def linear_arg(arg):
""" Test if arg is of form a*s+b, raise exception if not. """
if not arg.is_polynomial(s):
raise exception(fact)
p = Poly(arg, s)
if p.degree() != 1:
raise exception(fact)
return p.all_coeffs()
def linear_arg(arg):
""" Test if arg is of form a*s+b, raise exception if not. """
if not arg.is_polynomial(s):
raise exception(fact)
p = Poly(arg, s)
if p.degree() != 1:
raise exception(fact)
return p.all_coeffs()
def lower_degree_to(polynomial: Poly,
max_polynomial_degree: int) -> Poly:
"""
Lowers the degree of the polynomial by using Chebyshev polynomials.
:param polynomial: polynomial to lower the degree of
:param max_polynomial_degree: maximum polynomial degree to lower to
:return: polynomial with the degree less than or equal to `max_polynomial_degree`
:rtype: Poly
"""
while polynomial.degree() > max_polynomial_degree:
normalised_chebyshev_polynomial = get_normalised_nth_chebyshev_polynomial(polynomial.degree())
polynomial -= normalised_chebyshev_polynomial * polynomial.LC()
return polynomial
def fn(theta, phi, pa, tau):
x = Symbol('x')
pa = Symbol('pa')
phi = Symbol('phi')
theta = Symbol('theta')
tau = Symbol('tau')
f = (2 * pa * phi**2) / tau**5 + x**5 + (
2 * pa * phi**2 * cos(theta)
) / tau**5 - (2 * pa**2 * phi**2 * cos(theta)) / tau**5 - (
2 * pa * phi**2 * cos(theta)**2
) / tau**5 - (2 * pa * phi**2 * cos(theta)**3) / tau**5 + (
2 * pa**2 * phi**2 * cos(theta)**3) / tau**5 + x**4 * (
4 / tau + cos(theta) / tau -
(2 * pa * cos(theta)) / tau - cos(theta)**2 / tau) + x * (
tau**(-4) +
(4 * phi**2) / tau**4 +
(4 * pa * phi**2) / tau**4 + cos(theta) / tau**4 -
(2 * pa * cos(theta)) / tau**4 +
(4 * phi**2 * cos(theta)) / tau**4 -
(2 * pa * phi**2 * cos(theta)) / tau**4 -
(2 * pa**2 * phi**2 * cos(theta)) / tau**4 -
cos(theta)**2 / tau**4 -
(4 * phi**2 * cos(theta)**2) / tau**4 -
cos(theta)**3 / tau**4 + (2 * pa * cos(theta)**3) / tau**4 -
(4 * phi**2 * cos(theta)**3) / tau**4 +
(6 * pa * phi**2 * cos(theta)**3) / tau**4 -
(2 * pa**2 * phi**2 * cos(theta)**3) / tau**4) + x**2 * (
4 / tau**3 +
(8 * phi**2) / tau**3 + (2 * pa * phi**2) / tau**3 +
(3 * cos(theta)) / tau**3 -
(6 * pa * cos(theta)) / tau**3 +
(4 * phi**2 * cos(theta)) / tau**3 -
(4 * pa * phi**2 * cos(theta)) / tau**3 -
(3 * cos(theta)**2) / tau**3 -
(4 * phi**2 * cos(theta)**2) / tau**3 +
(2 * pa * phi**2 * cos(theta)**2) / tau**3 -
(2 * cos(theta)**3) / tau**3 +
(4 * pa * cos(theta)**3) / tau**3) + x**3 * (
6 / tau**2 +
(4 * phi**2) / tau**2 + (3 * cos(theta)) / tau**2 -
(6 * pa * cos(theta)) / tau**2 -
(3 * cos(theta)**2) / tau**2 - cos(theta)**3 / tau**2 +
(2 * pa * cos(theta)**3) / tau**2)
p = Poly(f, x)
ans = [p.root(k).evalf() for k in range(p.degree())]
for i, a in enumerate(ans):
print(i, a)
return ans
def check_convergence(numer, denom, n):
"""
Returns (h, g, p) where
-- h is:
> 0 for convergence of rate 1/factorial(n)**h
< 0 for divergence of rate factorial(n)**(-h)
= 0 for geometric or polynomial convergence or divergence
-- abs(g) is:
> 1 for geometric convergence of rate 1/h**n
< 1 for geometric divergence of rate h**n
