
term, order = Derivative(term, var), order - 1
if sym is None:
if rat is S.One:
product.append(term)
else:
product.append(Pow(term, rat))
else:
product.append(Pow(term, rat * sym))
return Mul(*product)
def parse_derivative(deriv):
# scan derivatives tower in the input expression and return
# underlying function and maximal differentiation order
expr, sym, order = deriv.expr, deriv.variables[0], 1
for s in deriv.variables[1:]:
if s == sym:
order += 1
else:
raise NotImplementedError(
'Improve MV Derivative support in collect')
while isinstance(expr, Derivative):
s0 = expr.variables[0]
for s in expr.variables:
if s != s0:
raise NotImplementedError(
'Improve MV Derivative support in collect')
if s0 == sym:
expr, order = expr.expr, order + len(expr.variables)
else:
break
return expr, (sym, Rational(order))
def parse_term(expr):
"""Parses expression expr and outputs tuple (sexpr, rat_expo,
sym_expo, deriv)
where:
- sexpr is the base expression
- rat_expo is the rational exponent that sexpr is raised to
- sym_expo is the symbolic exponent that sexpr is raised to
- deriv contains the derivatives the the expression
for example, the output of x would be (x, 1, None, None)
the output of 2**x would be (2, 1, x, None)
"""
rat_expo, sym_expo = S.One, None
sexpr, deriv = expr, None
if expr.is_Pow:
if isinstance(expr.base, Derivative):
sexpr, deriv = parse_derivative(expr.base)
else:
sexpr = expr.base
if expr.exp.is_Number:
rat_expo = expr.exp
else:
coeff, tail = expr.exp.as_coeff_Mul()
if coeff.is_Number:
rat_expo, sym_expo = coeff, tail
else:
sym_expo = expr.exp
elif expr.func is exp:
arg = expr.args[0]
if arg.is_Rational:
sexpr, rat_expo = S.Exp1, arg
elif arg.is_Mul:
coeff, tail = arg.as_coeff_Mul(rational=True)
sexpr, rat_expo = exp(tail), coeff
elif isinstance(expr, Derivative):
sexpr, deriv = parse_derivative(expr)
return sexpr, rat_expo, sym_expo, deriv
def parse_expression(terms, pattern):
"""Parse terms searching for a pattern.
terms is a list of tuples as returned by parse_terms;
pattern is an expression treated as a product of factors
"""
pattern = Mul.make_args(pattern)
if len(terms) < len(pattern):
# pattern is longer than matched product
# so no chance for positive parsing result
return None
else:
pattern = [parse_term(elem) for elem in pattern]
terms = terms[:] # need a copy
elems, common_expo, has_deriv = [], None, False
for elem, e_rat, e_sym, e_ord in pattern:
if elem.is_Number and e_rat == 1 and e_sym is None:
# a constant is a match for everything
continue
for j in range(len(terms)):
if terms[j] is None:
continue
term, t_rat, t_sym, t_ord = terms[j]
# keeping track of whether one of the terms had
# a derivative or not as this will require rebuilding
# the expression later
if t_ord is not None:
has_deriv = True
if (term.match(elem) is not None
and (t_sym == e_sym
or t_sym is not None and e_sym is not None
and t_sym.match(e_sym) is not None)):
if exact is False:
# we don't have to be exact so find common exponent
# for both expression's term and pattern's element
expo = t_rat / e_rat
if common_expo is None:
# first time
common_expo = expo
else:
# common exponent was negotiated before so
# there is no chance for a pattern match unless
# common and current exponents are equal
if common_expo != expo:
common_expo = 1
else:
# we ought to be exact so all fields of
# interest must match in every details
if e_rat != t_rat or e_ord != t_ord:
continue
# found common term so remove it from the expression
# and try to match next element in the pattern
elems.append(terms[j])
terms[j] = None
break
else:
# pattern element not found
return None
return [_f for _f in terms if _f], elems, common_expo, has_deriv
if evaluate:
if expr.is_Mul:
return expr.func(*[
collect(term, syms, func, True, exact, distribute_order_term)
for term in expr.args
])
elif expr.is_Pow:
b = collect(expr.base, syms, func, True, exact,
distribute_order_term)
return Pow(b, expr.exp)
if iterable(syms):
syms = [expand_power_base(i, deep=False) for i in syms]
else:
syms = [expand_power_base(syms, deep=False)]
expr = sympify(expr)
order_term = None
if distribute_order_term:
order_term = expr.getO()
if order_term is not None:
if order_term.has(*syms):
order_term = None
else:
expr = expr.removeO()
summa = [expand_power_base(i, deep=False) for i in Add.make_args(expr)]
collected, disliked = defaultdict(list), S.Zero
for product in summa:
terms = [parse_term(i) for i in Mul.make_args(product)]
for symbol in syms:
if SYMPY_DEBUG:
print("DEBUG: parsing of expression %s with symbol %s " %
(str(terms), str(symbol)))
result = parse_expression(terms, symbol)
if SYMPY_DEBUG:
print("DEBUG: returned %s" % str(result))
if result is not None:
terms, elems, common_expo, has_deriv = result
# when there was derivative in current pattern we
# will need to rebuild its expression from scratch
if not has_deriv:
index = 1
for elem in elems:
e = elem[1]
if elem[2] is not None:
e *= elem[2]
index *= Pow(elem[0], e)
else:
index = make_expression(elems)
terms = expand_power_base(make_expression(terms), deep=False)
index = expand_power_base(index, deep=False)
collected[index].append(terms)
break
else:
# none of the patterns matched
disliked += product
# add terms now for each key
collected = {k: Add(*v) for k, v in collected.items()}
if disliked is not S.Zero:
collected[S.One] = disliked
if order_term is not None:
for key, val in collected.items():
collected[key] = val + order_term
if func is not None:
collected = dict([(key, func(val)) for key, val in collected.items()])
if evaluate:
return Add(*[key * val for key, val in collected.items()])
else:
return collected
def solveset(f, symbol=None):
"""Solves a given inequality or equation with set as output
Parameters
==========
f : Expr or a relational.
The target equation or inequality
symbol : Symbol
The variable for which the equation is solved
Returns
=======
Set
A set of values for `symbol` for which `f` is True or is equal to
zero. An `EmptySet` is returned if no solution is found.
`solveset` claims to be complete in the solution set that it returns.
Raises
======
NotImplementedError
The algorithms for to find the solution of the given equation are
not yet implemented.
