def _eval_evalf(self, prec):
from sympy.mpmath import mp, workprec
from sympy import Expr
z = self.args[0]._to_mpmath(prec)
with workprec(prec):
res = mp.airybi(z, derivative=1)
return Expr._from_mpmath(res, prec)
def apply(self, z, evaluation):
'%(name)s[z__]'
args = z.get_sequence()
if len(args) != self.nargs:
return
# if no arguments are inexact attempt to use sympy
if all(not x.is_inexact() for x in args):
result = Expression(self.get_name(), *args).to_sympy()
result = self.prepare_mathics(result)
result = from_sympy(result)
# evaluate leaves to convert e.g. Plus[2, I] -> Complex[2, 1]
result = result.evaluate_leaves(evaluation)
else:
prec = min_prec(*args)
with mpmath.workprec(prec):
mpmath_args = [sympy2mpmath(x.to_sympy()) for x in args]
if None in mpmath_args:
return
try:
result = self.eval(*mpmath_args)
result = from_sympy(mpmath2sympy(result, prec))
except ValueError, exc:
text = str(exc)
if text == 'gamma function pole':
return Symbol('ComplexInfinity')
else:
raise
except ZeroDivisionError:
return
except SpecialValueError, exc:
return Symbol(exc.name)
def hypsum(expr, n, start, prec):
"""
Sum a rapidly convergent infinite hypergeometric series with
given general term, e.g. e = hypsum(1/factorial(n), n). The
quotient between successive terms must be a quotient of integer
polynomials.
"""
from sympy import hypersimp, lambdify
if start:
expr = expr.subs(n, n + start)
hs = hypersimp(expr, n)
if hs is None:
raise NotImplementedError("a hypergeometric series is required")
num, den = hs.as_numer_denom()
func1 = lambdify(n, num)
func2 = lambdify(n, den)
h, g, p = check_convergence(num, den, n)
if h < 0:
raise ValueError("Sum diverges like (n!)^%i" % (-h))
# Direct summation if geometric or faster
if h > 0 or (h == 0 and abs(g) > 1):
term = expr.subs(n, 0)
term = (MPZ(term.p) << prec) // term.q
s = term
k = 1
while abs(term) > 5:
term *= MPZ(func1(k - 1))
term //= MPZ(func2(k - 1))
s += term
k += 1
return from_man_exp(s, -prec)
else:
alt = g < 0
if abs(g) < 1:
raise ValueError("Sum diverges like (%i)^n" % abs(1 / g))
if p < 1 or (p == 1 and not alt):
raise ValueError("Sum diverges like n^%i" % (-p))
# We have polynomial convergence: use Richardson extrapolation
# Need to use at least quad precision because a lot of cancellation
# might occur in the extrapolation process
prec2 = 4 * prec
term = expr.subs(n, 0)
term = (MPZ(term.p) << prec2) // term.q
def summand(k, _term=[term]):
if k:
k = int(k)
_term[0] *= MPZ(func1(k - 1))
_term[0] //= MPZ(func2(k - 1))
return make_mpf(from_man_exp(_term[0], -prec2))
with workprec(prec):
v = nsum(summand, [0, mpmath_inf], method="richardson")
return v._mpf_
def _eval_evalf(self, prec):
from sympy.mpmath import mp, workprec
from sympy import Expr
z = self.args[0]._to_mpmath(prec)
with workprec(prec):
res = mp.airybi(z, derivative=1)
return Expr._from_mpmath(res, prec)
def hypsum(expr, n, start, prec):
"""
Sum a rapidly convergent infinite hypergeometric series with
given general term, e.g. e = hypsum(1/factorial(n), n). The
quotient between successive terms must be a quotient of integer
polynomials.
