def get_nullspace(n,q,eigv):
M = matrix_cache.get_reduced_matrix(n,q, True)
M.m = M.m._new(M.m.rows, M.m.cols,[nsimplify(v, rational=True) for v in M.m])
# N = M - lambdaI
N = M.m - sympy.Matrix.eye(M.get_size())* eigv
# rationalize entries
N = N._new(N.rows, N.cols,[nsimplify(v, rational=True) for v in N])
# find eigenspace - Kernel(M-lambdaI) = Kernel(N)
EigenSpaceBase = N.nullspace()
# orthogonal space = Kernel(TransposedEigenVectors)
TransposedEigenSpaceBase = [list(v) for v in EigenSpaceBase]
EigenSpaceTransposeMatrix = (sympy.Matrix(numpy.matrix(TransposedEigenSpaceBase)))
OrthogonalSubspaceBase = EigenSpaceTransposeMatrix.nullspace()
# need to transpose to put the vectors in columns, and get a matrix for the orthogonal operator
OrthogonalSubspaceMatrix = (sympy.Matrix(numpy.matrix(OrthogonalSubspaceBase))).transpose()
# multiply M * Ortohogonal, to get the reduction of M on the orthogonal subspace
OrthogonalSubspaceMatrix = OrthogonalSubspaceMatrix._new(OrthogonalSubspaceMatrix.rows, OrthogonalSubspaceMatrix.cols,
[nsimplify(v, rational=True) for v in OrthogonalSubspaceMatrix])
EigenValueReducedMatrix = M.m * OrthogonalSubspaceMatrix
EigenValueReducedMatrix = EigenValueReducedMatrix.rref()
def random_complex_number(a=2, b=-1, c=3, d=1, rational=False):
"""
Return a random complex number.
To reduce chance of hitting branch cuts or anything, we guarantee
b <= Im z <= d, a <= Re z <= c
"""
A, B = uniform(a, c), uniform(b, d)
if not rational:
return A + I*B
return nsimplify(A, rational=True) + I*nsimplify(B, rational=True)
def eval(cls, p, q):
from sympy.simplify.simplify import nsimplify
if q.is_Number:
float = not q.is_Rational
pnew = expand_mul(p)
if pnew.is_Number:
float = float or not pnew.is_Rational
if not float:
return pnew % q
return Float(nsimplify(pnew) % nsimplify(q))
elif pnew.is_Add and pnew.args[0].is_Number:
r, p = pnew.as_two_terms()
p += Mod(r, q)
return Mod(p, q, evaluate=False)
def random_complex_number(a=2, b=-1, c=3, d=1, rational=False, tolerance=None):
"""
Return a random complex number.
To reduce chance of hitting branch cuts or anything, we guarantee
b <= Im z <= d, a <= Re z <= c
When rational is True, a rational approximation to a random number
is obtained within specified tolerance, if any.
"""
A, B = uniform(a, c), uniform(b, d)
if not rational:
return A + I*B
return (nsimplify(A, rational=True, tolerance=tolerance) +
I*nsimplify(B, rational=True, tolerance=tolerance))
def test_minpoly_issue_7113():
# see discussion in https://github.com/sympy/sympy/pull/2234
from sympy.simplify.simplify import nsimplify
r = nsimplify(pi, tolerance=0.000000001)
mp = minimal_polynomial(r, x)
assert mp == 1768292677839237920489538677417507171630859375*x**109 - \
2734577732179183863586489182929671773182898498218854181690460140337930774573792597743853652058046464
def _pow_float(inter, power):
"""Evaluates an interval raised to a floating point."""