= 1 for polynomial convergence or divergence
(g < 0 indicates an alternating series)
-- p is:
> 1 for polynomial convergence of rate 1/n**h
<= 1 for polynomial divergence of rate n**(-h)
"""
from sympy import Poly
npol = Poly(numer, n)
dpol = Poly(denom, n)
p = npol.degree()
q = dpol.degree()
rate = q - p
if rate:
return rate, None, None
constant = dpol.LC() / npol.LC()
if abs(constant) != 1:
return rate, constant, None
if npol.degree() == dpol.degree() == 0:
return rate, constant, 0
pc = npol.all_coeffs()[1]
qc = dpol.all_coeffs()[1]
return rate, constant, (qc - pc) / dpol.LC()
def check_convergence(numer, denom, n):
"""
Returns (h, g, p) where
-- h is:
> 0 for convergence of rate 1/factorial(n)**h
< 0 for divergence of rate factorial(n)**(-h)
= 0 for geometric or polynomial convergence or divergence
-- abs(g) is:
> 1 for geometric convergence of rate 1/h**n
< 1 for geometric divergence of rate h**n
= 1 for polynomial convergence or divergence
(g < 0 indicates an alternating series)
-- p is:
> 1 for polynomial convergence of rate 1/n**h
<= 1 for polynomial divergence of rate n**(-h)
"""
from sympy import Poly
npol = Poly(numer, n)
dpol = Poly(denom, n)
p = npol.degree()
q = dpol.degree()
rate = q - p
if rate:
return rate, None, None
constant = dpol.LC() / npol.LC()
if abs(constant) != 1:
return rate, constant, None
if npol.degree() == dpol.degree() == 0:
return rate, constant, 0
pc = npol.all_coeffs()[1]
qc = dpol.all_coeffs()[1]
return rate, constant, (qc - pc)/dpol.LC()
def simplify(polynom):
"""Simplifies a function with binary variables
"""
polynom = Poly(polynom)
new_polynom = 0
variables = list(polynom.free_symbols)
for var_i in variables:
coefficient_i = polynom.as_expr().coeff(var_i)/2
for degree in range(polynom.degree()-1):
coefficient_i += polynom.as_expr().coeff(var_i ** (degree+2))
if coefficient_i.is_Float:
new_polynom += float(coefficient_i) * var_i
else:
new_polynom += float(coefficient_i.as_coefficients_dict()[1]) * var_i
for var_j in variables:
if var_j != var_i:
coefficient_j = coefficient_i.coeff(var_j)
if coefficient_j.is_Float:
new_polynom += float(coefficient_j) * var_i * var_j
else:
new_polynom += float(coefficient_j.as_coefficients_dict()[1]) * var_i * var_j
return new_polynom + polynom.as_expr().as_coefficients_dict()[1]
def _rewrite_gamma(f, s, a, b):
"""
Try to rewrite the product f(s) as a product of gamma functions,
so that the inverse mellin transform of f can be expressed as a meijer
G function.
Return (an, ap), (bm, bq), arg, exp, fac such that
G((an, ap), (bm, bq), arg/z**exp)*fac is the inverse mellin transform of f(s).
Raises IntegralTransformError or MellinTransformStripError on failure.
It is asserted that f has no poles in the fundamental strip designated by
(a, b). One of a and b is allowed to be None. The fundamental strip is
important, because it determines the inversion contour.
This function can handle exponentials, linear factors, trigonometric
functions.
This is a helper function for inverse_mellin_transform that will not
attempt any transformations on f.
