def _solve_as_poly(f, symbol, solveset_solver, invert_func):
"""
Solve the equation using polynomial techniques if it already is a
polynomial equation or, with a change of variables, can be made so.
"""
result = None
if f.is_polynomial(symbol):
solns = roots(f, symbol, cubics=True, quartics=True,
quintics=True, domain='EX')
num_roots = sum(solns.values())
if degree(f, symbol) <= num_roots:
result = FiniteSet(*solns.keys())
else:
poly = Poly(f, symbol)
solns = poly.all_roots()
if poly.degree() <= len(solns):
result = FiniteSet(*solns)
else:
result = ConditionSet(symbol, Eq(f, 0), S.Complexes)
else:
poly = Poly(f)
if poly is None:
result = ConditionSet(symbol, Eq(f, 0), S.Complexes)
gens = [g for g in poly.gens if g.has(symbol)]
if len(gens) == 1:
poly = Poly(poly, gens[0])
gen = poly.gen
deg = poly.degree()
poly = Poly(poly.as_expr(), poly.gen, composite=True)
poly_solns = FiniteSet(*roots(poly, cubics=True, quartics=True,
quintics=True).keys())
if len(poly_solns) < deg:
result = ConditionSet(symbol, Eq(f, 0), S.Complexes)
if gen != symbol:
y = Dummy('y')
lhs, rhs_s = invert_func(gen, y, symbol)
if lhs is symbol:
result = Union(*[rhs_s.subs(y, s) for s in poly_solns])
else:
result = ConditionSet(symbol, Eq(f, 0), S.Complexes)
else:
result = ConditionSet(symbol, Eq(f, 0), S.Complexes)
if result is not None:
if isinstance(result, FiniteSet):
# this is to simplify solutions like -sqrt(-I) to sqrt(2)/2
# - sqrt(2)*I/2. We are not expanding for solution with free
# variables because that makes the solution more complicated. For
# example expand_complex(a) returns re(a) + I*im(a)
if all([s.free_symbols == set() and not isinstance(s, RootOf)
for s in result]):
s = Dummy('s')
result = imageset(Lambda(s, expand_complex(s)), result)
return result
else:
return ConditionSet(symbol, Eq(f, 0), S.Complexes)
def _eval_conjugate(self):
from sympy.functions.elementary.complexes import conjugate as c
i, p = self.exp.is_integer, self.base.is_positive
if i:
return c(self.base)**self.exp
if p:
return self.base**c(self.exp)
if i is False and p is False:
expanded = expand_complex(self)
if expanded != self:
return c(expanded)
def _eval_adjoint(self):
from sympy.functions.elementary.complexes import adjoint
i, p = self.exp.is_integer, self.base.is_positive
if i:
return adjoint(self.base)**self.exp
if p:
return self.base**adjoint(self.exp)
if i is False and p is False:
expanded = expand_complex(self)
if expanded != self:
return adjoint(expanded)
def _eval_transpose(self):
from sympy.functions.elementary.complexes import transpose
i, p = self.exp.is_integer, self.base.is_complex
if p:
return self.base**self.exp
if i:
return transpose(self.base)**self.exp
if i is False and p is False:
expanded = expand_complex(self)
if expanded != self:
return transpose(expanded)
def _intersect(self, other):
from sympy.solvers.diophantine import diophantine
if self.base_set is S.Integers:
g = None
if isinstance(other, ImageSet) and other.base_set is S.Integers:
g = other.lamda.expr
m = other.lamda.variables[0]
elif other is S.Integers:
m = g = Dummy('x')
if g is not None:
f = self.lamda.expr
n = self.lamda.variables[0]
# Diophantine sorts the solutions according to the alphabetic
# order of the variable names, since the result should not depend
# on the variable name, they are replaced by the dummy variables
# below
a, b = Dummy('a'), Dummy('b')