ValueError
The input is not valid.
RuntimeError
It is a bug, please report to the github issue tracker.
`solveset` uses two underlying functions `solveset_real` and
`solveset_complex` to solve equations. They are
the solvers for real and complex domain respectively. The domain of
the solver is decided by the assumption on the variable for which the
equation is being solved.
See Also
========
solveset_real: solver for real domain
solveset_complex: solver for complex domain
Examples
========
>>> from sympy import exp, Symbol, Eq, pprint
>>> from sympy.solvers.solveset import solveset
>>> from sympy.abc import x
* Symbols in Sympy are complex by default. A complex variable
will lead to the solving of the equation in complex domain.
>>> pprint(solveset(exp(x) - 1, x), use_unicode=False)
{2*n*I*pi | n in Integers()}
* If you want to solve equation in real domain by the `solveset`
interface, then specify the variable to real. Alternatively use
`solveset\_real`.
>>> x = Symbol('x', real=True)
>>> solveset(exp(x) - 1, x)
{0}
>>> solveset(Eq(exp(x), 1), x)
{0}
* Inequalities are always solved in the real domain irrespective of
the assumption on the variable for which the inequality is solved.
>>> solveset(exp(x) > 1, x)
(0, oo)
"""
from sympy.solvers.inequalities import solve_univariate_inequality
if symbol is None:
free_symbols = f.free_symbols
if len(free_symbols) == 1:
symbol = free_symbols.pop()
else:
raise ValueError(
filldedent('''
The independent variable must be specified for a
multivariate equation.'''))
elif not symbol.is_Symbol:
raise ValueError('A Symbol must be given, not type %s: %s' %
(type(symbol), symbol))
real = (symbol.is_real is True)
f = sympify(f)
if f is S.false:
return EmptySet()
if f is S.true:
if real:
return S.Reals
else:
return S.Complexes
if isinstance(f, Eq):
from sympy.core import Add
f = Add(f.lhs, -f.rhs, evaluate=False)
if f.is_Relational:
if real is False:
warnings.warn(
filldedent('''
The variable you are solving for is complex
but will assumed to be real since solving complex
inequalities is not supported.
'''))
return solve_univariate_inequality(f, symbol, relational=False)
if isinstance(f, (Expr, Number)):
if real is True:
return solveset_real(f, symbol)
else:
return solveset_complex(f, symbol)
def eval(cls, *_args, **kwargs):
"""."""
if not _args:
return
if not len(_args) == 1:
raise ValueError('Expecting one argument')
expr = _args[0]
code = kwargs.pop('code', None)
if isinstance(expr, Add):
args = [cls.eval(a, code=code) for a in expr.args]
v = Add(*args)
return v
elif isinstance(expr, Mul):
args = [cls.eval(a, code=code) for a in expr.args]
v = Mul(*args)
return v
elif isinstance(expr, Pow):
b = expr.base
e = expr.exp
v = Pow(cls.eval(b, code=code), e)
return v
elif isinstance(expr, _coeffs_registery):
return expr
elif isinstance(expr, (list, tuple, Tuple)):
expr = [cls.eval(a, code=code) for a in expr]
return Tuple(*expr)
elif isinstance(expr, (Matrix, ImmutableDenseMatrix)):
lines = []
n_row, n_col = expr.shape
for i_row in range(0, n_row):
line = []
for i_col in range(0, n_col):
line.append(cls.eval(expr[i_row, i_col], code=code))
lines.append(line)
return type(expr)(lines)
elif isinstance(expr, (ScalarFunction, VectorFunction)):
if code:
name = '{name}_{code}'.format(name=expr.name, code=code)
else:
name = str(expr.name)
return Symbol(name)
elif isinstance(expr, (PlusInterfaceOperator, MinusInterfaceOperator)):
return cls.eval(expr.args[0], code=code)
elif isinstance(expr, Indexed):
base = expr.base
if isinstance(base, Mapping):
if expr.indices[0] == 0:
name = 'x'
elif expr.indices[0] == 1:
name = 'y'
elif expr.indices[0] == 2:
name = 'z'
else:
raise ValueError('Wrong index')
else:
name = '{base}_{i}'.format(base=base.name, i=expr.indices[0])
if code:
name = '{name}_{code}'.format(name=name, code=code)
return Symbol(name)
elif isinstance(expr, _partial_derivatives):
atom = get_atom_derivatives(expr)
indices = get_index_derivatives_atom(expr, atom)
code = None
if indices:
index = indices[0]
code = ''
index = dict(sorted(index.items()))
for k, n in list(index.items()):
code += k * n
return cls.eval(atom, code=code)
elif isinstance(expr, _logical_partial_derivatives):
atom = get_atom_logical_derivatives(expr)
indices = get_index_logical_derivatives_atom(expr, atom)
code = None
if indices:
index = indices[0]
code = ''
index = dict(sorted(index.items()))
for k, n in list(index.items()):
code += k * n
return cls.eval(atom, code=code)
elif isinstance(expr, Mapping):
return Symbol(expr.name)
# ... this must be done here, otherwise codegen for FEM will not work
elif isinstance(expr, Symbol):
return expr
elif isinstance(expr, IndexedBase):
return expr
elif isinstance(expr, Indexed):
return expr
elif isinstance(expr, Idx):
return expr
elif isinstance(expr, Function):
args = [cls.eval(a, code=code) for a in expr.args]
return type(expr)(*args)
elif isinstance(expr, ImaginaryUnit):
return expr
elif isinstance(expr, SymbolicWeightedVolume):
mapping = expr.args[0]
if isinstance(mapping, InterfaceMapping):
mapping = mapping.minus
name = 'wvol_{mapping}'.format(mapping=mapping)
return Symbol(name)
elif isinstance(expr, SymbolicDeterminant):
name = 'det_{}'.format(str(expr.args[0]))
return Symbol(name)
elif isinstance(expr, PullBack):
return cls.eval(expr.expr, code=code)
# Expression must always be translated to Sympy!
# TODO: check if we should use 'sympy.sympify(expr)' instead
else:
raise NotImplementedError(
'Cannot translate to Sympy: {}'.format(expr))
def classify_pde(eq, func=None, dict=False, **kwargs):
"""
Returns a tuple of possible pdsolve() classifications for a PDE.