>>> from sympy.integrals.transforms import _rewrite_gamma
>>> from sympy.abc import s
>>> from sympy import oo
>>> _rewrite_gamma(s*(s+3)*(s-1), s, -oo, oo)
(([], [-3, 0, 1]), ([-2, 1, 2], []), 1, 1, -1)
>>> _rewrite_gamma((s-1)**2, s, -oo, oo)
(([], [1, 1]), ([2, 2], []), 1, 1, 1)
Importance of the fundamental strip:
>>> _rewrite_gamma(1/s, s, 0, oo)
(([1], []), ([], [0]), 1, 1, 1)
>>> _rewrite_gamma(1/s, s, None, oo)
(([1], []), ([], [0]), 1, 1, 1)
>>> _rewrite_gamma(1/s, s, 0, None)
(([1], []), ([], [0]), 1, 1, 1)
>>> _rewrite_gamma(1/s, s, -oo, 0)
(([], [1]), ([0], []), 1, 1, -1)
>>> _rewrite_gamma(1/s, s, None, 0)
(([], [1]), ([0], []), 1, 1, -1)
>>> _rewrite_gamma(1/s, s, -oo, None)
(([], [1]), ([0], []), 1, 1, -1)
>>> _rewrite_gamma(2**(-s+3), s, -oo, oo)
(([], []), ([], []), 1/2, 1, 8)
"""
from itertools import repeat
from sympy import (Poly, gamma, Mul, re, RootOf, exp as exp_, E, expand,
roots, ilcm, pi, sin, cos, tan, cot, igcd, exp_polar)
# Our strategy will be as follows:
# 1) Guess a constant c such that the inversion integral should be
# performed wrt s'=c*s (instead of plain s). Write s for s'.
# 2) Process all factors, rewrite them independently as gamma functions in
# argument s, or exponentials of s.
# 3) Try to transform all gamma functions s.t. they have argument
# a+s or a-s.
# 4) Check that the resulting G function parameters are valid.
# 5) Combine all the exponentials.
a_, b_ = S([a, b])
def left(c, is_numer):
"""
Decide whether pole at c lies to the left of the fundamental strip.
"""
# heuristically, this is the best chance for us to solve the inequalities
c = expand(re(c))
if a_ is None:
return c < b_
if b_ is None:
return c <= a_
if (c >= b_) is True:
return False
if (c <= a_) is True:
return True
if is_numer:
return None
if a_.free_symbols or b_.free_symbols or c.free_symbols:
return None # XXX
#raise IntegralTransformError('Inverse Mellin', f,
# 'Could not determine position of singularity %s'
# ' relative to fundamental strip' % c)
raise MellinTransformStripError('Pole inside critical strip?')
# 1)
s_multipliers = []
for g in f.atoms(gamma):
if not g.has(s):
continue
arg = g.args[0]
if arg.is_Add:
arg = arg.as_independent(s)[1]
coeff, _ = arg.as_coeff_mul(s)
s_multipliers += [coeff]
for g in f.atoms(sin, cos, tan, cot):
if not g.has(s):
continue
arg = g.args[0]
if arg.is_Add:
arg = arg.as_independent(s)[1]
coeff, _ = arg.as_coeff_mul(s)
s_multipliers += [coeff/pi]
s_multipliers = [abs(x) for x in s_multipliers if x.is_real]
common_coefficient = S(1)
for x in s_multipliers:
if not x.is_Rational:
common_coefficient = x
break
s_multipliers = [x/common_coefficient for x in s_multipliers]
if any(not x.is_Rational for x in s_multipliers):
raise NotImplementedError
s_multiplier = common_coefficient/reduce(ilcm, [S(x.q) for x in s_multipliers], S(1))
if s_multiplier == common_coefficient:
if len(s_multipliers) == 0:
s_multiplier = common_coefficient
else:
s_multiplier = common_coefficient \
*reduce(igcd, [S(x.p) for x in s_multipliers])
exponent = S(1)
fac = S(1)
f = f.subs(s, s/s_multiplier)
fac /= s_multiplier
exponent = 1/s_multiplier
if a_ is not None:
a_ *= s_multiplier
if b_ is not None:
b_ *= s_multiplier
# 2)
numer, denom = f.as_numer_denom()
numer = Mul.make_args(numer)
denom = Mul.make_args(denom)
args = zip(numer, repeat(True)) + zip(denom, repeat(False))
facs = []
dfacs = []
# *_gammas will contain pairs (a, c) representing Gamma(a*s + c)
numer_gammas = []
denom_gammas = []
# exponentials will contain bases for exponentials of s
exponentials = []
def exception(fact):
return IntegralTransformError("Inverse Mellin", f, "Unrecognised form '%s'." % fact)
while args:
fact, is_numer = args.pop()
if is_numer:
ugammas, lgammas = numer_gammas, denom_gammas
ufacs, lfacs = facs, dfacs
else:
ugammas, lgammas = denom_gammas, numer_gammas
ufacs, lfacs = dfacs, facs
def linear_arg(arg):
""" Test if arg is of form a*s+b, raise exception if not. """
if not arg.is_polynomial(s):
raise exception(fact)
p = Poly(arg, s)
if p.degree() != 1:
raise exception(fact)
return p.all_coeffs()
# constants
if not fact.has(s):
ufacs += [fact]
# exponentials
elif fact.is_Pow or isinstance(fact, exp_):
if fact.is_Pow:
base = fact.base
exp = fact.exp
else:
base = exp_polar(1)
exp = fact.args[0]
if exp.is_Integer:
cond = is_numer
if exp < 0:
cond = not cond
args += [(base, cond)]*abs(exp)
continue
elif not base.has(s):
a, b = linear_arg(exp)
if not is_numer:
base = 1/base
exponentials += [base**a]
facs += [base**b]
else:
raise exception(fact)
# linear factors
elif fact.is_polynomial(s):
p = Poly(fact, s)
if p.degree() != 1:
# We completely factor the poly. For this we need the roots.
# Now roots() only works in some cases (low degree), and RootOf
# only works without parameters. So try both...
coeff = p.LT()[1]
rs = roots(p, s)
if len(rs) != p.degree():
rs = RootOf.all_roots(p)
ufacs += [coeff]
args += [(s - c, is_numer) for c in rs]
continue
a, c = p.all_coeffs()
ufacs += [a]
c /= -a
# Now need to convert s - c
if left(c, is_numer):
ugammas += [(S(1), -c + 1)]
lgammas += [(S(1), -c)]
else:
ufacs += [-1]
ugammas += [(S(-1), c + 1)]
lgammas += [(S(-1), c)]
elif isinstance(fact, gamma):
a, b = linear_arg(fact.args[0])
if is_numer:
if (a > 0 and (left(-b/a, is_numer) is False)) or \
(a < 0 and (left(-b/a, is_numer) is True)):
raise NotImplementedError('Gammas partially over the strip.')
ugammas += [(a, b)]
elif isinstance(fact, sin):
# We try to re-write all trigs as gammas. This is not in
# general the best strategy, since sometimes this is impossible,
# but rewriting as exponentials would work. However trig functions
# in inverse mellin transforms usually all come from simplifying
# gamma terms, so this should work.
a = fact.args[0]
if is_numer:
# No problem with the poles.
gamma1, gamma2, fac_ = gamma(a/pi), gamma(1 - a/pi), pi
else:
gamma1, gamma2, fac_ = _rewrite_sin(linear_arg(a), s, a_, b_)
args += [(gamma1, not is_numer), (gamma2, not is_numer)]
ufacs += [fac_]
elif isinstance(fact, tan):
a = fact.args[0]
args += [(sin(a, evaluate=False), is_numer),
(sin(pi/2 - a, evaluate=False), not is_numer)]
elif isinstance(fact, cos):
a = fact.args[0]
args += [(sin(pi/2 - a, evaluate=False), is_numer)]
elif isinstance(fact, cot):
a = fact.args[0]
args += [(sin(pi/2 - a, evaluate=False), is_numer),
(sin(a, evaluate=False), not is_numer)]
else:
raise exception(fact)
fac *= Mul(*facs)/Mul(*dfacs)
# 3)
an, ap, bm, bq = [], [], [], []
for gammas, plus, minus, is_numer in [(numer_gammas, an, bm, True),
(denom_gammas, bq, ap, False)]:
while gammas:
a, c = gammas.pop()
if a != -1 and a != +1:
# We use the gamma function multiplication theorem.
p = abs(S(a))
newa = a/p
newc = c/p
assert a.is_Integer
for k in range(p):
gammas += [(newa, newc + k/p)]
if is_numer:
fac *= (2*pi)**((1 - p)/2) * p**(c - S(1)/2)
exponentials += [p**a]
else:
fac /= (2*pi)**((1 - p)/2) * p**(c - S(1)/2)
exponentials += [p**(-a)]
continue
if a == +1:
plus.append(1 - c)
else:
minus.append(c)
# 4)
# TODO
# 5)
arg = Mul(*exponentials)
# for testability, sort the arguments
an.sort()
ap.sort()
bm.sort()
bq.sort()
return (an, ap), (bm, bq), arg, exponent, fac
def _rewrite_gamma(f, s, a, b):
"""
Try to rewrite the product f(s) as a product of gamma functions,
so that the inverse mellin transform of f can be expressed as a meijer
G function.