f, g = f.subs(n, a), g.subs(m, b)
solns_set = diophantine(f - g)
if solns_set == set():
return EmptySet()
solns = list(diophantine(f - g))
if len(solns) != 1:
return
# since 'a' < 'b', select soln for n
nsol = solns[0][0]
t = nsol.free_symbols.pop()
return imageset(Lambda(n, f.subs(a, nsol.subs(t, n))), S.Integers)
if other == S.Reals:
from sympy.solvers.solveset import solveset_real
from sympy.core.function import expand_complex
if len(self.lamda.variables) > 1:
return None
f = self.lamda.expr
n = self.lamda.variables[0]
n_ = Dummy(n.name, real=True)
f_ = f.subs(n, n_)
re, im = f_.as_real_imag()
im = expand_complex(im)
return imageset(Lambda(n_, re),
self.base_set.intersect(
solveset_real(im, n_)))
def _intersect(self, other):
from sympy import Dummy
from sympy.solvers.diophantine import diophantine
from sympy.sets.sets import imageset
if self.base_set is S.Integers:
if isinstance(other, ImageSet) and other.base_set is S.Integers:
f, g = self.lamda.expr, other.lamda.expr
n, m = self.lamda.variables[0], other.lamda.variables[0]
# Diophantine sorts the solutions according to the alphabetic
# order of the variable names, since the result should not depend
# on the variable name, they are replaced by the dummy variables
# below
a, b = Dummy("a"), Dummy("b")
f, g = f.subs(n, a), g.subs(m, b)
solns_set = diophantine(f - g)
if solns_set == set():
return EmptySet()
solns = list(diophantine(f - g))
if len(solns) == 1:
t = list(solns[0][0].free_symbols)[0]
else:
return None
# since 'a' < 'b'
return imageset(Lambda(t, f.subs(a, solns[0][0])), S.Integers)
if other == S.Reals:
from sympy.solvers.solveset import solveset_real
from sympy.core.function import expand_complex
if len(self.lamda.variables) > 1:
return None
f = self.lamda.expr
n = self.lamda.variables[0]
n_ = Dummy(n.name, real=True)
f_ = f.subs(n, n_)
re, im = f_.as_real_imag()
im = expand_complex(im)
return imageset(Lambda(n_, re), self.base_set.intersect(solveset_real(im, n_)))
def _intersect(self, other):
from sympy import Dummy
from sympy.solvers.diophantine import diophantine
from sympy.sets.sets import imageset
if self.base_set is S.Integers:
if isinstance(other, ImageSet) and other.base_set is S.Integers:
f, g = self.lamda.expr, other.lamda.expr
n, m = self.lamda.variables[0], other.lamda.variables[0]
# Diophantine sorts the solutions according to the alphabetic
# order of the variable names, since the result should not depend
# on the variable name, they are replaced by the dummy variables
# below
a, b = Dummy('a'), Dummy('b')
f, g = f.subs(n, a), g.subs(m, b)
solns_set = diophantine(f - g)
if solns_set == set():
return EmptySet()
solns = list(diophantine(f - g))
if len(solns) == 1:
t = list(solns[0][0].free_symbols)[0]
else:
return None
# since 'a' < 'b'
return imageset(Lambda(t, f.subs(a, solns[0][0])), S.Integers)
if other == S.Reals:
from sympy.solvers.solveset import solveset_real
from sympy.core.function import expand_complex
if len(self.lamda.variables) > 1:
return None
f = self.lamda.expr
n = self.lamda.variables[0]
n_ = Dummy(n.name, real=True)
f_ = f.subs(n, n_)
re, im = f_.as_real_imag()
im = expand_complex(im)
return imageset(Lambda(n_, re),
self.base_set.intersect(solveset_real(im, n_)))
def _eval_nseries(self, x, n, logx, cdir=0):