"""
from sympy import hypersimp, lambdify
if start:
expr = expr.subs(n, n + start)
hs = hypersimp(expr, n)
if hs is None:
raise NotImplementedError("a hypergeometric series is required")
num, den = hs.as_numer_denom()
func1 = lambdify(n, num)
func2 = lambdify(n, den)
h, g, p = check_convergence(num, den, n)
if h < 0:
raise ValueError("Sum diverges like (n!)^%i" % (-h))
# Direct summation if geometric or faster
if h > 0 or (h == 0 and abs(g) > 1):
term = expr.subs(n, 0)
term = (MPZ(term.p) << prec) // term.q
s = term
k = 1
while abs(term) > 5:
term *= MPZ(func1(k - 1))
term //= MPZ(func2(k - 1))
s += term
k += 1
return from_man_exp(s, -prec)
else:
alt = g < 0
if abs(g) < 1:
raise ValueError("Sum diverges like (%i)^n" % abs(1/g))
if p < 1 or (p == 1 and not alt):
raise ValueError("Sum diverges like n^%i" % (-p))
# We have polynomial convergence: use Richardson extrapolation
# Need to use at least quad precision because a lot of cancellation
# might occur in the extrapolation process
prec2 = 4*prec
term = expr.subs(n, 0)
term = (MPZ(term.p) << prec2) // term.q
def summand(k, _term=[term]):
if k:
k = int(k)
_term[0] *= MPZ(func1(k - 1))
_term[0] //= MPZ(func2(k - 1))
return make_mpf(from_man_exp(_term[0], -prec2))
with workprec(prec):
v = nsum(summand, [0, mpmath_inf], method='richardson')
return v._mpf_
def apply(self, z, evaluation):
'%(name)s[z__]'
args = z.get_sequence()
if len(args) != self.nargs:
return
# if no arguments are inexact attempt to use sympy
if all(not x.is_inexact() for x in args):
result = Expression(self.get_name(), *args).to_sympy()
result = self.prepare_mathics(result)
result = from_sympy(result)
# evaluate leaves to convert e.g. Plus[2, I] -> Complex[2, 1]
result = result.evaluate_leaves(evaluation)
else:
prec = min_prec(*args)
with mpmath.workprec(prec):
mpmath_args = [sympy2mpmath(x.to_sympy()) for x in args]
if None in mpmath_args:
return
try:
result = self.eval(*mpmath_args)
result = from_sympy(mpmath2sympy(result, prec))
except ValueError, exc:
text = str(exc)
if text == 'gamma function pole':
return Symbol('ComplexInfinity')
else:
raise
except ZeroDivisionError:
return
except SpecialValueError, exc:
return Symbol(exc.name)
def _eval_evalf(self, prec):
# The default code is insufficient for polar arguments.
# mpmath provides an optional argument "r", which evaluates
# G(z**(1/r)). I am not sure what its intended use is, but we hijack it
# here in the following way: to evaluate at a number z of |argument|
# less than (say) n*pi, we put r=1/n, compute z' = root(z, n)
# (carefully so as not to loose the branch information), and evaluate
# G(z'**(1/r)) = G(z'**n) = G(z).
from sympy.functions import exp_polar, ceiling
from sympy import mpmath, Expr
z = self.argument
znum = self.argument._eval_evalf(prec)
if znum.has(exp_polar):
znum, branch = znum.as_coeff_mul(exp_polar)
if len(branch) != 1:
return
branch = branch[0].args[0]/I
else:
branch = S(0)
n = ceiling(abs(branch/S.Pi)) + 1
znum = znum**(S(1)/n)*exp(I*branch / n)
#print znum, branch, n
# Convert all args to mpf or mpc
try:
[z, r, ap, bq] = [arg._to_mpmath(prec)
for arg in [znum, 1/n, self.args[0], self.args[1]]]
except ValueError:
return
with mpmath.workprec(prec):
v = mpmath.meijerg(ap, bq, z, r)
return Expr._from_mpmath(v, prec)
def _eval_evalf(self, prec):
from sympy.mpmath import mp, workprec
from sympy import Expr
a = self.args[0]._to_mpmath(prec)
z = self.args[1]._to_mpmath(prec)
with workprec(prec):
res = mp.gammainc(a, z, mp.inf)
return Expr._from_mpmath(res, prec)
def _eval_evalf(self, prec):
from sympy.mpmath import mp, workprec
from sympy import Expr
a = self.args[0]._to_mpmath(prec)
z = self.args[1]._to_mpmath(prec)
with workprec(prec):
res = mp.gammainc(a, z, mp.inf)
return Expr._from_mpmath(res, prec)
def _eval_evalf(self, prec):
"""Evaluate this complex root to the given precision. """
with workprec(prec):
g = self.poly.gen
if not g.is_Symbol:
d = Dummy('x')
func = lambdify(d, self.expr.subs(g, d))
else:
func = lambdify(g, self.expr)
interval = self._get_interval()
if not self.is_real:
# For complex intervals, we need to keep refining until the
# imaginary interval is disjunct with other roots, that is,
# until both ends get refined.
ay = interval.ay
by = interval.by
while interval.ay == ay or interval.by == by:
interval = interval.refine()
while True:
if self.is_real:
x0 = mpf(str(interval.center))
else:
x0 = mpc(*map(str, interval.center))
try:
root = findroot(func, x0, verify=False)
# If the (real or complex) root is not in the 'interval',
# then keep refining the interval. This happens if findroot
# accidentally finds a different root outside of this
# interval because our initial estimate 'x0' was not close
# enough.
if self.is_real:
a = mpf(str(interval.a))
b = mpf(str(interval.b))
if a == b:
root = a
break
if not (a < root < b):
raise ValueError("Root not in the interval.")
else:
ax = mpf(str(interval.ax))
bx = mpf(str(interval.bx))
ay = mpf(str(interval.ay))
by = mpf(str(interval.by))
if ax == bx and ay == by:
root = ax + S.ImaginaryUnit * by
break
if not (ax < root.real < bx and ay < root.imag < by):
raise ValueError("Root not in the interval.")
except ValueError:
interval = interval.refine()
continue
else:
break
return Float._new(root.real._mpf_,
prec) + I * Float._new(root.imag._mpf_, prec)
def _eval_evalf(self, prec):
"""Evaluate this complex root to the given precision. """
with workprec(prec):
g = self.poly.gen
if not g.is_Symbol:
d = Dummy('x')
func = lambdify(d, self.expr.subs(g, d))
else:
func = lambdify(g, self.expr)
interval = self._get_interval()
if not self.is_real:
# For complex intervals, we need to keep refining until the
# imaginary interval is disjunct with other roots, that is,
# until both ends get refined.
ay = interval.ay
by = interval.by
while interval.ay == ay or interval.by == by:
interval = interval.refine()
while True:
if self.is_real:
x0 = mpf(str(interval.center))
else:
x0 = mpc(*map(str, interval.center))
try:
root = findroot(func, x0, verify=False)
# If the (real or complex) root is not in the 'interval',
# then keep refining the interval. This happens if findroot
# accidentally finds a different root outside of this
# interval because our initial estimate 'x0' was not close
# enough.
if self.is_real:
a = mpf(str(interval.a))
b = mpf(str(interval.b))
if a == b:
root = a
break
if not (a < root < b):
raise ValueError("Root not in the interval.")
else:
ax = mpf(str(interval.ax))
bx = mpf(str(interval.bx))
ay = mpf(str(interval.ay))
by = mpf(str(interval.by))
if ax == bx and ay == by:
root = ax + S.ImaginaryUnit*by
break
if not (ax < root.real < bx and ay < root.imag < by):
raise ValueError("Root not in the interval.")
except ValueError:
interval = interval.refine()
continue
else:
break
return Float._new(root.real._mpf_, prec) + I*Float._new(root.imag._mpf_, prec)
def apply_N(self, k, precision, evaluation):
'N[AiryBiZero[k_Integer], precision_]'
prec = get_precision(precision, evaluation)
k_int = k.get_int_value()
with mpmath.workprec(prec):
result = mpmath2sympy(mpmath.airybizero(k_int), prec)
return from_sympy(result)
def apply_N(self, k, precision, evaluation):
'N[AiryBiZero[k_Integer], precision_]'
prec = get_precision(precision, evaluation)
k_int = k.get_int_value()
with mpmath.workprec(prec):
result = mpmath2sympy(mpmath.airybizero(k_int), prec)
return from_sympy(result)
def _eval_evalf(self, prec):
m = self.args[0]
if m.is_Integer and m.is_nonnegative:
from sympy.mpmath import mp
from sympy import Expr
m = m._to_mpmath(prec)
with workprec(prec):
res = mp.eulernum(m)
return Expr._from_mpmath(res, prec)
def _eval_evalf(self, prec):
m = self.args[0]
if m.is_Integer and m.is_nonnegative:
from sympy.mpmath import mp
from sympy import Expr
m = m._to_mpmath(prec)
with workprec(prec):
res = mp.eulernum(m)
return Expr._from_mpmath(res, prec)
def _eval_evalf(self, prec):