power_rational = nsimplify(power)
num, denom = power_rational.as_numer_denom()
if num % 2 == 0:
start = abs(inter.start)**power
end = abs(inter.end)**power
if start < 0:
ret = interval(0, max(start, end))
else:
ret = interval(start, end)
return ret
elif denom % 2 == 0:
if inter.end < 0:
return interval(-float('inf'), float('inf'), is_valid=False)
elif inter.start < 0:
return interval(0, inter.end**power, is_valid=None)
else:
return interval(inter.start**power, inter.end**power)
else:
if inter.start < 0:
start = -abs(inter.start)**power
else:
start = inter.start**power
if inter.end < 0:
end = -abs(inter.end)**power
else:
end = inter.end**power
return interval(start, end, is_valid=inter.is_valid)
def test_issue_2877():
f = Float(2.0)
assert (x + f).subs({f: 2}) == x + 2
def r(a, b, c):
return factor(a * x**2 + b * x + c)
e = r(5.0 / 6, 10, 5)
assert nsimplify(e) == 5 * x**2 / 6 + 10 * x + 5
def test_minpoly_issue_7113():
# see discussion in https://github.com/sympy/sympy/pull/2234
from sympy.simplify.simplify import nsimplify
r = nsimplify(pi, tolerance=0.000000001)
mp = minimal_polynomial(r, x)
assert (
mp == 1768292677839237920489538677417507171630859375 * x**109 -
2734577732179183863586489182929671773182898498218854181690460140337930774573792597743853652058046464
)
def _eval_sum_hyper(f, i, a):
""" Returns (res, cond). Sums from a to oo. """
from sympy.functions import hyper
from sympy.simplify import hyperexpand, hypersimp, fraction, simplify
from sympy.polys.polytools import Poly, factor
from sympy.core.numbers import Float
if a != 0:
return _eval_sum_hyper(f.subs(i, i + a), i, 0)
if f.subs(i, 0) == 0:
if simplify(f.subs(i, Dummy("i", integer=True, positive=True))) == 0:
return S.Zero, True
return _eval_sum_hyper(f.subs(i, i + 1), i, 0)
hs = hypersimp(f, i)
if hs is None:
return None
if isinstance(hs, Float):
from sympy.simplify.simplify import nsimplify
hs = nsimplify(hs)
numer, denom = fraction(factor(hs))
top, topl = numer.as_coeff_mul(i)
bot, botl = denom.as_coeff_mul(i)
ab = [top, bot]
factors = [topl, botl]
params = [[], []]
for k in range(2):
for fac in factors[k]:
mul = 1
if fac.is_Pow:
mul = fac.exp
fac = fac.base
if not mul.is_Integer:
return None
p = Poly(fac, i)
if p.degree() != 1:
return None
m, n = p.all_coeffs()
ab[k] *= m ** mul
params[k] += [n / m] * mul
# Add "1" to numerator parameters, to account for implicit n! in
# hypergeometric series.
ap = params[0] + [1]
bq = params[1]
x = ab[0] / ab[1]
h = hyper(ap, bq, x)
f = combsimp(f)
return f.subs(i, 0) * hyperexpand(h), h.convergence_statement
def _eval_sum_hyper(f, i, a):
""" Returns (res, cond). Sums from a to oo. """
from sympy.functions import hyper
from sympy.simplify import hyperexpand, hypersimp, fraction, simplify
from sympy.polys.polytools import Poly, factor
from sympy.core.numbers import Float
if a != 0:
return _eval_sum_hyper(f.subs(i, i + a), i, 0)
if f.subs(i, 0) == 0:
if simplify(f.subs(i, Dummy('i', integer=True, positive=True))) == 0:
return S(0), True
return _eval_sum_hyper(f.subs(i, i + 1), i, 0)
hs = hypersimp(f, i)
if hs is None:
return None
if isinstance(hs, Float):
from sympy.simplify.simplify import nsimplify
hs = nsimplify(hs)
numer, denom = fraction(factor(hs))
top, topl = numer.as_coeff_mul(i)
bot, botl = denom.as_coeff_mul(i)
ab = [top, bot]
factors = [topl, botl]
params = [[], []]
for k in range(2):
for fac in factors[k]:
mul = 1
if fac.is_Pow:
mul = fac.exp
fac = fac.base
if not mul.is_Integer:
return None
p = Poly(fac, i)
if p.degree() != 1:
return None
m, n = p.all_coeffs()