# NOTE Please see the comment at the beginning of this file, labelled
# IMPORTANT.
from sympy.functions.elementary.complexes import sign
from sympy.functions.elementary.integers import ceiling
from sympy.series.limits import limit
from sympy.series.order import Order
from sympy.simplify.powsimp import powsimp
arg = self.exp
arg_series = arg._eval_nseries(x, n=n, logx=logx)
if arg_series.is_Order:
return 1 + arg_series
arg0 = limit(arg_series.removeO(), x, 0)
if arg0 is S.NegativeInfinity:
return Order(x**n, x)
if arg0 is S.Infinity:
return self
# checking for indecisiveness/ sign terms in arg0
if any(isinstance(arg, (sign, ImaginaryUnit)) for arg in arg0.args):
return self
t = Dummy("t")
nterms = n
try:
cf = Order(arg.as_leading_term(x, logx=logx), x).getn()
except (NotImplementedError, PoleError):
cf = 0
if cf and cf > 0:
nterms = ceiling(n/cf)
exp_series = exp(t)._taylor(t, nterms)
r = exp(arg0)*exp_series.subs(t, arg_series - arg0)
if cf and cf > 1:
r += Order((arg_series - arg0)**n, x)/x**((cf-1)*n)
else:
r += Order((arg_series - arg0)**n, x)
r = r.expand()
r = powsimp(r, deep=True, combine='exp')
# powsimp may introduce unexpanded (-1)**Rational; see PR #17201
simplerat = lambda x: x.is_Rational and x.q in [3, 4, 6]
w = Wild('w', properties=[simplerat])
r = r.replace(S.NegativeOne**w, expand_complex(S.NegativeOne**w))
return r
def _solve_as_poly(f, symbol, domain=S.Complexes):
"""
Solve the equation using polynomial techniques if it already is a
polynomial equation or, with a change of variables, can be made so.
"""
result = None
if f.is_polynomial(symbol):
solns = roots(f, symbol, cubics=True, quartics=True,
quintics=True, domain='EX')
num_roots = sum(solns.values())
if degree(f, symbol) <= num_roots:
result = FiniteSet(*solns.keys())
else:
poly = Poly(f, symbol)
solns = poly.all_roots()
if poly.degree() <= len(solns):
result = FiniteSet(*solns)
else:
result = ConditionSet(symbol, Eq(f, 0), domain)
else:
poly = Poly(f)
if poly is None:
result = ConditionSet(symbol, Eq(f, 0), domain)
gens = [g for g in poly.gens if g.has(symbol)]
if len(gens) == 1:
poly = Poly(poly, gens[0])
gen = poly.gen
deg = poly.degree()
poly = Poly(poly.as_expr(), poly.gen, composite=True)
poly_solns = FiniteSet(*roots(poly, cubics=True, quartics=True,
quintics=True).keys())
if len(poly_solns) < deg:
result = ConditionSet(symbol, Eq(f, 0), domain)
if gen != symbol:
y = Dummy('y')
inverter = invert_real if domain.is_subset(S.Reals) else invert_complex
lhs, rhs_s = inverter(gen, y, symbol)
if lhs == symbol:
result = Union(*[rhs_s.subs(y, s) for s in poly_solns])
else:
result = ConditionSet(symbol, Eq(f, 0), domain)
else:
result = ConditionSet(symbol, Eq(f, 0), domain)
if result is not None:
if isinstance(result, FiniteSet):
# this is to simplify solutions like -sqrt(-I) to sqrt(2)/2
# - sqrt(2)*I/2. We are not expanding for solution with free
# variables because that makes the solution more complicated. For
# example expand_complex(a) returns re(a) + I*im(a)
if all([s.free_symbols == set() and not isinstance(s, RootOf)
for s in result]):
s = Dummy('s')
result = imageset(Lambda(s, expand_complex(s)), result)
if isinstance(result, FiniteSet):
result = result.intersection(domain)
return result
else:
return ConditionSet(symbol, Eq(f, 0), domain)
def intersection_sets(self, other):
from sympy.solvers.diophantine import diophantine
# Only handle the straight-forward univariate case
if (len(self.lamda.variables) > 1
or self.lamda.signature != self.lamda.variables):
return None
base_set = self.base_sets[0]
