def _to_mpmath(self, prec, allow_ints=True):
# mpmath functions accept ints as input
errmsg = "cannot convert to mpmath number"
if allow_ints and self.is_Integer:
return self.p
try:
re, im, _, _ = evalf(self, prec, {})
if im:
return make_mpc((re, im))
else:
return make_mpf(re)
except NotImplementedError:
v = self._eval_evalf(prec)
if v is None:
raise ValueError(errmsg)
if v.is_Float:
return make_mpf(v._mpf_)
# Number + Number*I is also fine
re, im = v.as_real_imag()
if allow_ints and re.is_Integer:
re = from_int(re.p)
elif re.is_Float:
re = re._mpf_
else:
raise ValueError(errmsg)
if allow_ints and im.is_Integer:
im = from_int(im.p)
elif im.is_Float:
im = im._mpf_
else:
raise ValueError(errmsg)
return make_mpc((re, im))
def _to_mpmath(self, prec, allow_ints=True):
# mpmath functions accept ints as input
errmsg = "cannot convert to mpmath number"
if allow_ints and self.is_Integer:
return self.p
if hasattr(self, '_as_mpf_val'):
return make_mpf(self._as_mpf_val(prec))
try:
re, im, _, _ = evalf(self, prec, {})
if im:
if not re:
re = fzero
return make_mpc((re, im))
elif re:
return make_mpf(re)
else:
return make_mpf(fzero)
except NotImplementedError:
v = self._eval_evalf(prec)
if v is None:
raise ValueError(errmsg)
if v.is_Float:
return make_mpf(v._mpf_)
# Number + Number*I is also fine
re, im = v.as_real_imag()
if allow_ints and re.is_Integer:
re = from_int(re.p)
elif re.is_Float:
re = re._mpf_
else:
raise ValueError(errmsg)
if allow_ints and im.is_Integer:
im = from_int(im.p)
elif im.is_Float:
im = im._mpf_
else:
raise ValueError(errmsg)
return make_mpc((re, im))
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))
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))
def _mpmath_(self, prec, rnd):
return mpmath.make_mpf(self._as_mpf_val(prec))
def _mpmath_(self, prec, rnd):
return mpmath.make_mpf(mlib.from_rational(self.p, self.q, prec, rnd))
def _mpmath_(self, prec, rnd):
return mpmath.make_mpf(self._as_mpf_val(prec))
def _mpmath_(self, prec, rnd):
return mpmath.make_mpf(mlib.from_rational(self.p, self.q, prec, rnd))
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):
one = MP_BASE(1) << prec
term = expr.subs(n, 0)
term = (MP_BASE(term.p) << prec) // term.q
s = term
k = 1
while abs(term) > 5:
term *= MP_BASE(func1(k - 1))
term //= MP_BASE(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 Shanks extrapolation for alternating series,
# Richardson extrapolation for nonalternating series
if alt:
# XXX: better parameters for Shanks transformation
# This tends to get bad somewhere > 50 digits
N = 5 + int(prec * 0.36)
M = 2 + N // 3
NTERMS = M + N + 2
else:
N = 3 + int(prec * 0.15)
M = 2 * N
NTERMS = M + N + 2
# Need to use at least double precision because a lot of cancellation
# might occur in the extrapolation process
prec2 = 2 * prec
one = MP_BASE(1) << prec2
term = expr.subs(n, 0)
term = (MP_BASE(term.p) << prec2) // term.q
s = term
table = [make_mpf(from_man_exp(s, -prec2))]
for k in xrange(1, NTERMS):
term *= MP_BASE(func1(k - 1))
term //= MP_BASE(func2(k - 1))
s += term
table.append(make_mpf(from_man_exp(s, -prec2)))
k += 1
orig = mp.prec
try:
mp.prec = prec
if alt:
v = shanks_extrapolation(table, N, M)
else:
v = richardson_extrapolation(table, N, M)
finally:
mp.prec = orig
return v._mpf_
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):
one = MP_BASE(1) << prec
term = expr.subs(n, 0)
term = (MP_BASE(term.p) << prec) // term.q
s = term
k = 1
while abs(term) > 5:
term *= MP_BASE(func1(k-1))
term //= MP_BASE(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 Shanks extrapolation for alternating series,
# Richardson extrapolation for nonalternating series
if alt:
# XXX: better parameters for Shanks transformation
# This tends to get bad somewhere > 50 digits
N = 5 + int(prec*0.36)
M = 2 + N//3
NTERMS = M + N + 2
else:
N = 3 + int(prec*0.15)
M = 2*N
NTERMS = M + N + 2
# Need to use at least double precision because a lot of cancellation
# might occur in the extrapolation process
prec2 = 2*prec
one = MP_BASE(1) << prec2
term = expr.subs(n, 0)
term = (MP_BASE(term.p) << prec2) // term.q
s = term
table = [make_mpf(from_man_exp(s, -prec2))]
for k in xrange(1, NTERMS):
term *= MP_BASE(func1(k-1))
term //= MP_BASE(func2(k-1))
s += term
table.append(make_mpf(from_man_exp(s, -prec2)))
k += 1
orig = mp.prec
try:
mp.prec = prec
if alt:
v = shanks_extrapolation(table, N, M)
else:
v = richardson_extrapolation(table, N, M)
finally:
mp.prec = orig
return v._mpf_