The tuple is ordered so that first item is the classification that
pdsolve() uses to solve the PDE by default. In general,
classifications near the beginning of the list will produce
better solutions faster than those near the end, though there are
always exceptions. To make pdsolve use a different classification,
use pdsolve(PDE, func, hint=<classification>). See also the pdsolve()
docstring for different meta-hints you can use.
If ``dict`` is true, classify_pde() will return a dictionary of
hint:match expression terms. This is intended for internal use by
pdsolve(). Note that because dictionaries are ordered arbitrarily,
this will most likely not be in the same order as the tuple.
You can get help on different hints by doing help(pde.pde_hintname),
where hintname is the name of the hint without "_Integral".
See sympy.pde.allhints or the sympy.pde docstring for a list of all
supported hints that can be returned from classify_pde.
Examples
========
>>> from sympy.solvers.pde import classify_pde
>>> from sympy import Function, diff, Eq
>>> from sympy.abc import x, y
>>> f = Function('f')
>>> u = f(x, y)
>>> ux = u.diff(x)
>>> uy = u.diff(y)
>>> eq = Eq(1 + (2*(ux/u)) + (3*(uy/u)))
>>> classify_pde(eq)
('1st_linear_constant_coeff_homogeneous',)
"""
prep = kwargs.pop('prep', True)
if func and len(func.args) != 2:
raise NotImplementedError(
"Right now only partial "
"differential equations of two variables are supported")
if prep or func is None:
prep, func_ = _preprocess(eq, func)
if func is None:
func = func_
if isinstance(eq, Equality):
if eq.rhs != 0:
return classify_pde(eq.lhs - eq.rhs, func)
eq = eq.lhs
f = func.func
x = func.args[0]
y = func.args[1]
fx = f(x, y).diff(x)
fy = f(x, y).diff(y)
# TODO : For now pde.py uses support offered by the ode_order function
# to find the order with respect to a multi-variable function. An
# improvement could be to classify the order of the PDE on the basis of
# individual variables.
order = ode_order(eq, f(x, y))
# hint:matchdict or hint:(tuple of matchdicts)
# Also will contain "default":<default hint> and "order":order items.
matching_hints = {'order': order}
if not order:
if dict:
matching_hints["default"] = None
return matching_hints
else:
return ()
eq = expand(eq)
a = Wild('a', exclude=[f(x, y)])
b = Wild('b', exclude=[f(x, y), fx, fy, x, y])
c = Wild('c', exclude=[f(x, y), fx, fy, x, y])
d = Wild('d', exclude=[f(x, y), fx, fy, x, y])
e = Wild('e', exclude=[f(x, y), fx, fy])
n = Wild('n', exclude=[x, y])
# Try removing the smallest power of f(x,y)
# from the highest partial derivatives of f(x,y)
reduced_eq = None
if eq.is_Add:
var = set(combinations_with_replacement((x, y), order))
dummyvar = var.copy()
power = None
for i in var:
coeff = eq.coeff(f(x, y).diff(*i))
if coeff != 1:
match = coeff.match(a * f(x, y)**n)
if match and match[a]:
power = match[n]
dummyvar.remove(i)
break
dummyvar.remove(i)
for i in dummyvar:
coeff = eq.coeff(f(x, y).diff(*i))
if coeff != 1:
match = coeff.match(a * f(x, y)**n)
if match and match[a] and match[n] < power:
power = match[n]
if power:
den = f(x, y)**power
reduced_eq = Add(*[arg / den for arg in eq.args])
if not reduced_eq:
reduced_eq = eq
if order == 1:
reduced_eq = collect(reduced_eq, f(x, y))
r = reduced_eq.match(b * fx + c * fy + d * f(x, y) + e)
if r:
if not r[e]:
## Linear first-order homogeneous partial-differential
## equation with constant coefficients
r.update({'b': b, 'c': c, 'd': d})
matching_hints["1st_linear_constant_coeff_homogeneous"] = r
else:
if r[b]**2 + r[c]**2 != 0:
## Linear first-order general partial-differential
## equation with constant coefficients
r.update({'b': b, 'c': c, 'd': d, 'e': e})
matching_hints["1st_linear_constant_coeff"] = r
matching_hints["1st_linear_constant_coeff_Integral"] = r
else:
b = Wild('b', exclude=[f(x, y), fx, fy])
c = Wild('c', exclude=[f(x, y), fx, fy])
d = Wild('d', exclude=[f(x, y), fx, fy])
r = reduced_eq.match(b * fx + c * fy + d * f(x, y) + e)
if r:
r.update({'b': b, 'c': c, 'd': d, 'e': e})
matching_hints["1st_linear_variable_coeff"] = r
# Order keys based on allhints.
retlist = []
for i in allhints:
if i in matching_hints:
retlist.append(i)
if dict:
# Dictionaries are ordered arbitrarily, so make note of which
# hint would come first for pdsolve(). Use an ordered dict in Py 3.
matching_hints["default"] = None
matching_hints["ordered_hints"] = tuple(retlist)
for i in allhints:
if i in matching_hints:
matching_hints["default"] = i
break
return matching_hints
else:
return tuple(retlist)
def test_MatAdd_postprocessor_xfail():
# This is difficult to get working because of the way that Add processes
# its args.
z = zeros(2)
assert Add(z, S.NaN) == Add(S.NaN, z)
def analyse_gens(gens, hints):
"""
Analyse the generators ``gens``, using the hints ``hints``.
The meaning of ``hints`` is described in the main docstring.
Return a new list of generators, and also the ideal we should
work with.
"""
# First parse the hints
n, funcs, iterables, extragens = parse_hints(hints)
debug('n=%s' % n, 'funcs:', funcs, 'iterables:', iterables,
'extragens:', extragens)
# We just add the extragens to gens and analyse them as before
gens = list(gens)
gens.extend(extragens)
# remove duplicates
funcs = list(set(funcs))
iterables = list(set(iterables))
gens = list(set(gens))
# all the functions we can do anything with
allfuncs = {sin, cos, tan, sinh, cosh, tanh}
# sin(3*x) -> ((3, x), sin)
trigterms = [(g.args[0].as_coeff_mul(), g.func) for g in gens
if g.func in allfuncs]
# Our list of new generators - start with anything that we cannot
# work with (i.e. is not a trigonometric term)
freegens = [g for g in gens if g.func not in allfuncs]
newgens = []
trigdict = {}
for (coeff, var), fn in trigterms:
trigdict.setdefault(var, []).append((coeff, fn))
res = [] # the ideal
for key, val in trigdict.items():