# Intersection between ImageSets with Integers as base set
# For {f(n) : n in Integers} & {g(m) : m in Integers} we solve the
# diophantine equations f(n)=g(m).
# If the solutions for n are {h(t) : t in Integers} then we return
# {f(h(t)) : t in integers}.
if base_set is S.Integers:
gm = None
if isinstance(other, ImageSet) and other.base_sets == (S.Integers, ):
gm = other.lamda.expr
m = other.lamda.variables[0]
elif other is S.Integers:
m = gm = Dummy('x')
if gm is not None:
fn = self.lamda.expr
n = self.lamda.variables[0]
solns = list(diophantine(fn - gm, syms=(n, m)))
if len(solns) == 0:
return EmptySet
elif len(solns) != 1:
return
else:
soln, solm = solns[0]
(t, ) = soln.free_symbols
expr = fn.subs(n, soln.subs(t, n))
return imageset(Lambda(n, expr), S.Integers)
if other == S.Reals:
from sympy.solvers.solveset import solveset_real
from sympy.core.function import expand_complex
f = self.lamda.expr
n = self.lamda.variables[0]
n_ = Dummy(n.name, real=True)
f_ = f.subs(n, n_)
re, im = f_.as_real_imag()
im = expand_complex(im)
re = re.subs(n_, n)
im = im.subs(n_, n)
ifree = im.free_symbols
lam = Lambda(n, re)
if not im:
# allow re-evaluation
# of self in this case to make
# the result canonical
pass
elif im.is_zero is False:
return S.EmptySet
elif ifree != {n}:
return None
else:
# univarite imaginary part in same variable
base_set = base_set.intersect(solveset_real(im, n))
return imageset(lam, base_set)
elif isinstance(other, Interval):
from sympy.solvers.solveset import (invert_real, invert_complex,
solveset)
f = self.lamda.expr
n = self.lamda.variables[0]
new_inf, new_sup = None, None
new_lopen, new_ropen = other.left_open, other.right_open
if f.is_real:
inverter = invert_real
else:
inverter = invert_complex
g1, h1 = inverter(f, other.inf, n)
g2, h2 = inverter(f, other.sup, n)
if all(isinstance(i, FiniteSet) for i in (h1, h2)):
if g1 == n:
if len(h1) == 1:
new_inf = h1.args[0]
if g2 == n:
if len(h2) == 1:
new_sup = h2.args[0]
# TODO: Design a technique to handle multiple-inverse
# functions
# Any of the new boundary values cannot be determined
if any(i is None for i in (new_sup, new_inf)):
return
range_set = S.EmptySet
if all(i.is_real for i in (new_sup, new_inf)):
# this assumes continuity of underlying function
# however fixes the case when it is decreasing
if new_inf > new_sup:
new_inf, new_sup = new_sup, new_inf
new_interval = Interval(new_inf, new_sup, new_lopen, new_ropen)
range_set = base_set.intersect(new_interval)
else:
if other.is_subset(S.Reals):
solutions = solveset(f, n, S.Reals)
if not isinstance(range_set, (ImageSet, ConditionSet)):
range_set = solutions.intersect(other)
else:
return
if range_set is S.EmptySet:
return S.EmptySet
elif isinstance(range_set,
Range) and range_set.size is not S.Infinity:
range_set = FiniteSet(*list(range_set))
if range_set is not None:
return imageset(Lambda(n, f), range_set)
return
else:
return
def _intersect(self, other):
from sympy.solvers.diophantine import diophantine
if self.base_set is S.Integers:
g = None
if isinstance(other, ImageSet) and other.base_set is S.Integers:
g = other.lamda.expr
m = other.lamda.variables[0]
elif other is S.Integers:
m = g = Dummy('x')
if g is not None:
f = self.lamda.expr
n = self.lamda.variables[0]
# Diophantine sorts the solutions according to the alphabetic
# order of the variable names, since the result should not depend
# on the variable name, they are replaced by the dummy variables
# below
a, b = Dummy('a'), Dummy('b')
f, g = f.subs(n, a), g.subs(m, b)