ab[k] *= m**mul
params[k] += [n/m]*mul
# Add "1" to numerator parameters, to account for implicit n! in
# hypergeometric series.
ap = params[0] + [1]
bq = params[1]
x = ab[0]/ab[1]
h = hyper(ap, bq, x)
return f.subs(i, 0)*hyperexpand(h), h.convergence_statement
def sparse_sympy_matrix(self, triqs_operator_expression):
Hsp = self.sparse_matrix(triqs_operator_expression)
from sympy.matrices import SparseMatrix
from sympy.simplify.simplify import nsimplify
d = dict([((i, j), nsimplify(val))
for (i, j), val in Hsp.todok().items()])
H = SparseMatrix(Hsp.shape[0], Hsp.shape[1], d)
return H
def __new__(cls, *args, **kwargs):
if iterable(args[0]):
coords = Tuple(*args[0])
elif isinstance(args[0], Point):
coords = args[0].args
else:
coords = Tuple(*args)
if len(coords) != 2:
raise NotImplementedError("Only two dimensional points currently supported")
if kwargs.get('evaluate', True):
coords = [nsimplify(c) for c in coords]
return GeometryEntity.__new__(cls, *coords)
def __new__(cls, *args, **kwargs):
if iterable(args[0]):
coords = Tuple(*args[0])
elif isinstance(args[0], Point):
coords = args[0].args
else:
coords = Tuple(*args)
if len(coords) != 2:
raise NotImplementedError(
"Only two dimensional points currently supported")
if kwargs.get('evaluate', True):
coords = [nsimplify(c) for c in coords]
return GeometryEntity.__new__(cls, *coords)
def custom_charpoly(self, **flags):
"""
custom charpoly
"""
if (self.isSymbolic == True):
self.m = self.m._new(self.m.rows, self.m.cols,[nsimplify(v, rational=True) for v in self.m])
max_denom = 0;
for i in range (0,self.m.rows):
for j in range (0,self.m.cols):
if self.m[i,j] > max_denom:
max_denom = self.m[i,j].q
print max_denom
self.m *= max_denom
flags.pop('simplify', None) # pop unsupported flag
return self.m.berkowitz_charpoly(Dummy('x'))
else:
numpy.rint(self.m)
return numpy.rint(numpy.poly(self.m))
def eval(cls, n, p, t):
# ...
if not 0 <= p:
raise ValueError("must have 0 <= p")
if not 0 <= n:
raise ValueError("must have 0 <= n")
# ...
# ...
r = Symbol('r')
pp = 2 * p + 1
N = pp + 1
L = list(range(0, N + pp + 1))
b0 = bspline_basis(pp, L, 0, r)
b0_r = diff(b0, r)
b0_rr = diff(b0_r, r)
b0_rrr = diff(b0_rr, r)
b0_rrrr = diff(b0_rrr, r)
bsp = lambdify(r, b0_rrrr)
# ...
# ... we use nsimplify to get the rational number
phi = []
for i in range(0, p + 1):
y = bsp(p + 1 - i)
y = nsimplify(y, tolerance=TOLERANCE, rational=True)
phi.append(y)
# ...
# ...
m = phi[0] * cos(S.Zero)
for i in range(1, p + 1):
m += 2 * phi[i] * cos(i * t)
# ...
# ... scaling
m *= n**3
# ...
return m
def eval(cls, p, t):
if p is S.Infinity:
raise NotImplementedError('Add symbol limit for p -> oo')
elif isinstance(p, Symbol):
return Bilaplacian(p, t, evaluate=False)
elif isinstance(p, int):
# ...
r = Symbol('r')
pp = 2*p + 1
N = pp + 1
L = list(range(0, N + pp + 1))
b0 = bspline_basis(pp, L, 0, r)
b0_r = diff(b0, r)
b0_rr = diff(b0_r, r)
b0_rrr = diff(b0_rr, r)
b0_rrrr = diff(b0_rrr, r)
bsp = lambdify(r, b0_rrrr)
# ...
# ... we use nsimplify to get the rational number
phi = []
for i in range(0, p+1):
y = bsp(p+1-i)
y = nsimplify(y, tolerance=TOLERANCE, rational=True)
phi.append(y)