Return (an, ap), (bm, bq), arg, exp, fac such that
G((an, ap), (bm, bq), arg/z**exp)*fac is the inverse mellin transform of f(s).
Raises IntegralTransformError or ValueError on failure.
It is asserted that f has no poles in the fundamental strip designated by
(a, b). One of a and b is allowed to be None. The fundamental strip is
important, because it determines the inversion contour.
This function can handle exponentials, linear factors, trigonometric
functions.
This is a helper function for inverse_mellin_transform that will not
attempt any transformations on f.
>>> from sympy.integrals.transforms import _rewrite_gamma
>>> from sympy.abc import s
>>> from sympy import oo
>>> _rewrite_gamma(s*(s+3)*(s-1), s, -oo, oo)
(([], [-3, 0, 1]), ([-2, 1, 2], []), 1, 1, -1)
>>> _rewrite_gamma((s-1)**2, s, -oo, oo)
(([], [1, 1]), ([2, 2], []), 1, 1, 1)
Importance of the fundamental strip:
>>> _rewrite_gamma(1/s, s, 0, oo)
(([1], []), ([], [0]), 1, 1, 1)
>>> _rewrite_gamma(1/s, s, None, oo)
(([1], []), ([], [0]), 1, 1, 1)
>>> _rewrite_gamma(1/s, s, 0, None)
(([1], []), ([], [0]), 1, 1, 1)
>>> _rewrite_gamma(1/s, s, -oo, 0)
(([], [1]), ([0], []), 1, 1, -1)
>>> _rewrite_gamma(1/s, s, None, 0)
(([], [1]), ([0], []), 1, 1, -1)
>>> _rewrite_gamma(1/s, s, -oo, None)
(([], [1]), ([0], []), 1, 1, -1)
>>> _rewrite_gamma(2**(-s+3), s, -oo, oo)
(([], []), ([], []), 1/2, 1, 8)
"""
from itertools import repeat
from sympy import (Poly, gamma, Mul, re, RootOf, exp as exp_, E, expand,
roots, ilcm, pi, sin, cos, tan, cot, igcd)
# Our strategy will be as follows:
# 1) Guess a constant c such that the inversion integral should be
# performed wrt s'=c*s (instead of plain s). Write s for s'.
# 2) Process all factors, rewrite them independently as gamma functions in
# argument s, or exponentials of s.
# 3) Try to transform all gamma functions s.t. they have argument
# a+s or a-s.
# 4) Check that the resulting G function parameters are valid.
# 5) Combine all the exponentials.
a_, b_ = S([a, b])
def left(c, is_numer):
"""
Decide whether pole at c lies to the left of the fundamental strip.
"""
# heuristically, this is the best chance for us to solve the inequalities
c = expand(re(c))
if a_ is None:
return c < b_
if b_ is None:
return c <= a_
if (c >= b_) is True:
return False
if (c <= a_) is True:
return True
if is_numer:
return None
if a_.free_symbols or b_.free_symbols or c.free_symbols:
return None # XXX
#raise IntegralTransformError('Inverse Mellin', f,
# 'Could not determine position of singularity %s'
# ' relative to fundamental strip' % c)
raise ValueError('Pole inside critical strip?')