solns_set = diophantine(f - g)
if solns_set == set():
return EmptySet()
solns = list(diophantine(f - g))
if len(solns) != 1:
return
# since 'a' < 'b', select soln for n
nsol = solns[0][0]
t = nsol.free_symbols.pop()
return imageset(Lambda(n, f.subs(a, nsol.subs(t, n))),
S.Integers)
if other == S.Reals:
from sympy.solvers.solveset import solveset_real
from sympy.core.function import expand_complex
if len(self.lamda.variables) > 1:
return None
f = self.lamda.expr
n = self.lamda.variables[0]
n_ = Dummy(n.name, real=True)
f_ = f.subs(n, n_)
re, im = f_.as_real_imag()
im = expand_complex(im)
return imageset(Lambda(n_, re),
self.base_set.intersect(solveset_real(im, n_)))
elif isinstance(other, Interval):
from sympy.solvers.solveset import (invert_real, invert_complex,
solveset)
f = self.lamda.expr
n = self.lamda.variables[0]
base_set = self.base_set
new_inf, new_sup = None, None
new_lopen, new_ropen = other.left_open, other.right_open
if f.is_real:
inverter = invert_real
else:
inverter = invert_complex
g1, h1 = inverter(f, other.inf, n)
g2, h2 = inverter(f, other.sup, n)
if all(isinstance(i, FiniteSet) for i in (h1, h2)):
if g1 == n:
if len(h1) == 1:
new_inf = h1.args[0]
if g2 == n:
if len(h2) == 1:
new_sup = h2.args[0]
# TODO: Design a technique to handle multiple-inverse
# functions
# Any of the new boundary values cannot be determined
if any(i is None for i in (new_sup, new_inf)):
return
range_set = S.EmptySet
if all(i.is_real for i in (new_sup, new_inf)):
# this assumes continuity of underlying function
# however fixes the case when it is decreasing
if new_inf > new_sup:
new_inf, new_sup = new_sup, new_inf
new_interval = Interval(new_inf, new_sup, new_lopen,
new_ropen)
range_set = base_set._intersect(new_interval)
else:
if other.is_subset(S.Reals):
solutions = solveset(f, n, S.Reals)
if not isinstance(range_set, (ImageSet, ConditionSet)):
range_set = solutions._intersect(other)
else:
return
if range_set is S.EmptySet:
return S.EmptySet
elif isinstance(range_set,
Range) and range_set.size is not S.Infinity:
range_set = FiniteSet(*list(range_set))
if range_set is not None:
return imageset(Lambda(n, f), range_set)
return
else:
return
def intersection_sets(self, other):
from sympy.solvers.diophantine import diophantine
# Only handle the straight-forward univariate case
if (len(self.lamda.variables) > 1
or self.lamda.signature != self.lamda.variables):
return None
base_set = self.base_sets[0]
if base_set is S.Integers:
g = None
if isinstance(other, ImageSet) and other.base_sets == (S.Integers, ):
g = other.lamda.expr
m = other.lamda.variables[0]
elif other is S.Integers:
m = g = Dummy('x')
if g is not None:
f = self.lamda.expr
n = self.lamda.variables[0]
# Diophantine sorts the solutions according to the alphabetic
# order of the variable names, since the result should not depend
# on the variable name, they are replaced by the dummy variables
# below
a, b = Dummy('a'), Dummy('b')
fa, ga = f.subs(n, a), g.subs(m, b)
solns = list(diophantine(fa - ga))
if not solns:
return EmptySet()
if len(solns) != 1:
return
nsol = solns[0][0] # since 'a' < 'b', nsol is first
t = nsol.free_symbols.pop() # diophantine supplied symbol
nsol = nsol.subs(t, n)
if nsol != n:
# if nsol == n and we know were are working with
# a base_set of Integers then this was an unevaluated
# ImageSet representation of Integers, otherwise
# it is a new ImageSet intersection with a subset
# of integers
nsol = f.subs(n, nsol)
return imageset(Lambda(n, nsol), S.Integers)