# Note: works without this function by just calling
# mpmath for Legendre polynomials. But using
# the dedicated function directly is cleaner.
from sympy.mpmath import mp, workprec
from sympy import Expr
n = self.args[0]._to_mpmath(prec)
m = self.args[1]._to_mpmath(prec)
theta = self.args[2]._to_mpmath(prec)
phi = self.args[3]._to_mpmath(prec)
with workprec(prec):
res = mp.spherharm(n, m, theta, phi)
return Expr._from_mpmath(res, prec)
def _eval_evalf(self, prec):
# Note: works without this function by just calling
# mpmath for Legendre polynomials. But using
# the dedicated function directly is cleaner.
from sympy.mpmath import mp, workprec
from sympy import Expr
n = self.args[0]._to_mpmath(prec)
m = self.args[1]._to_mpmath(prec)
theta = self.args[2]._to_mpmath(prec)
phi = self.args[3]._to_mpmath(prec)
with workprec(prec):
res = mp.spherharm(n, m, theta, phi)
return Expr._from_mpmath(res, prec)
def _eval_evalf(self, prec):
# The default code is insufficient for polar arguments.
# mpmath provides an optional argument "r", which evaluates
# G(z**(1/r)). I am not sure what its intended use is, but we hijack it
# here in the following way: to evaluate at a number z of |argument|
# less than (say) n*pi, we put r=1/n, compute z' = root(z, n)
# (carefully so as not to loose the branch information), and evaluate
# G(z'**(1/r)) = G(z'**n) = G(z).
from sympy.functions import exp_polar, ceiling
from sympy import mpmath, Expr
z = self.argument
znum = self.argument._eval_evalf(prec)
if znum.has(exp_polar):
znum, branch = znum.as_coeff_mul(exp_polar)
if len(branch) != 1:
return
branch = branch[0].args[0] / I
else:
branch = S(0)
n = ceiling(abs(branch / S.Pi)) + 1
znum = znum**(S(1) / n) * exp(I * branch / n)
#print znum, branch, n
# Convert all args to mpf or mpc
try:
[z, r, ap, bq] = [
arg._to_mpmath(prec)
for arg in [znum, 1 / n, self.args[0], self.args[1]]
]
except ValueError:
return
with mpmath.workprec(prec):
v = mpmath.meijerg(ap, bq, z, r)
return Expr._from_mpmath(v, prec)
def do_integral(expr, prec, options):
func = expr.args[0]
x, xlow, xhigh = expr.args[1]
if xlow == xhigh:
xlow = xhigh = 0
elif x not in func.free_symbols:
# only the difference in limits matters in this case
# so if there is a symbol in common that will cancel
# out when taking the difference, then use that
# difference
if xhigh.free_symbols & xlow.free_symbols:
diff = xhigh - xlow
if not diff.free_symbols:
xlow, xhigh = 0, diff
oldmaxprec = options.get('maxprec', DEFAULT_MAXPREC)
options['maxprec'] = min(oldmaxprec, 2*prec)
with workprec(prec + 5):
xlow = as_mpmath(xlow, prec + 15, options)
xhigh = as_mpmath(xhigh, prec + 15, options)
# Integration is like summation, and we can phone home from
# the integrand function to update accuracy summation style
# Note that this accuracy is inaccurate, since it fails
# to account for the variable quadrature weights,
# but it is better than nothing
have_part = [False, False]
max_real_term = [MINUS_INF]
max_imag_term = [MINUS_INF]
def f(t):
re, im, re_acc, im_acc = evalf(func, mp.prec, {'subs': {x: t}})
have_part[0] = re or have_part[0]
have_part[1] = im or have_part[1]
max_real_term[0] = max(max_real_term[0], fastlog(re))
max_imag_term[0] = max(max_imag_term[0], fastlog(im))
if im:
return mpc(re or fzero, im)
return mpf(re or fzero)
if options.get('quad') == 'osc':
A = C.Wild('A', exclude=[x])
B = C.Wild('B', exclude=[x])
D = C.Wild('D')
m = func.match(C.cos(A*x + B)*D)