# ...
# ...
m = phi[0] * cos(S.Zero)
for i in range(1, p+1):
m += 2 * phi[i] * cos(i * t)
# ...
return m
def __rpow__(self, other):
if isinstance(other, (float, int)):
if not self.is_valid:
#Don't do anything
return self
elif other < 0:
if self.width > 0:
return interval(-float('inf'), float('inf'), is_valid=False)
else:
power_rational = nsimplify(self.start)
num, denom = power_rational.as_numer_denom()
if denom % 2 == 0:
return interval(-float('inf'), float('inf'),
is_valid=False)
else:
start = -abs(other)**self.start
end = start
return interval(start, end)
else:
return interval(other**self.start, other**self.end)
elif isinstance(other, interval):
return other.__pow__(self)
else:
return NotImplemented
def __rpow__(self, other):
if isinstance(other, (float, int)):
if not self.is_valid:
#Don't do anything
return self
elif other < 0:
if self.width > 0:
return interval(-float('inf'), float('inf'), is_valid=False)
else:
power_rational = nsimplify(self.start)
num, denom = power_rational.as_numer_denom()
if denom % 2 == 0:
return interval(-float('inf'), float('inf'),
is_valid=False)
else:
start = -abs(other)**self.start
end = start
return interval(start, end)
else:
return interval(other**self.start, other**self.end)
elif isinstance(other, interval):
return other.__pow__(self)
else:
return NotImplemented
def polytope_integrate(poly, expr=None, **kwargs):
"""Integrates polynomials over 2/3-Polytopes.
This function accepts the polytope in `poly` and the function in `expr`
(uni/bi/trivariate polynomials are implemented) and returns
the exact integral of `expr` over `poly`.
Parameters
==========
poly : The input Polygon.
expr : The input polynomial.
clockwise : Binary value to sort input points of 2-Polytope clockwise.(Optional)
max_degree : The maximum degree of any monomial of the input polynomial.(Optional)
Examples
========
>>> from sympy.abc import x, y
>>> from sympy.geometry.polygon import Polygon
>>> from sympy.geometry.point import Point
>>> from sympy.integrals.intpoly import polytope_integrate
>>> polygon = Polygon(Point(0,0), Point(0,1), Point(1,1), Point(1,0))
>>> polys = [1, x, y, x*y, x**2*y, x*y**2]
>>> expr = x*y
>>> polytope_integrate(polygon, expr)
1/4
>>> polytope_integrate(polygon, polys, max_degree=3)
{1: 1, x: 1/2, y: 1/2, x*y: 1/4, x*y**2: 1/6, x**2*y: 1/6}
"""
clockwise = kwargs.get('clockwise', False)
max_degree = kwargs.get('max_degree', None)
if clockwise is True:
if isinstance(poly, Polygon):
poly = point_sort(poly)
else:
raise TypeError("clockwise=True works for only 2-Polytope"
"V-representation input")
if isinstance(poly, Polygon):
# For Vertex Representation(2D case)
hp_params = hyperplane_parameters(poly)
facets = poly.sides
elif len(poly[0]) == 2:
# For Hyperplane Representation(2D case)
plen = len(poly)
if len(poly[0][0]) == 2:
intersections = [intersection(poly[(i - 1) % plen], poly[i],
"plane2D")
for i in range(0, plen)]
hp_params = poly
lints = len(intersections)
facets = [Segment2D(intersections[i],
intersections[(i + 1) % lints])
for i in range(0, lints)]
else:
raise NotImplementedError("Integration for H-representation 3D"
"case not implemented yet.")
else:
# For Vertex Representation(3D case)
vertices = poly[0]
facets = poly[1:]
hp_params = hyperplane_parameters(facets, vertices)
if max_degree is None:
if expr is None:
raise TypeError('Input expression be must'
'be a valid SymPy expression')
return main_integrate3d(expr, facets, vertices, hp_params)
if max_degree is not None:
result = {}
if not isinstance(expr, list) and expr is not None:
raise TypeError('Input polynomials must be list of expressions')
if len(hp_params[0][0]) == 3:
result_dict = main_integrate3d(0, facets, vertices, hp_params,
max_degree)
else:
result_dict = main_integrate(0, facets, hp_params, max_degree)
if expr is None:
return result_dict
for poly in expr:
if poly not in result:
if poly is S.Zero:
result[S.Zero] = S.Zero
continue
integral_value = S.Zero
monoms = decompose(poly, separate=True)
for monom in monoms:
monom = nsimplify(monom)
coeff, m = strip(monom)
integral_value += result_dict[m] * coeff
result[poly] = integral_value
return result
if expr is None:
raise TypeError('Input expression be must'
'be a valid SymPy expression')
return main_integrate(expr, facets, hp_params)
def polytope_integrate(poly, expr=None, **kwargs):
"""Integrates polynomials over 2/3-Polytopes.