if other == S.Reals:
from sympy.solvers.solveset import solveset_real
from sympy.core.function import expand_complex
f = self.lamda.expr
n = self.lamda.variables[0]
n_ = Dummy(n.name, real=True)
f_ = f.subs(n, n_)
re, im = f_.as_real_imag()
im = expand_complex(im)
re = re.subs(n_, n)
im = im.subs(n_, n)
ifree = im.free_symbols
lam = Lambda(n, re)
if not im:
# allow re-evaluation
# of self in this case to make
# the result canonical
pass
elif im.is_zero is False:
return S.EmptySet
elif ifree != {n}:
return None
else:
# univarite imaginary part in same variable
base_set = base_set.intersect(solveset_real(im, n))
return imageset(lam, base_set)
elif isinstance(other, Interval):
from sympy.solvers.solveset import (invert_real, invert_complex,
solveset)
f = self.lamda.expr
n = self.lamda.variables[0]
new_inf, new_sup = None, None
new_lopen, new_ropen = other.left_open, other.right_open
if f.is_real:
inverter = invert_real
else:
inverter = invert_complex
g1, h1 = inverter(f, other.inf, n)
g2, h2 = inverter(f, other.sup, n)
if all(isinstance(i, FiniteSet) for i in (h1, h2)):
if g1 == n:
if len(h1) == 1:
new_inf = h1.args[0]
if g2 == n:
if len(h2) == 1:
new_sup = h2.args[0]
# TODO: Design a technique to handle multiple-inverse
# functions
# Any of the new boundary values cannot be determined
if any(i is None for i in (new_sup, new_inf)):
return
range_set = S.EmptySet
if all(i.is_real for i in (new_sup, new_inf)):
# this assumes continuity of underlying function
# however fixes the case when it is decreasing
if new_inf > new_sup:
new_inf, new_sup = new_sup, new_inf
new_interval = Interval(new_inf, new_sup, new_lopen, new_ropen)
range_set = base_set.intersect(new_interval)
else:
if other.is_subset(S.Reals):
solutions = solveset(f, n, S.Reals)
if not isinstance(range_set, (ImageSet, ConditionSet)):
range_set = solutions.intersect(other)
else:
return
if range_set is S.EmptySet:
return S.EmptySet
elif isinstance(range_set,
Range) and range_set.size is not S.Infinity:
range_set = FiniteSet(*list(range_set))
if range_set is not None:
return imageset(Lambda(n, f), range_set)
return
else:
return
def test_issue_4124():
from sympy.core.numbers import oo
assert expand_complex(I * oo) == oo * I
def _intersect(self, other):
from sympy.solvers.diophantine import diophantine
if self.base_set is S.Integers:
g = None
if isinstance(other, ImageSet) and other.base_set is S.Integers:
g = other.lamda.expr
m = other.lamda.variables[0]
elif other is S.Integers:
m = g = Dummy('x')
if g is not None:
f = self.lamda.expr
n = self.lamda.variables[0]
# Diophantine sorts the solutions according to the alphabetic
# order of the variable names, since the result should not depend
# on the variable name, they are replaced by the dummy variables
# below
a, b = Dummy('a'), Dummy('b')
f, g = f.subs(n, a), g.subs(m, b)
solns_set = diophantine(f - g)
if solns_set == set():
return EmptySet()
solns = list(diophantine(f - g))
if len(solns) != 1:
return
# since 'a' < 'b', select soln for n
nsol = solns[0][0]
t = nsol.free_symbols.pop()
return imageset(Lambda(n, f.subs(a, nsol.subs(t, n))), S.Integers)
if other == S.Reals:
from sympy.solvers.solveset import solveset_real
from sympy.core.function import expand_complex
if len(self.lamda.variables) > 1:
return None
f = self.lamda.expr
n = self.lamda.variables[0]
n_ = Dummy(n.name, real=True)
f_ = f.subs(n, n_)
re, im = f_.as_real_imag()
im = expand_complex(im)
return imageset(Lambda(n_, re),
self.base_set.intersect(
solveset_real(im, n_)))