if not m:
m = func.match(C.sin(A*x + B)*D)
if not m:
raise ValueError("An integrand of the form sin(A*x+B)*f(x) "
"or cos(A*x+B)*f(x) is required for oscillatory quadrature")
period = as_mpmath(2*S.Pi/m[A], prec + 15, options)
result = quadosc(f, [xlow, xhigh], period=period)
# XXX: quadosc does not do error detection yet
quadrature_error = MINUS_INF
else:
result, quadrature_error = quadts(f, [xlow, xhigh], error=1)
quadrature_error = fastlog(quadrature_error._mpf_)
options['maxprec'] = oldmaxprec
if have_part[0]:
re = result.real._mpf_
if re == fzero:
re, re_acc = scaled_zero(
min(-prec, -max_real_term[0], -quadrature_error))
re = scaled_zero(re) # handled ok in evalf_integral
else:
re_acc = -max(max_real_term[0] - fastlog(re) -
prec, quadrature_error)
else:
re, re_acc = None, None
if have_part[1]:
im = result.imag._mpf_
if im == fzero:
im, im_acc = scaled_zero(
min(-prec, -max_imag_term[0], -quadrature_error))
im = scaled_zero(im) # handled ok in evalf_integral
else:
im_acc = -max(max_imag_term[0] - fastlog(im) -
prec, quadrature_error)
else:
im, im_acc = None, None
result = re, im, re_acc, im_acc
return result
def apply(self, items, evaluation):
'Power[items__]'
items_sequence = items.get_sequence()
if len(items_sequence) == 2:
x, y = items_sequence
else:
return Expression('Power', *items_sequence)
if y.get_int_value() == 1:
return x
elif x.get_int_value() == 1:
return x
elif y.get_int_value() == 0:
if x.get_int_value() == 0:
evaluation.message('Power', 'indet', Expression('Power', x, y))
return Symbol('Indeterminate')
else:
return Integer(1)
elif x.has_form('Power', 2) and isinstance(y, Integer):
return Expression('Power', x.leaves[0],
Expression('Times', x.leaves[1], y))
elif x.has_form('Times', None) and isinstance(y, Integer):
return Expression('Times', *[
Expression('Power', leaf, y) for leaf in x.leaves])
elif (isinstance(x, Number) and isinstance(y, Number)
and not (x.is_inexact() or y.is_inexact())):
sym_x, sym_y = x.to_sympy(), y.to_sympy()
try:
if sym_y >= 0:
result = sym_x ** sym_y
else:
if sym_x == 0:
evaluation.message('Power', 'infy')
return Symbol('ComplexInfinity')
result = sympy.Integer(1) / (sym_x ** (-sym_y))
if isinstance(result, sympy.Pow):
result = result.simplify()
args = [from_sympy(expr) for expr in result.as_base_exp()]
result = Expression('Power', *args)
result = result.evaluate_leaves(evaluation)
return result
return from_sympy(result)
except ValueError:
return Expression('Power', x, y)
except ZeroDivisionError:
evaluation.message('Power', 'infy')
return Symbol('ComplexInfinity')
elif (isinstance(x, Number) and isinstance(y, Number)
and (x.is_inexact() or y.is_inexact())):
try:
prec = min(max(x.get_precision(), 64), max(
y.get_precision(), 64))
with mpmath.workprec(prec):
mp_x = sympy2mpmath(x.to_sympy())
mp_y = sympy2mpmath(y.to_sympy())
result = mp_x ** mp_y
if isinstance(result, mpmath.mpf):
return Real(str(result), prec)
elif isinstance(result, mpmath.mpc):
return Complex(str(result.real),
str(result.imag), prec)
except ZeroDivisionError:
evaluation.message('Power', 'infy')
return Symbol('ComplexInfinity')
else:
numerified_items = items.numerify(evaluation)
return Expression('Power', *numerified_items.get_sequence())
def hypsum(expr, n, start, prec):
"""
Sum a rapidly convergent infinite hypergeometric series with
given general term, e.g. e = hypsum(1/factorial(n), n). The
quotient between successive terms must be a quotient of integer
polynomials.