This function accepts the polytope in `poly` and the function in `expr`
(uni/bi/trivariate polynomials are implemented) and returns
the exact integral of `expr` over `poly`.
Parameters
==========
poly : The input Polygon.
expr : The input polynomial.
clockwise : Binary value to sort input points of 2-Polytope clockwise.(Optional)
max_degree : The maximum degree of any monomial of the input polynomial.(Optional)
Examples
========
>>> from sympy.abc import x, y
>>> from sympy.geometry.polygon import Polygon
>>> from sympy.geometry.point import Point
>>> from sympy.integrals.intpoly import polytope_integrate
>>> polygon = Polygon(Point(0, 0), Point(0, 1), Point(1, 1), Point(1, 0))
>>> polys = [1, x, y, x*y, x**2*y, x*y**2]
>>> expr = x*y
>>> polytope_integrate(polygon, expr)
1/4
>>> polytope_integrate(polygon, polys, max_degree=3)
{1: 1, x: 1/2, y: 1/2, x*y: 1/4, x*y**2: 1/6, x**2*y: 1/6}
"""
clockwise = kwargs.get('clockwise', False)
max_degree = kwargs.get('max_degree', None)
if clockwise:
if isinstance(poly, Polygon):
poly = Polygon(*point_sort(poly.vertices), evaluate=False)
else:
raise TypeError("clockwise=True works for only 2-Polytope"
"V-representation input")
if isinstance(poly, Polygon):
# For Vertex Representation(2D case)
hp_params = hyperplane_parameters(poly)
facets = poly.sides
elif len(poly[0]) == 2:
# For Hyperplane Representation(2D case)
plen = len(poly)
if len(poly[0][0]) == 2:
intersections = [
intersection(poly[(i - 1) % plen], poly[i], "plane2D")
for i in range(0, plen)
]
hp_params = poly
lints = len(intersections)
facets = [
Segment2D(intersections[i], intersections[(i + 1) % lints])
for i in range(0, lints)
]
else:
raise NotImplementedError("Integration for H-representation 3D"
"case not implemented yet.")
else:
# For Vertex Representation(3D case)
vertices = poly[0]
facets = poly[1:]
hp_params = hyperplane_parameters(facets, vertices)
if max_degree is None:
if expr is None:
raise TypeError('Input expression be must'
'be a valid SymPy expression')
return main_integrate3d(expr, facets, vertices, hp_params)
if max_degree is not None:
result = {}
if not isinstance(expr, list) and expr is not None:
raise TypeError('Input polynomials must be list of expressions')
if len(hp_params[0][0]) == 3:
result_dict = main_integrate3d(0, facets, vertices, hp_params,
max_degree)
else:
result_dict = main_integrate(0, facets, hp_params, max_degree)
if expr is None:
return result_dict
for poly in expr:
if poly not in result:
if poly is S.Zero:
result[S.Zero] = S.Zero
continue
integral_value = S.Zero
monoms = decompose(poly, separate=True)
for monom in monoms:
monom = nsimplify(monom)
coeff, m = strip(monom)
integral_value += result_dict[m] * coeff
result[poly] = integral_value
return result
if expr is None:
raise TypeError('Input expression be must'
'be a valid SymPy expression')
return main_integrate(expr, facets, hp_params)
def doit(self, **hints):
"""Evaluates the limit.
Parameters
==========
deep : bool, optional (default: True)
Invoke the ``doit`` method of the expressions involved before
taking the limit.
hints : optional keyword arguments
To be passed to ``doit`` methods; only used if deep is True.