elif isinstance(other, Interval):
from sympy.solvers.solveset import (invert_real, invert_complex,
solveset)
f = self.lamda.expr
n = self.lamda.variables[0]
base_set = self.base_set
new_inf, new_sup = None, None
new_lopen, new_ropen = other.left_open, other.right_open
if f.is_real:
inverter = invert_real
else:
inverter = invert_complex
g1, h1 = inverter(f, other.inf, n)
g2, h2 = inverter(f, other.sup, n)
if all(isinstance(i, FiniteSet) for i in (h1, h2)):
if g1 == n:
if len(h1) == 1:
new_inf = h1.args[0]
if g2 == n:
if len(h2) == 1:
new_sup = h2.args[0]
# TODO: Design a technique to handle multiple-inverse
# functions
# Any of the new boundary values cannot be determined
if any(i is None for i in (new_sup, new_inf)):
return
range_set = S.EmptySet
if all(i.is_real for i in (new_sup, new_inf)):
new_interval = Interval(new_inf, new_sup, new_lopen, new_ropen)
range_set = base_set._intersect(new_interval)
else:
if other.is_subset(S.Reals):
solutions = solveset(f, n, S.Reals)
if not isinstance(range_set, (ImageSet, ConditionSet)):
range_set = solutions._intersect(other)
else:
return
if range_set is S.EmptySet:
return S.EmptySet
elif isinstance(range_set, Range) and range_set.size is not S.Infinity:
range_set = FiniteSet(*list(range_set))
if range_set is not None:
return imageset(Lambda(n, f), range_set)
return
else:
return
def _solve_as_poly(f, symbol, solveset_solver, invert_func):
"""
Solve the equation using polynomial techniques if it already is a
polynomial equation or, with a change of variables, can be made so.
"""
result = None
if f.is_polynomial(symbol):
solns = roots(f,
symbol,
cubics=True,
quartics=True,
quintics=True,
domain='EX')
num_roots = sum(solns.values())
if degree(f, symbol) <= num_roots:
result = FiniteSet(*solns.keys())
else:
poly = Poly(f, symbol)
solns = poly.all_roots()
if poly.degree() <= len(solns):
result = FiniteSet(*solns)
else:
raise NotImplementedError("Couldn't find all roots "
"of the equation %s" % f)
else:
poly = Poly(f)
if poly is None:
raise NotImplementedError("Could not convert %s to Poly" % f)
gens = [g for g in poly.gens if g.has(symbol)]
if len(gens) == 1:
poly = Poly(poly, gens[0])
gen = poly.gen
deg = poly.degree()
poly = Poly(poly.as_expr(), poly.gen, composite=True)
poly_solns = FiniteSet(
*roots(poly, cubics=True, quartics=True, quintics=True).keys())
if len(poly_solns) < deg:
raise NotImplementedError("Couldn't find all the roots of "
"the equation %s" % f)
if gen != symbol:
y = Dummy('y')
lhs, rhs_s = invert_func(gen, y, symbol)
if lhs is symbol:
result = Union(*[rhs_s.subs(y, s) for s in poly_solns])
else:
raise NotImplementedError("inversion of %s not handled" %
gen)
else:
raise NotImplementedError("multiple generators not handled"
" by solveset")
if result is not None:
if isinstance(result, FiniteSet):
# this is to simplify solutions like -sqrt(-I) to sqrt(2)/2
# - sqrt(2)*I/2. We are not expanding for solution with free
# variables because that makes the solution more complicated. For
# example expand_complex(a) returns re(a) + I*im(a)
if all([
s.free_symbols == set() and not isinstance(s, RootOf)
for s in result
]):
s = Dummy('s')
result = imageset(Lambda(s, expand_complex(s)), result)
return result
else:
raise NotImplementedError
def intersection_sets(self, other): # noqa:F811
from sympy.solvers.diophantine import diophantine
# Only handle the straight-forward univariate case
if (len(self.lamda.variables) > 1
or self.lamda.signature != self.lamda.variables):
return None
base_set = self.base_sets[0]