# 1)
s_multipliers = []
for g in f.atoms(gamma):
if not g.has(s):
continue
arg = g.args[0]
if arg.is_Add:
arg = arg.as_independent(s)[1]
coeff, _ = arg.as_coeff_mul(s)
s_multipliers += [coeff]
for g in f.atoms(sin, cos, tan, cot):
if not g.has(s):
continue
arg = g.args[0]
if arg.is_Add:
arg = arg.as_independent(s)[1]
coeff, _ = arg.as_coeff_mul(s)
s_multipliers += [coeff / pi]
s_multipliers = [abs(x) for x in s_multipliers if x.is_real]
common_coefficient = S(1)
for x in s_multipliers:
if not x.is_Rational:
common_coefficient = x
break
s_multipliers = [x / common_coefficient for x in s_multipliers]
if any(not x.is_Rational for x in s_multipliers):
raise NotImplementedError
s_multiplier = common_coefficient / reduce(
ilcm, [S(x.q) for x in s_multipliers], S(1))
if s_multiplier == common_coefficient:
if len(s_multipliers) == 0:
s_multiplier = common_coefficient
else:
s_multiplier = common_coefficient \
*reduce(igcd, [S(x.p) for x in s_multipliers])
exponent = S(1)
fac = S(1)
f = f.subs(s, s / s_multiplier)
fac /= s_multiplier
exponent = 1 / s_multiplier
if a_ is not None:
a_ *= s_multiplier
if b_ is not None:
b_ *= s_multiplier
# 2)
numer, denom = f.as_numer_denom()
numer = Mul.make_args(numer)
denom = Mul.make_args(denom)
args = zip(numer, repeat(True)) + zip(denom, repeat(False))
facs = []
dfacs = []
# *_gammas will contain pairs (a, c) representing Gamma(a*s + c)
numer_gammas = []
denom_gammas = []
# exponentials will contain bases for exponentials of s
exponentials = []
def exception(fact):
return IntegralTransformError("Inverse Mellin", f,
"Unrecognised form '%s'." % fact)
while args:
fact, is_numer = args.pop()
if is_numer:
ugammas, lgammas = numer_gammas, denom_gammas
ufacs, lfacs = facs, dfacs
else:
ugammas, lgammas = denom_gammas, numer_gammas
ufacs, lfacs = dfacs, facs
def linear_arg(arg):
""" Test if arg is of form a*s+b, raise exception if not. """
if not arg.is_polynomial(s):
raise exception(fact)
p = Poly(arg, s)
if p.degree() != 1:
raise exception(fact)
return p.all_coeffs()
# constants
if not fact.has(s):
ufacs += [fact]
# exponentials
elif fact.is_Pow or isinstance(fact, exp_):
if fact.is_Pow:
base = fact.base
exp = fact.exp
else:
base = E
exp = fact.args[0]
if exp.is_Integer:
cond = is_numer
if exp < 0:
cond = not cond
args += [(base, cond)] * abs(exp)
continue
elif not base.has(s):
a, b = linear_arg(exp)
if not is_numer:
base = 1 / base
exponentials += [base**a]
facs += [base**b]
else:
raise exception(fact)
# linear factors
elif fact.is_polynomial(s):
p = Poly(fact, s)
if p.degree() != 1:
# We completely factor the poly. For this we need the roots.
# Now roots() only works in some cases (low degree), and RootOf
# only works without parameters. So try both...
coeff = p.LT()[1]
rs = roots(p, s)
if len(rs) != p.degree():
rs = RootOf.all_roots(p)
ufacs += [coeff]
args += [(s - c, is_numer) for c in rs]
continue
a, c = p.all_coeffs()
ufacs += [a]
c /= -a
# Now need to convert s - c
if left(c, is_numer):
ugammas += [(S(1), -c + 1)]
lgammas += [(S(1), -c)]
else:
ufacs += [-1]
ugammas += [(S(-1), c + 1)]
lgammas += [(S(-1), c)]
elif isinstance(fact, gamma):
a, b = linear_arg(fact.args[0])
if is_numer:
if (a > 0 and (left(-b/a, is_numer) is False)) or \
(a < 0 and (left(-b/a, is_numer) is True)):
raise NotImplementedError(
'Gammas partially over the strip.')