# We have now assembeled a dictionary. Its keys are common
# arguments in trigonometric expressions, and values are lists of
# pairs (fn, coeff). x0, (fn, coeff) in trigdict means that we
# need to deal with fn(coeff*x0). We take the rational gcd of the
# coeffs, call it ``gcd``. We then use x = x0/gcd as "base symbol",
# all other arguments are integral multiples thereof.
# We will build an ideal which works with sin(x), cos(x).
# If hint tan is provided, also work with tan(x). Moreover, if
# n > 1, also work with sin(k*x) for k <= n, and similarly for cos
# (and tan if the hint is provided). Finally, any generators which
# the ideal does not work with but we need to accommodate (either
# because it was in expr or because it was provided as a hint)
# we also build into the ideal.
# This selection process is expressed in the list ``terms``.
# build_ideal then generates the actual relations in our ideal,
# from this list.
fns = [x[1] for x in val]
val = [x[0] for x in val]
gcd = reduce(igcd, val)
terms = [(fn, v / gcd) for (fn, v) in zip(fns, val)]
fs = set(funcs + fns)
for c, s, t in ([cos, sin, tan], [cosh, sinh, tanh]):
if any(x in fs for x in (c, s, t)):
fs.add(c)
fs.add(s)
for fn in fs:
for k in range(1, n + 1):
terms.append((fn, k))
extra = []
for fn, v in terms:
if fn == tan:
extra.append((sin, v))
extra.append((cos, v))
if fn in [sin, cos] and tan in fs:
extra.append((tan, v))
if fn == tanh:
extra.append((sinh, v))
extra.append((cosh, v))
if fn in [sinh, cosh] and tanh in fs:
extra.append((tanh, v))
terms.extend(extra)
x = gcd * Mul(*key)
r = build_ideal(x, terms)
res.extend(r)
newgens.extend(set(fn(v * x) for fn, v in terms))
# Add generators for compound expressions from iterables
for fn, args in iterables:
if fn == tan:
# Tan expressions are recovered from sin and cos.
iterables.extend([(sin, args), (cos, args)])
elif fn == tanh:
# Tanh expressions are recovered from sihn and cosh.
iterables.extend([(sinh, args), (cosh, args)])
else:
dummys = symbols('d:%i' % len(args), cls=Dummy)
expr = fn(Add(*dummys)).expand(trig=True).subs(
list(zip(dummys, args)))
res.append(fn(Add(*args)) - expr)
if myI in gens:
res.append(myI**2 + 1)
freegens.remove(myI)
newgens.append(myI)
return res, freegens, newgens
def test_MatAdd_postprocessor():
# Some of these are nonsensical, but we do not raise errors for Add
# because that breaks algorithms that want to replace matrices with dummy
# symbols.
z = zeros(2)
assert Add(0, z) == Add(z, 0) == z
a = Add(S.Infinity, z)
assert a == Add(z, S.Infinity)
assert isinstance(a, Add)
assert a.args == (S.Infinity, z)
a = Add(S.ComplexInfinity, z)
assert a == Add(z, S.ComplexInfinity)
assert isinstance(a, Add)
assert a.args == (S.ComplexInfinity, z)
a = Add(z, S.NaN)
# assert a == Add(S.NaN, z) # See the XFAIL above
assert isinstance(a, Add)
assert a.args == (S.NaN, z)
M = Matrix([[1, 2], [3, 4]])
a = Add(x, M)
assert a == Add(M, x)
assert isinstance(a, Add)
assert a.args == (x, M)
A = MatrixSymbol("A", 2, 2)
assert Add(A, M) == Add(M, A) == A + M
# Scalars should be absorbed into constant matrices (producing an error)
a = Add(x, M, A)
assert a == Add(M, x, A) == Add(M, A, x) == Add(x, A, M) == Add(A, x, M) == Add(A, M, x)
assert isinstance(a, Add)
assert a.args == (x, A + M)
assert Add(M, M) == 2*M
assert Add(M, A, M) == Add(M, M, A) == Add(A, M, M) == A + 2*M
a = Add(A, x, M, M, x)
assert isinstance(a, Add)
assert a.args == (2*x, A + 2*M)
def __new__(cls, expr, *args, **kwargs):
expr = sympify(expr)
if not args:
if expr.is_Order:
variables = expr.variables
point = expr.point
else:
variables = list(expr.free_symbols)
point = [S.Zero]*len(variables)
else:
args = list(args if is_sequence(args) else [args])
variables, point = [], []
if is_sequence(args[0]):
for a in args:
v, p = list(map(sympify, a))
variables.append(v)
point.append(p)
else:
variables = list(map(sympify, args))
point = [S.Zero]*len(variables)
if not all(v.is_Symbol for v in variables):
raise TypeError('Variables are not symbols, got %s' % variables)
if len(list(uniq(variables))) != len(variables):
raise ValueError('Variables are supposed to be unique symbols, got %s' % variables)
if expr.is_Order:
expr_vp = dict(expr.args[1:])
new_vp = dict(expr_vp)
vp = dict(zip(variables, point))
for v, p in vp.items():
if v in new_vp.keys():
if p != new_vp[v]:
raise NotImplementedError(
"Mixing Order at different points is not supported.")
else:
new_vp[v] = p
if set(expr_vp.keys()) == set(new_vp.keys()):
return expr
else:
variables = list(new_vp.keys())
point = [new_vp[v] for v in variables]
if expr is S.NaN:
return S.NaN
if any(x in p.free_symbols for x in variables for p in point):
raise ValueError('Got %s as a point.' % point)
if variables:
if any(p != point[0] for p in point):
raise NotImplementedError
if point[0] is S.Infinity:
s = {k: 1/Dummy() for k in variables}
rs = {1/v: 1/k for k, v in s.items()}
elif point[0] is not S.Zero:
s = dict((k, Dummy() + point[0]) for k in variables)
rs = dict((v - point[0], k - point[0]) for k, v in s.items())
else:
s = ()
rs = ()
expr = expr.subs(s)
if expr.is_Add:
from sympy import expand_multinomial
expr = expand_multinomial(expr)
if s:
args = tuple([r[0] for r in rs.items()])
else:
args = tuple(variables)
if len(variables) > 1:
# XXX: better way? We need this expand() to
# workaround e.g: expr = x*(x + y).
# (x*(x + y)).as_leading_term(x, y) currently returns
# x*y (wrong order term!). That's why we want to deal with
# expand()'ed expr (handled in "if expr.is_Add" branch below).
expr = expr.expand()
if expr.is_Add:
lst = expr.extract_leading_order(args)
expr = Add(*[f.expr for (e, f) in lst])
elif expr:
expr = expr.as_leading_term(*args)
expr = expr.as_independent(*args, as_Add=False)[1]
expr = expand_power_base(expr)
expr = expand_log(expr)
if len(args) == 1:
# The definition of O(f(x)) symbol explicitly stated that
# the argument of f(x) is irrelevant. That's why we can
# combine some power exponents (only "on top" of the
# expression tree for f(x)), e.g.:
# x**p * (-x)**q -> x**(p+q) for real p, q.
x = args[0]
margs = list(Mul.make_args(
expr.as_independent(x, as_Add=False)[1]))
for i, t in enumerate(margs):
if t.is_Pow:
b, q = t.args
if b in (x, -x) and q.is_real and not q.has(x):
margs[i] = x**q
elif b.is_Pow and not b.exp.has(x):
b, r = b.args
if b in (x, -x) and r.is_real:
margs[i] = x**(r*q)
elif b.is_Mul and b.args[0] is S.NegativeOne:
b = -b
if b.is_Pow and not b.exp.has(x):
b, r = b.args
if b in (x, -x) and r.is_real:
margs[i] = x**(r*q)
expr = Mul(*margs)
expr = expr.subs(rs)
if expr is S.Zero:
return expr
if expr.is_Order:
expr = expr.expr
if not expr.has(*variables):
expr = S.One
# create Order instance:
vp = dict(zip(variables, point))
variables.sort(key=default_sort_key)
point = [vp[v] for v in variables]
args = (expr,) + Tuple(*zip(variables, point))
obj = Expr.__new__(cls, *args)
return obj
def limit(e, z, z0, dir="+"):
"""
Compute the limit of e(z) at the point z0.
z0 can be any expression, including oo and -oo.
For dir="+" (default) it calculates the limit from the right
(z->z0+) and for dir="-" the limit from the left (z->z0-). For infinite z0
(oo or -oo), the dir argument doesn't matter.
Examples
========
>>> from sympy import limit, sin, Symbol, oo
>>> from sympy.abc import x
>>> limit(sin(x)/x, x, 0)
1
>>> limit(1/x, x, 0, dir="+")
oo
>>> limit(1/x, x, 0, dir="-")
-oo
>>> limit(1/x, x, oo)
0
Notes
=====
First we try some heuristics for easy and frequent cases like "x", "1/x",
"x**2" and similar, so that it's fast. For all other cases, we use the
Gruntz algorithm (see the gruntz() function).
"""
from sympy import Wild, log
e = sympify(e)
z = sympify(z)
z0 = sympify(z0)
if e == z:
return z0
if e.is_Rational:
return e
if not e.has(z):
return e
if e.func is tan:
# discontinuity at odd multiples of pi/2; 0 at even
disc = S.Pi/2
sign = 1
if dir == '-':
sign *= -1
i = limit(sign*e.args[0], z, z0)/disc
if i.is_integer:
if i.is_even:
return S.Zero
elif i.is_odd:
if dir == '+':
return S.NegativeInfinity
else:
return S.Infinity
if e.func is cot:
# discontinuity at multiples of pi; 0 at odd pi/2 multiples
disc = S.Pi
sign = 1
if dir == '-':
sign *= -1
i = limit(sign*e.args[0], z, z0)/disc
if i.is_integer:
if dir == '-':
return S.NegativeInfinity
else:
return S.Infinity
elif (2*i).is_integer:
return S.Zero
if e.is_Pow:
b, ex = e.args
c = None # records sign of b if b is +/-z or has a bounded value
if b.is_Mul:
c, b = b.as_two_terms()
if c is S.NegativeOne and b == z:
c = '-'
elif b == z:
c = '+'
if ex.is_number:
if c is None:
base = b.subs(z, z0)
if base.is_finite and (ex.is_bounded or base is not S.One):
return base**ex
else:
if z0 == 0 and ex < 0:
if dir != c:
# integer
if ex.is_even:
return S.Infinity
elif ex.is_odd:
return S.NegativeInfinity
# rational
elif ex.is_Rational:
return (S.NegativeOne**ex)*S.Infinity
else:
return S.ComplexInfinity
return S.Infinity
return z0**ex
if e.is_Mul or not z0 and e.is_Pow and b.func is log:
if e.is_Mul:
if abs(z0) is S.Infinity:
n, d = e.as_numer_denom()
# XXX todo: this should probably be stated in the
# negative -- i.e. to exclude expressions that should
# not be handled this way but I'm not sure what that
# condition is; when ok is True it means that the leading
# term approach is going to succeed (hopefully)
ok = lambda w: (z in w.free_symbols and
any(a.is_polynomial(z) or
any(z in m.free_symbols and m.is_polynomial(z)
for m in Mul.make_args(a))
for a in Add.make_args(w)))
if all(ok(w) for w in (n, d)):
u = C.Dummy(positive=(z0 is S.Infinity))
inve = (n/d).subs(z, 1/u)
return limit(inve.as_leading_term(u), u,
S.Zero, "+" if z0 is S.Infinity else "-")
# weed out the z-independent terms
i, d = e.as_independent(z)
if i is not S.One and i.is_bounded:
return i*limit(d, z, z0, dir)
else:
i, d = S.One, e
if not z0:
# look for log(z)**q or z**p*log(z)**q
p, q = Wild("p"), Wild("q")
r = d.match(z**p * log(z)**q)
if r:
p, q = [r.get(w, w) for w in [p, q]]
if q and q.is_number and p.is_number:
if q > 0:
if p > 0:
return S.Zero
else:
return -oo*i
else:
if p >= 0:
return S.Zero
else:
return -oo*i
if e.is_Add:
if e.is_polynomial() and not z0.is_unbounded:
return Add(*[limit(term, z, z0, dir) for term in e.args])
# this is a case like limit(x*y+x*z, z, 2) == x*y+2*x
# but we need to make sure, that the general gruntz() algorithm is
# executed for a case like "limit(sqrt(x+1)-sqrt(x),x,oo)==0"
unbounded = []
unbounded_result = []
unbounded_const = []
unknown = []
unknown_result = []
finite = []
zero = []
def _sift(term):
if z not in term.free_symbols:
if term.is_unbounded:
unbounded_const.append(term)
else:
finite.append(term)
else:
result = term.subs(z, z0)
bounded = result.is_bounded
if bounded is False or result is S.NaN:
unbounded.append(term)
if result != S.NaN:
# take result from direction given
result = limit(term, z, z0, dir)
unbounded_result.append(result)
elif bounded:
if result:
finite.append(result)
else:
zero.append(term)
else:
unknown.append(term)
unknown_result.append(result)
for term in e.args:
_sift(term)
bad = bool(unknown and unbounded)
if bad or len(unknown) > 1 or len(unbounded) > 1 and not zero:
uu = unknown + unbounded
# we won't be able to resolve this with unbounded
# terms, e.g. Sum(1/k, (k, 1, n)) - log(n) as n -> oo:
# since the Sum is unevaluated it's boundedness is
# unknown and the log(n) is oo so you get Sum - oo
# which is unsatisfactory. BUT...if there are both
# unknown and unbounded terms (condition 'bad') or
# there are multiple terms that are unknown, or
# there are multiple symbolic unbounded terms they may
# respond better if they are made into a rational
# function, so give them a chance to do so before
# reporting failure.
u = Add(*uu)
f = u.normal()
if f != u:
unknown = []
unbounded = []
unbounded_result = []
unknown_result = []
_sift(limit(f, z, z0, dir))
# We came in with a) unknown and unbounded terms or b) had multiple
# unknown terms
# At this point we've done one of 3 things.
# (1) We did nothing with f so we now report the error
# showing the troublesome terms which are now in uu. OR
# (2) We did something with f but the result came back as unknown.
# Normally this wouldn't be a problem,
# but we had either multiple terms that were troublesome (unk and
# unbounded or multiple unknown terms) so if we
# weren't able to resolve the boundedness by now, that indicates a
# problem so we report the error showing the troublesome terms which are
# now in uu.
if unknown:
if bad:
msg = 'unknown and unbounded terms present in %s'
elif unknown:
msg = 'multiple terms with unknown boundedness in %s'
raise NotImplementedError(msg % uu)
# OR
# (3) the troublesome terms have been identified as finite or unbounded
# and we proceed with the non-error code since the lists have been updated.
u = Add(*unknown_result)
if unbounded_result or unbounded_const:
unbounded.extend(zero)
inf_limit = Add(*(unbounded_result + unbounded_const))
if inf_limit is not S.NaN:
return inf_limit + u
if finite:
return Add(*finite) + limit(Add(*unbounded), z, z0, dir) + u
else:
return Add(*finite) + u
if e.is_Order:
args = e.args
return C.Order(limit(args[0], z, z0), *args[1:])
try:
r = gruntz(e, z, z0, dir)
if r is S.NaN:
raise PoleError()
except (PoleError, ValueError):
r = heuristics(e, z, z0, dir)
return r
def is_deriv_k(fa, fd, DE):
"""
Checks if Df/f is the derivative of an element of k(t).
a in k(t) is the derivative of an element of k(t) if there exists b in k(t)
such that a = Db. Either returns (ans, u), such that Df/f == Du, or None,
which means that Df/f is not the derivative of an element of k(t). ans is
a list of tuples such that Add(*[i*j for i, j in ans]) == u. This is useful
for seeing exactly which elements of k(t) produce u.
This function uses the structure theorem approach, which says that for any
f in K, Df/f is the derivative of a element of K if and only if there are ri
in QQ such that::
--- --- Dt
\ r * Dt + \ r * i Df
/ i i / i --- = --.
--- --- t f
i in L i in E i
K/C(x) K/C(x)
Where C = Const(K), L_K/C(x) = { i in {1, ..., n} such that t_i is
transcendental over C(x)(t_1, ..., t_i-1) and Dt_i = Da_i/a_i, for some a_i
in C(x)(t_1, ..., t_i-1)* } (i.e., the set of all indices of logarithmic
monomials of K over C(x)), and E_K/C(x) = { i in {1, ..., n} such that t_i
is transcendental over C(x)(t_1, ..., t_i-1) and Dt_i/t_i = Da_i, for some
a_i in C(x)(t_1, ..., t_i-1) } (i.e., the set of all indices of
hyperexponential monomials of K over C(x)). If K is an elementary extension
over C(x), then the cardinality of L_K/C(x) U E_K/C(x) is exactly the
transcendence degree of K over C(x). Furthermore, because Const_D(K) ==
Const_D(C(x)) == C, deg(Dt_i) == 1 when t_i is in E_K/C(x) and
deg(Dt_i) == 0 when t_i is in L_K/C(x), implying in particular that E_K/C(x)
and L_K/C(x) are disjoint.
The sets L_K/C(x) and E_K/C(x) must, by their nature, be computed
recursively using this same function. Therefore, it is required to pass
them as indices to D (or T). E_args are the arguments of the
hyperexponentials indexed by E_K (i.e., if i is in E_K, then T[i] ==
exp(E_args[i])). This is needed to compute the final answer u such that
Df/f == Du.
log(f) will be the same as u up to a additive constant. This is because
they will both behave the same as monomials. For example, both log(x) and
log(2*x) == log(x) + log(2) satisfy Dt == 1/x, because log(2) is constant.
Therefore, the term const is returned. const is such that
log(const) + f == u. This is calculated by dividing the arguments of one
logarithm from the other. Therefore, it is necessary to pass the arguments
of the logarithmic terms in L_args.
To handle the case where we are given Df/f, not f, use is_deriv_k_in_field().
"""
# Compute Df/f
dfa, dfd = fd * (fd * derivation(fa, DE) -
fa * derivation(fd, DE)), fd**2 * fa
dfa, dfd = dfa.cancel(dfd, include=True)
# Our assumption here is that each monomial is recursively transcendental
if len(DE.L_K) + len(DE.E_K) != len(DE.D) - 1:
if [i for i in DE.cases if i == 'tan'] or \
set([i for i in DE.cases if i == 'primitive']) - set(DE.L_K):
raise NotImplementedError(
"Real version of the structure "
"theorems with hypertangent support is not yet implemented.")
# TODO: What should really be done in this case?
raise NotImplementedError("Nonelementary extensions not supported "
"in the structure theorems.")