# Intersection between ImageSets with Integers as base set
# For {f(n) : n in Integers} & {g(m) : m in Integers} we solve the
# diophantine equations f(n)=g(m).
# If the solutions for n are {h(t) : t in Integers} then we return
# {f(h(t)) : t in integers}.
# If the solutions for n are {n_1, n_2, ..., n_k} then we return
# {f(n_i) : 1 <= i <= k}.
if base_set is S.Integers:
gm = None
if isinstance(other, ImageSet) and other.base_sets == (S.Integers, ):
gm = other.lamda.expr
var = other.lamda.variables[0]
# Symbol of second ImageSet lambda must be distinct from first
m = Dummy('m')
gm = gm.subs(var, m)
elif other is S.Integers:
m = gm = Dummy('m')
if gm is not None:
fn = self.lamda.expr
n = self.lamda.variables[0]
try:
solns = list(diophantine(fn - gm, syms=(n, m), permute=True))
except (TypeError, NotImplementedError):
# TypeError if equation not polynomial with rational coeff.
# NotImplementedError if correct format but no solver.
return
# 3 cases are possible for solns:
# - empty set,
# - one or more parametric (infinite) solutions,
# - a finite number of (non-parametric) solution couples.
# Among those, there is one type of solution set that is
# not helpful here: multiple parametric solutions.
if len(solns) == 0:
return EmptySet
elif any(not isinstance(s, int) and s.free_symbols
for tupl in solns for s in tupl):
if len(solns) == 1:
soln, solm = solns[0]
(t, ) = soln.free_symbols
expr = fn.subs(n, soln.subs(t, n)).expand()
return imageset(Lambda(n, expr), S.Integers)
else:
return
else:
return FiniteSet(*(fn.subs(n, s[0]) for s in solns))
if other == S.Reals:
from sympy.solvers.solveset import solveset_real
from sympy.core.function import expand_complex
f = self.lamda.expr
n = self.lamda.variables[0]
n_ = Dummy(n.name, real=True)
f_ = f.subs(n, n_)
re, im = f_.as_real_imag()
im = expand_complex(im)
re = re.subs(n_, n)
im = im.subs(n_, n)
ifree = im.free_symbols
lam = Lambda(n, re)
if not im:
# allow re-evaluation
# of self in this case to make
# the result canonical
pass
elif im.is_zero is False:
return S.EmptySet
elif ifree != {n}:
return None
else:
# univarite imaginary part in same variable
base_set = base_set.intersect(solveset_real(im, n))
return imageset(lam, base_set)
elif isinstance(other, Interval):
from sympy.solvers.solveset import (invert_real, invert_complex,
solveset)
f = self.lamda.expr
n = self.lamda.variables[0]
new_inf, new_sup = None, None
new_lopen, new_ropen = other.left_open, other.right_open
if f.is_real:
inverter = invert_real
else:
inverter = invert_complex
g1, h1 = inverter(f, other.inf, n)
g2, h2 = inverter(f, other.sup, n)
if all(isinstance(i, FiniteSet) for i in (h1, h2)):
if g1 == n:
if len(h1) == 1:
new_inf = h1.args[0]
if g2 == n:
if len(h2) == 1:
new_sup = h2.args[0]
# TODO: Design a technique to handle multiple-inverse
# functions
# Any of the new boundary values cannot be determined
if any(i is None for i in (new_sup, new_inf)):
return
range_set = S.EmptySet
if all(i.is_real for i in (new_sup, new_inf)):
# this assumes continuity of underlying function
# however fixes the case when it is decreasing
if new_inf > new_sup:
new_inf, new_sup = new_sup, new_inf
new_interval = Interval(new_inf, new_sup, new_lopen, new_ropen)
range_set = base_set.intersect(new_interval)
else:
if other.is_subset(S.Reals):
solutions = solveset(f, n, S.Reals)
if not isinstance(range_set, (ImageSet, ConditionSet)):
range_set = solutions.intersect(other)
else:
return
if range_set is S.EmptySet:
return S.EmptySet
elif isinstance(range_set,
Range) and range_set.size is not S.Infinity:
range_set = FiniteSet(*list(range_set))
if range_set is not None:
return imageset(Lambda(n, f), range_set)
return
else:
return
def simp(x):
return simplify(expand_trig(expand_complex(expand(x))))