ugammas += [(a, b)]
elif isinstance(fact, sin):
# We try to re-write all trigs as gammas. This is not in
# general the best strategy, since sometimes this is impossible,
# but rewriting as exponentials would work. However trig functions
# in inverse mellin transforms usually all come from simplifying
# gamma terms, so this should work.
a = fact.args[0]
if is_numer:
# No problem with the poles.
gamma1, gamma2, fac_ = gamma(a / pi), gamma(1 - a / pi), pi
else:
gamma1, gamma2, fac_ = _rewrite_sin(linear_arg(a), s, a_, b_)
args += [(gamma1, not is_numer), (gamma2, not is_numer)]
ufacs += [fac_]
elif isinstance(fact, tan):
a = fact.args[0]
args += [(sin(a, evaluate=False), is_numer),
(sin(pi / 2 - a, evaluate=False), not is_numer)]
elif isinstance(fact, cos):
a = fact.args[0]
args += [(sin(pi / 2 - a, evaluate=False), is_numer)]
elif isinstance(fact, cot):
a = fact.args[0]
args += [(sin(pi / 2 - a, evaluate=False), is_numer),
(sin(a, evaluate=False), not is_numer)]
else:
raise exception(fact)
fac *= Mul(*facs) / Mul(*dfacs)
# 3)
an, ap, bm, bq = [], [], [], []
for gammas, plus, minus, is_numer in [(numer_gammas, an, bm, True),
(denom_gammas, bq, ap, False)]:
while gammas:
a, c = gammas.pop()
if a != -1 and a != +1:
# We use the gamma function multiplication theorem.
p = abs(S(a))
newa = a / p
newc = c / p
assert a.is_Integer
for k in range(p):
gammas += [(newa, newc + k / p)]
if is_numer:
fac *= (2 * pi)**((1 - p) / 2) * p**(c - S(1) / 2)
exponentials += [p**a]
else:
fac /= (2 * pi)**((1 - p) / 2) * p**(c - S(1) / 2)
exponentials += [p**(-a)]
continue
if a == +1:
plus.append(1 - c)
else:
minus.append(c)
# 4)
# TODO
# 5)
arg = Mul(*exponentials)
# for testability, sort the arguments
an.sort()
ap.sort()
bm.sort()
bq.sort()
return (an, ap), (bm, bq), arg, exponent, fac
def _expansion_search(expr, rep_list, lib=None):
"""
Search for and substitute terms that match a series expansion of
fundamental math functions.
e: expression
rep_list: list containing dummy variables
"""
if debug:
print("_expansion_search: ", expr)
try:
if isinstance(expr, Mul):
exprs = [expr]
elif isinstance(expr, Add):
exprs = expr.args
else:
return expr
if lib is not None:
fncs, series = lib
else:
fncs, series = _fncs, _series
newargs = []
for expr in exprs:
# Try to simplify the expression
if isinstance(expr, Mul):
# Split the expression into a LIST of commutative and a LIST of non-commutative factor (i.e. operators).
c, nc = expr.args_cnc()
if nc and c:
# Construct an expression of commutative elements ONLY.
expr = Mul(*c).expand()
#out = find_expansion(expr, *rep_list)
eq = Poly(expr, *rep_list)
bdict = eq.as_dict()
if any([
n > 1 for n in
[sum([i != 0 for i in k]) for k in bdict.keys()]
]):
raise ValueError(
"Expressions cross-products of symbols are not supported (yet?).."
)
# Construct the list of 'vectors'
vecl = []
fncl = []
for x in rep_list:
for n in range(3): #range(eq.degree()-1):
X = pow(x, n)
vecl.extend([((X * p.subs(_s, x)).expand() +
O(pow(x,
eq.degree() + 1))).removeO()
for p in series])
fncl.extend([X * f(x) for f in fncs])
vecl.extend(
[pow(x, n) for n in range(max(eq.degree(), 3))])
fncl.extend(
[pow(x, n) for n in range(max(eq.degree(), 3))])
# Create vector coefficients
q = symbols(f'q:{len(vecl)}')
# Take inner product of 'vector' list and vector coefficients
A = Add(*[x * y
for x, y in zip(q, vecl)]).as_poly(*rep_list)
terms = []
for k, eq in A.as_dict().items():
terms.append(eq - bdict.get(k, 0))
H = linsolve(terms, *q)
h = H.subs({k: 0 for k in q})
out = Add(*[c * fnc for fnc, c in zip(fncl, list(h)[0])])
newargs.append(out * Mul(*nc))
else:
newargs.append(expr)
else:
newargs.append(expr)
return Add(*newargs)
except Exception as e:
print("Failed to identify series expansions: " + str(e))
return e
if len(uniq_b) == max_size:
break
if flag_all:
break
if not flag_all:
ans.append(p)
if __name__ == '__main__':
print('Enter the polynomial: ', end='')
from sympy import Poly, Symbol
polynom = Poly(input())
d = polynom.degree() + 1
a = Symbol('a')
symbols = []
for symbol in polynom.free_symbols:
symbols.append(str(symbol))
symbols.sort()
replace_symbols = {}
j = 0
for symbol in symbols:
replace_symbols[symbol] = a**(d**j)
j += 1
for symbol in replace_symbols:
def sturm():
divid = [fx, diff(fx)] # Array where equations will be stored
while True:
test = Poly(divid[-1], x)
if len(test.coeffs()) > 1:
q, r = div(divid[-2], divid[-1],
x) # Taking quotient and remainder of dividing
divid.append(-1 * r)
else:
break
print('\n Let\'s calculate dividing of a polynomials: ')
for i in divid:
print('f' + str(divid.index(i)) + '(x): ' + str(i))
# Create information table
table = [[' '], ['-inf'], ['+inf']]
for i in divid:
table[0].append('f' + str(divid.index(i)) +
'(x)') # Better information association
test = Poly(i, x)
# -inf
if (test.degree() % 2 == 0 and test.coeffs()[0] > 0) or \
(test.degree() % 2 != 0 and test.coeffs()[0] < 0):
table[1].append('+')
else:
table[1].append('-')
# +inf
if test.coeffs()[0] > 0:
table[2].append('+')
else:
table[2].append('-')
qwe = changes(table)
mininf = qwe[0]
plusinf = qwe[1]
# Add found data to a table for better user experience
table[0].append('Sign changes')
table[1].append(mininf)
table[2].append(plusinf)
print('---------')
print(
'\n\n Then we need to calculate number of sign changes for (-inf) and (+inf) cases. '
)
print(' Let\'s create a table to visualize our results: ')
for i in table:
print()
for j in i:
if table.index(i) == 0:
print(' ' + j, end='')
else:
print(j, end=' ' + ' ' * i.index(j))
print(
'\n\n As we see, our equation has {0} - {1} = {2} real roots.'.format(
mininf, plusinf, mininf - plusinf))
print('---------')
print('\n\n Now we need to define possible intervals for roots. ')
print(
' Let\'s create a table with calculated functions in possible root interval. '
)
# Creating a table
results = [[' ']]
for i in divid:
results[0].append('f' + str(divid.index(i)) +
'(x) ') # Better information association
for i in range(floor(Rlo), ceil(Rup + 1)):
results.append(['x = ' + str(i)])
# Calculate value for each function and possible variable
count = 1
for j in range(floor(Rlo), ceil(Rup + 1)):
for i in divid:
if i.evalf(subs={x: j}) > 0:
results[count].append('+')
else:
results[count].append('-')
count += 1
# Check number of sign changes
ert = changes(results)
results[0].append('Sign changes')
for i in range(1, len(results)):
results[i].append(ert[i - 1])
# Printing table
for i in results:
print()
for j in i:
if results.index(i) == 0:
print(' ' + j, end='')
else:
print(j, end=' ' + ' ' * i.index(j))
send = []
for i in range(1, len(results) - 1):
if results[i][-1] != results[i + 1][-1]:
temp = [i for i in range(floor(Rlo), ceil(Rup + 1))]
send.append((temp[i - 1], temp[i]))
# Check if we need extended computations
analys = False
for i in range(1, len(send)):
if send[i][0] == send[i - 1][1]:
analys = True
break
if len(send) != mininf - plusinf or analys is True:
print(
'\n\nWARNING!!! PROGRAM CALCULATE VALUES FROM {0} TO {1} with step 0.1. IT\'S NOT SHOWN IN A TABLE.'
.format(floor(Rlo), ceil(Rup + 1)) +
'\nIT HAPPENS BECAUSE TWO OR MORE ROOTS LIE IN INTERVAL WITH SIZE LESS THAN 1'
+
'\nOR THEIR INTERVALS STARTS AND BEGINS AT ONE POINT RESPECTIVELY.'
+ '\nWITHOUT THIS CHECKING CALCULATIONS WILL BE WRONG.')
send = deepanalysis(divid)
print('\n As we can see, roots of equation lie in intervals: ')
for i in send:
print(i)
return send # Return intervals of roots