E_part = [DE.D[i].quo(Poly(DE.T[i], DE.T[i])).as_expr() for i in DE.E_K]
L_part = [DE.D[i].as_expr() for i in DE.L_K]
lhs = Matrix([E_part + L_part])
rhs = Matrix([dfa.as_expr() / dfd.as_expr()])
A, u = constant_system(lhs, rhs, DE)
if not all(derivation(i, DE, basic=True).is_zero for i in u) or not A:
# If the elements of u are not all constant
# Note: See comment in constant_system
# Also note: derivation(basic=True) calls cancel()
return None
else:
if not all(i.is_Rational for i in u):
raise NotImplementedError("Cannot work with non-rational "
"coefficients in this case.")
else:
terms = DE.E_args + [DE.T[i] for i in DE.L_K]
ans = list(zip(terms, u))
result = Add(*[Mul(i, j) for i, j in ans])
argterms = [DE.T[i] for i in DE.E_K] + DE.L_args
l = []
ld = []
for i, j in zip(argterms, u):
# We need to get around things like sqrt(x**2) != x
# and also sqrt(x**2 + 2*x + 1) != x + 1
# Issue 10798: i need not be a polynomial
i, d = i.as_numer_denom()
icoeff, iterms = sqf_list(i)
l.append(
Mul(*([Pow(icoeff, j)] +
[Pow(b, e * j) for b, e in iterms])))
dcoeff, dterms = sqf_list(d)
ld.append(
Mul(*([Pow(dcoeff, j)] +
[Pow(b, e * j) for b, e in dterms])))
const = cancel(fa.as_expr() / fd.as_expr() / Mul(*l) * Mul(*ld))
return (ans, result, const)
def f(rv):
if not (rv.is_Add or rv.is_Mul):
return rv
def gooda(a):
# bool to tell whether the leading ``a`` in ``a*log(x)``
# could appear as log(x**a)
return (a is not S.NegativeOne and # -1 *could* go, but we disallow
(a.is_real or force and a.is_real is not False))
def goodlog(l):
# bool to tell whether log ``l``'s argument can combine with others
a = l.args[0]
return a.is_positive or force and a.is_nonpositive is not False
other = []
logs = []
log1 = defaultdict(list)
for a in Add.make_args(rv):
if a.func is log and goodlog(a):
log1[()].append(([], a))
elif not a.is_Mul:
other.append(a)
else:
ot = []
co = []
lo = []
for ai in a.args:
if ai.is_Rational and ai < 0:
ot.append(S.NegativeOne)
co.append(-ai)
elif ai.func is log and goodlog(ai):
lo.append(ai)
elif gooda(ai):
co.append(ai)
else:
ot.append(ai)
if len(lo) > 1:
logs.append((ot, co, lo))
elif lo:
log1[tuple(ot)].append((co, lo[0]))
else:
other.append(a)
# if there is only one log at each coefficient and none have
# an exponent to place inside the log then there is nothing to do
if not logs and all(len(log1[k]) == 1 and log1[k][0] == [] for k in log1):
return rv
# collapse multi-logs as far as possible in a canonical way
# TODO: see if x*log(a)+x*log(a)*log(b) -> x*log(a)*(1+log(b))?
# -- in this case, it's unambiguous, but if it were were a log(c) in
# each term then it's arbitrary whether they are grouped by log(a) or
# by log(c). So for now, just leave this alone; it's probably better to
# let the user decide
for o, e, l in logs:
l = list(ordered(l))
e = log(l.pop(0).args[0]**Mul(*e))
while l:
li = l.pop(0)
e = log(li.args[0]**e)
c, l = Mul(*o), e
if l.func is log: # it should be, but check to be sure
log1[(c,)].append(([], l))
else:
other.append(c*l)
# logs that have the same coefficient can multiply
for k in list(log1.keys()):
log1[Mul(*k)] = log(logcombine(Mul(*[
l.args[0]**Mul(*c) for c, l in log1.pop(k)]),
force=force))
# logs that have oppositely signed coefficients can divide
for k in ordered(list(log1.keys())):
if not k in log1: # already popped as -k
continue
if -k in log1:
# figure out which has the minus sign; the one with
# more op counts should be the one
num, den = k, -k
if num.count_ops() > den.count_ops():
num, den = den, num
other.append(num*log(log1.pop(num).args[0]/log1.pop(den).args[0]))
else:
other.append(k*log1.pop(k))
return Add(*other)
def _doprint_loops(self, expr, assign_to=None):
# Here we print an expression that contains Indexed objects, they
# correspond to arrays in the generated code. The low-level implementation
# involves looping over array elements and possibly storing results in temporary
# variables or accumulate it in the assign_to object.
if self._settings.get('contract', True):
from sympy.tensor import get_contraction_structure
# Setup loops over non-dummy indices -- all terms need these
indices = self._get_expression_indices(expr, assign_to)
# Setup loops over dummy indices -- each term needs separate treatment
dummies = get_contraction_structure(expr)
else:
indices = []
dummies = {None: (expr,)}
openloop, closeloop = self._get_loop_opening_ending(indices)
# terms with no summations first
if None in dummies:
text = StrPrinter.doprint(self, Add(*dummies[None]))
else:
# If all terms have summations we must initialize array to Zero
text = StrPrinter.doprint(self, 0)
# skip redundant assignments (where lhs == rhs)
lhs_printed = self._print(assign_to)
lines = []
if text != lhs_printed:
lines.extend(openloop)
if assign_to is not None:
text = self._get_statement("%s = %s" % (lhs_printed, text))
lines.append(text)
lines.extend(closeloop)
# then terms with summations
for d in dummies:
if isinstance(d, tuple):
indices = self._sort_optimized(d, expr)
openloop_d, closeloop_d = self._get_loop_opening_ending(
indices)
for term in dummies[d]:
if term in dummies and not ([list(f.keys()) for f in dummies[term]]
== [[None] for f in dummies[term]]):