"""
e, z, z0, dir = self.args
if z0 is S.ComplexInfinity:
raise NotImplementedError("Limits at complex "
"infinity are not implemented")
if hints.get('deep', True):
e = e.doit(**hints)
z = z.doit(**hints)
z0 = z0.doit(**hints)
if e == z:
return z0
if not e.has(z):
return e
if z0 is S.NaN:
return S.NaN
if e.has(*_illegal):
return self
if e.is_Order:
return Order(limit(e.expr, z, z0), *e.args[1:])
cdir = 0
if str(dir) == "+":
cdir = 1
elif str(dir) == "-":
cdir = -1
def set_signs(expr):
if not expr.args:
return expr
newargs = tuple(set_signs(arg) for arg in expr.args)
if newargs != expr.args:
expr = expr.func(*newargs)
abs_flag = isinstance(expr, Abs)
sign_flag = isinstance(expr, sign)
if abs_flag or sign_flag:
sig = limit(expr.args[0], z, z0, dir)
if sig.is_zero:
sig = limit(1 / expr.args[0], z, z0, dir)
if sig.is_extended_real:
if (sig < 0) == True:
return -expr.args[0] if abs_flag else S.NegativeOne
elif (sig > 0) == True:
return expr.args[0] if abs_flag else S.One
return expr
if e.has(Float):
# Convert floats like 0.5 to exact SymPy numbers like S.Half, to
# prevent rounding errors which can lead to unexpected execution
# of conditional blocks that work on comparisons
# Also see comments in https://github.com/sympy/sympy/issues/19453
from sympy.simplify.simplify import nsimplify
e = nsimplify(e)
e = set_signs(e)
if e.is_meromorphic(z, z0):
if abs(z0) is S.Infinity:
newe = e.subs(z, 1 / z)
# cdir changes sign as oo- should become 0+
cdir = -cdir
else:
newe = e.subs(z, z + z0)
try:
coeff, ex = newe.leadterm(z, cdir=cdir)
except ValueError:
pass
else:
if ex > 0:
return S.Zero
elif ex == 0:
return coeff
if cdir == 1 or not (int(ex) & 1):
return S.Infinity * sign(coeff)
elif cdir == -1:
return S.NegativeInfinity * sign(coeff)
else:
return S.ComplexInfinity
if abs(z0) is S.Infinity:
if e.is_Mul:
e = factor_terms(e)
newe = e.subs(z, 1 / z)
# cdir changes sign as oo- should become 0+
cdir = -cdir
else:
newe = e.subs(z, z + z0)
try:
coeff, ex = newe.leadterm(z, cdir=cdir)
except (ValueError, NotImplementedError, PoleError):
# The NotImplementedError catching is for custom functions
from sympy.simplify.powsimp import powsimp
e = powsimp(e)
if e.is_Pow:
r = self.pow_heuristics(e)
if r is not None:
return r
else:
if isinstance(coeff, AccumBounds) and ex == S.Zero:
return coeff
if coeff.has(S.Infinity, S.NegativeInfinity, S.ComplexInfinity,
S.NaN):
return self
if not coeff.has(z):
if ex.is_positive:
return S.Zero
elif ex == 0:
return coeff
elif ex.is_negative:
if ex.is_integer:
if cdir == 1 or ex.is_even:
return S.Infinity * sign(coeff)
elif cdir == -1:
return S.NegativeInfinity * sign(coeff)
else:
return S.ComplexInfinity
else:
if cdir == 1:
return S.Infinity * sign(coeff)
elif cdir == -1:
return S.Infinity * sign(coeff) * S.NegativeOne**ex
else:
return S.ComplexInfinity
# gruntz fails on factorials but works with the gamma function
# If no factorial term is present, e should remain unchanged.
# factorial is defined to be zero for negative inputs (which
# differs from gamma) so only rewrite for positive z0.
if z0.is_extended_positive:
e = e.rewrite(factorial, gamma)
l = None
try:
if str(dir) == '+-':
r = gruntz(e, z, z0, '+')
l = gruntz(e, z, z0, '-')
if l != r:
raise ValueError(
"The limit does not exist since "
"left hand limit = %s and right hand limit = %s" %
(l, r))
else:
r = gruntz(e, z, z0, dir)
if r is S.NaN or l is S.NaN:
raise PoleError()
except (PoleError, ValueError):
if l is not None:
raise
r = heuristics(e, z, z0, dir)
if r is None:
return self
return r
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))**numer(d.exp)
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 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))**numer(d.exp)
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)