"""
from sympy import hypersimp, lambdify
if prec == float('inf'):
raise NotImplementedError('does not support inf prec')
if start:
expr = expr.subs(n, n + start)
hs = hypersimp(expr, n)
if hs is None:
raise NotImplementedError("a hypergeometric series is required")
num, den = hs.as_numer_denom()
func1 = lambdify(n, num)
func2 = lambdify(n, den)
h, g, p = check_convergence(num, den, n)
if h < 0:
raise ValueError("Sum diverges like (n!)^%i" % (-h))
term = expr.subs(n, 0)
if not term.is_Rational:
raise NotImplementedError("Non rational term functionality is not implemented.")
# Direct summation if geometric or faster
if h > 0 or (h == 0 and abs(g) > 1):
term = (MPZ(term.p) << prec) // term.q
s = term
k = 1
while abs(term) > 5:
term *= MPZ(func1(k - 1))
term //= MPZ(func2(k - 1))
s += term
k += 1
return from_man_exp(s, -prec)
else:
alt = g < 0
if abs(g) < 1:
raise ValueError("Sum diverges like (%i)^n" % abs(1/g))
if p < 1 or (p == 1 and not alt):
raise ValueError("Sum diverges like n^%i" % (-p))
# We have polynomial convergence: use Richardson extrapolation
vold = None
ndig = prec_to_dps(prec)
while True:
# Need to use at least quad precision because a lot of cancellation
# might occur in the extrapolation process; we check the answer to
# make sure that the desired precision has been reached, too.
prec2 = 4*prec
term0 = (MPZ(term.p) << prec2) // term.q
def summand(k, _term=[term0]):
if k:
k = int(k)
_term[0] *= MPZ(func1(k - 1))
_term[0] //= MPZ(func2(k - 1))
return make_mpf(from_man_exp(_term[0], -prec2))
with workprec(prec):
v = nsum(summand, [0, mpmath_inf], method='richardson')
vf = C.Float(v, ndig)
if vold is not None and vold == vf:
break
prec += prec # double precision each time
vold = vf
return v._mpf_
def apply(self, items, evaluation):
'Power[items__]'
items_sequence = items.get_sequence()
if len(items_sequence) == 2:
x, y = items_sequence
else:
return Expression('Power', *items_sequence)
if y.get_int_value() == 1:
return x
elif x.get_int_value() == 1:
return x
elif y.get_int_value() == 0:
if x.get_int_value() == 0:
evaluation.message('Power', 'indet', Expression('Power', x, y))
return Symbol('Indeterminate')
else:
return Integer(1)
elif x.has_form('Power', 2) and isinstance(y, Integer):
return Expression('Power', x.leaves[0],
Expression('Times', x.leaves[1], y))
elif x.has_form('Times', None) and isinstance(y, Integer):
return Expression(
'Times', *[Expression('Power', leaf, y) for leaf in x.leaves])
elif (isinstance(x, Number) and isinstance(y, Number)
and not (x.is_inexact() or y.is_inexact())):
sym_x, sym_y = x.to_sympy(), y.to_sympy()
try:
if sym_y >= 0:
result = sym_x**sym_y
else:
if sym_x == 0:
evaluation.message('Power', 'infy')
return Symbol('ComplexInfinity')
result = sympy.Integer(1) / (sym_x**(-sym_y))
if isinstance(result, sympy.Pow):
result = result.simplify()
args = [from_sympy(expr) for expr in result.as_base_exp()]
result = Expression('Power', *args)
result = result.evaluate_leaves(evaluation)
return result
return from_sympy(result)
except ValueError:
return Expression('Power', x, y)
except ZeroDivisionError:
evaluation.message('Power', 'infy')
return Symbol('ComplexInfinity')
elif (isinstance(x, Number) and isinstance(y, Number)