# If one factor in the term has it's own internal
# contractions, those must be computed first.
# (temporary variables?)
raise NotImplementedError(
"FIXME: no support for contractions in factor yet")
else:
# We need the lhs expression as an accumulator for
# the loops, i.e
#
# for (int d=0; d < dim; d++){
# lhs[] = lhs[] + term[][d]
# } ^.................. the accumulator
#
# We check if the expression already contains the
# lhs, and raise an exception if it does, as that
# syntax is currently undefined. FIXME: What would be
# a good interpretation?
if assign_to is None:
raise AssignmentError(
"need assignment variable for loops")
if term.has(assign_to):
raise ValueError("FIXME: lhs present in rhs,\
this is undefined in CodePrinter")
lines.extend(openloop)
lines.extend(openloop_d)
text = "%s = %s" % (lhs_printed, StrPrinter.doprint(
self, assign_to + term))
lines.append(self._get_statement(text))
lines.extend(closeloop_d)
lines.extend(closeloop)
return "\n".join(lines)
def test_zero_matrix_add():
assert Add(ZeroMatrix(2, 2), ZeroMatrix(2, 2)) == ZeroMatrix(2, 2)
def _gelatize(a, degrees=None, evaluate=False, verbose=False):
if isinstance(a, BilinearForm) and not(isinstance(a, BilinearAtomicForm)):
expr = tensorize(a)
if verbose:
print('> tensorized = ', expr)
else:
expr = a
if isinstance(expr, Add):
args = [_gelatize(i, degrees=degrees, evaluate=evaluate) for i in expr.args]
return Add(*args)
elif isinstance(expr, Mul):
coeffs = [i for i in expr.args if isinstance(i, _coeffs_registery)]
vectors = [i for i in expr.args if not(i in coeffs)]
i = S.One
if coeffs:
i = Mul(*coeffs)
j = S.One
if vectors:
args = [_gelatize(i, degrees=degrees, evaluate=evaluate) for i in vectors]
j = Mul(*args)
return Mul(i, j)
elif isinstance(expr, (Matrix, ImmutableDenseMatrix)):
n_rows, n_cols = expr.shape
lines = []
for i in range(0, n_rows):
line = []
for j in range(0, n_cols):
eij = _gelatize(expr[i,j], degrees=degrees, evaluate=evaluate)
line.append(eij)
lines.append(line)
return Matrix(lines)
elif isinstance(expr, BilinearAtomicForm):
coord = expr.trial_spaces[0].coordinates
# ... construct the fourier variable and the number of elements
t_name = 't{}'.format(coord)
n_name = 'n{}'.format(coord)
t = Symbol(t_name)
n = Symbol(n_name, integer=True)
# ...
# ...
_coordinates = ['x', 'y', 'z']
index = _coordinates.index(str(coord.name))
# ...
if evaluate and ( degrees is None ):
raise ValueError('> degrees must be provided')
# ... get the degree
if not( degrees is None ):
if not isinstance(degrees, (tuple, list, Tuple)):
degrees = [degrees]
p = degrees[index]
else:
p_name = 'p{}'.format(coord)
p = Symbol(p_name, integer=True)
# ...
if isinstance(expr, MassForm):
symbol = Mass(p, t, evaluate=evaluate)
return symbol / n
elif isinstance(expr, StiffnessForm):
symbol = Stiffness(p, t, evaluate=evaluate)
return symbol * n
elif isinstance(expr, AdvectionForm):
symbol = sympy_I * Advection(p, t, evaluate=evaluate)
return symbol
elif isinstance(expr, AdvectionTForm):
symbol = - sympy_I * Advection(p, t, evaluate=evaluate)
return symbol
else:
raise NotImplementedError('TODO')
return expr
def test_matrix_add_with_scalar():
raises(TypeError, lambda: Add(0, ZeroMatrix(2, 2)))
def refine_Pow(expr, assumptions):
"""
Handler for instances of Pow.
>>> from sympy import Symbol, Q
>>> from sympy.assumptions.refine import refine_Pow
>>> from sympy.abc import x,y,z
>>> refine_Pow((-1)**x, Q.real(x))
>>> refine_Pow((-1)**x, Q.even(x))
1
>>> refine_Pow((-1)**x, Q.odd(x))
-1
For powers of -1, even parts of the exponent can be simplified:
>>> refine_Pow((-1)**(x+y), Q.even(x))
(-1)**y
>>> refine_Pow((-1)**(x+y+z), Q.odd(x) & Q.odd(z))
(-1)**y
>>> refine_Pow((-1)**(x+y+2), Q.odd(x))
(-1)**(y + 1)
>>> refine_Pow((-1)**(x+3), True)
(-1)**(x + 1)
"""
from sympy.core import Pow, Rational
from sympy.functions.elementary.complexes import Abs
from sympy.functions import sign
if isinstance(expr.base, Abs):
if ask(Q.real(expr.base.args[0]), assumptions) and \
ask(Q.even(expr.exp), assumptions):
return expr.base.args[0]**expr.exp
if ask(Q.real(expr.base), assumptions):
if expr.base.is_number:
if ask(Q.even(expr.exp), assumptions):
return abs(expr.base)**expr.exp
if ask(Q.odd(expr.exp), assumptions):
return sign(expr.base) * abs(expr.base)**expr.exp
if isinstance(expr.exp, Rational):
if type(expr.base) is Pow:
return abs(expr.base.base)**(expr.base.exp * expr.exp)
if expr.base is S.NegativeOne:
if expr.exp.is_Add:
old = expr
# For powers of (-1) we can remove
# - even terms
# - pairs of odd terms
# - a single odd term + 1
# - A numerical constant N can be replaced with mod(N,2)
coeff, terms = expr.exp.as_coeff_add()
terms = set(terms)
even_terms = set([])
odd_terms = set([])
initial_number_of_terms = len(terms)
for t in terms:
if ask(Q.even(t), assumptions):
even_terms.add(t)
elif ask(Q.odd(t), assumptions):
odd_terms.add(t)
terms -= even_terms
if len(odd_terms) % 2:
terms -= odd_terms
new_coeff = (coeff + S.One) % 2
else:
terms -= odd_terms
new_coeff = coeff % 2
if new_coeff != coeff or len(terms) < initial_number_of_terms:
terms.add(new_coeff)
expr = expr.base**(Add(*terms))
# Handle (-1)**((-1)**n/2 + m/2)
e2 = 2 * expr.exp
if ask(Q.even(e2), assumptions):
if e2.could_extract_minus_sign():
e2 *= expr.base
if e2.is_Add:
i, p = e2.as_two_terms()
if p.is_Pow and p.base is S.NegativeOne:
if ask(Q.integer(p.exp), assumptions):
i = (i + 1) / 2
if ask(Q.even(i), assumptions):
return expr.base**p.exp
elif ask(Q.odd(i), assumptions):
return expr.base**(p.exp + 1)
else:
return expr.base**(p.exp + i)
if old != expr:
return expr
def test_OneMatrix_doit():
Unn = OneMatrix(Add(n, n, evaluate=False), n)
assert isinstance(Unn.rows, Add)
assert Unn.doit() == OneMatrix(2 * n, n)
assert isinstance(Unn.doit().rows, Mul)
