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))
def evalf_mul(v, prec, options):
res = pure_complex(v)
if res:
# the only pure complex that is a mul is h*I
_, h = res
im, _, im_acc, _ = evalf(h, prec, options)
return None, im, None, im_acc
args = list(v.args)
# see if any argument is NaN or oo and thus warrants a special return
special = []
from sympy.core.numbers import Float
for arg in args:
arg = evalf(arg, prec, options)
if arg[0] is None:
continue
arg = Float._new(arg[0], 1)
if arg is S.NaN or arg.is_infinite:
special.append(arg)
if special:
from sympy.core.mul import Mul
special = Mul(*special)
return evalf(special, prec + 4, {})
# With guard digits, multiplication in the real case does not destroy
# accuracy. This is also true in the complex case when considering the
# total accuracy; however accuracy for the real or imaginary parts
# separately may be lower.
acc = prec
# XXX: big overestimate
working_prec = prec + len(args) + 5
# Empty product is 1
start = man, exp, bc = MPZ(1), 0, 1
# First, we multiply all pure real or pure imaginary numbers.
# direction tells us that the result should be multiplied by
# I**direction; all other numbers get put into complex_factors
# to be multiplied out after the first phase.
last = len(args)
direction = 0
args.append(S.One)
complex_factors = []
for i, arg in enumerate(args):
if i != last and pure_complex(arg):
args[-1] = (args[-1] * arg).expand()
continue
elif i == last and arg is S.One:
continue
re, im, re_acc, im_acc = evalf(arg, working_prec, options)
if re and im:
complex_factors.append((re, im, re_acc, im_acc))
continue
elif re:
(s, m, e, b), w_acc = re, re_acc
elif im:
(s, m, e, b), w_acc = im, im_acc
direction += 1
else:
return None, None, None, None
direction += 2 * s
man *= m
exp += e
bc += b
if bc > 3 * working_prec:
man >>= working_prec
exp += working_prec
acc = min(acc, w_acc)
sign = (direction & 2) >> 1
if not complex_factors:
v = normalize(sign, man, exp, bitcount(man), prec, rnd)
# multiply by i
if direction & 1:
return None, v, None, acc
else:
return v, None, acc, None
else:
# initialize with the first term
if (man, exp, bc) != start:
# there was a real part; give it an imaginary part
re, im = (sign, man, exp, bitcount(man)), (0, MPZ(0), 0, 0)
i0 = 0
else:
# there is no real part to start (other than the starting 1)
wre, wim, wre_acc, wim_acc = complex_factors[0]
acc = min(acc, complex_accuracy((wre, wim, wre_acc, wim_acc)))
re = wre
im = wim
i0 = 1
for wre, wim, wre_acc, wim_acc in complex_factors[i0:]:
# acc is the overall accuracy of the product; we aren't
# computing exact accuracies of the product.
acc = min(acc, complex_accuracy((wre, wim, wre_acc, wim_acc)))
use_prec = working_prec
A = mpf_mul(re, wre, use_prec)
B = mpf_mul(mpf_neg(im), wim, use_prec)
C = mpf_mul(re, wim, use_prec)
D = mpf_mul(im, wre, use_prec)
re = mpf_add(A, B, use_prec)
im = mpf_add(C, D, use_prec)
if options.get('verbose'):
print("MUL: wanted", prec, "accurate bits, got", acc)
# multiply by I
if direction & 1:
re, im = mpf_neg(im), re
return re, im, acc, acc
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 Float, 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 = Float(v, ndig)
if vold is not None and vold == vf:
break
prec += prec # double precision each time
vold = vf
return v._mpf_