and (x.is_inexact() or y.is_inexact())):
try:
prec = min(max(x.get_precision(), 64),
max(y.get_precision(), 64))
with mpmath.workprec(prec):
mp_x = sympy2mpmath(x.to_sympy())
mp_y = sympy2mpmath(y.to_sympy())
result = mp_x**mp_y
if isinstance(result, mpmath.mpf):
return Real(str(result), prec)
elif isinstance(result, mpmath.mpc):
return Complex(str(result.real), str(result.imag),
prec)
except ZeroDivisionError:
evaluation.message('Power', 'infy')
return Symbol('ComplexInfinity')
else:
numerified_items = items.numerify(evaluation)
return Expression('Power', *numerified_items.get_sequence())
def do_integral(expr, prec, options):
func = expr.args[0]
x, xlow, xhigh = expr.args[1]
if xlow == xhigh:
xlow = xhigh = 0
elif x not in func.free_symbols:
# only the difference in limits matters in this case
# so if there is a symbol in common that will cancel
# out when taking the difference, then use that
# difference
if xhigh.free_symbols & xlow.free_symbols:
diff = xhigh - xlow
if not diff.free_symbols:
xlow, xhigh = 0, diff
oldmaxprec = options.get('maxprec', DEFAULT_MAXPREC)
options['maxprec'] = min(oldmaxprec, 2*prec)
with workprec(prec + 5):
xlow = as_mpmath(xlow, prec + 15, options)
xhigh = as_mpmath(xhigh, prec + 15, options)
# Integration is like summation, and we can phone home from
# the integrand function to update accuracy summation style
# Note that this accuracy is inaccurate, since it fails
# to account for the variable quadrature weights,
# but it is better than nothing
have_part = [False, False]
max_real_term = [MINUS_INF]
max_imag_term = [MINUS_INF]
def f(t):
re, im, re_acc, im_acc = evalf(func, mp.prec, {'subs': {x: t}})
have_part[0] = re or have_part[0]
have_part[1] = im or have_part[1]
max_real_term[0] = max(max_real_term[0], fastlog(re))
max_imag_term[0] = max(max_imag_term[0], fastlog(im))
if im:
return mpc(re or fzero, im)
return mpf(re or fzero)
if options.get('quad') == 'osc':
A = C.Wild('A', exclude=[x])
B = C.Wild('B', exclude=[x])
D = C.Wild('D')
m = func.match(C.cos(A*x + B)*D)
if not m:
m = func.match(C.sin(A*x + B)*D)
if not m:
raise ValueError("An integrand of the form sin(A*x+B)*f(x) "
"or cos(A*x+B)*f(x) is required for oscillatory quadrature")
period = as_mpmath(2*S.Pi/m[A], prec + 15, options)
result = quadosc(f, [xlow, xhigh], period=period)
# XXX: quadosc does not do error detection yet
quadrature_error = MINUS_INF
else:
result, quadrature_error = quadts(f, [xlow, xhigh], error=1)
quadrature_error = fastlog(quadrature_error._mpf_)
options['maxprec'] = oldmaxprec
if have_part[0]:
re = result.real._mpf_
if re == fzero:
re, re_acc = scaled_zero(
min(-prec, -max_real_term[0], -quadrature_error))
re = scaled_zero(re) # handled ok in evalf_integral
else:
re_acc = -max(max_real_term[0] - fastlog(re) -
prec, quadrature_error)
else:
re, re_acc = None, None
if have_part[1]:
im = result.imag._mpf_
if im == fzero:
im, im_acc = scaled_zero(
min(-prec, -max_imag_term[0], -quadrature_error))
im = scaled_zero(im) # handled ok in evalf_integral
else:
im_acc = -max(max_imag_term[0] - fastlog(im) -
prec, quadrature_error)
else:
im, im_acc = None, None
result = re, im, re_acc, im_acc
return result