def A(n, j, prec):
"""Compute the inner sum in the HRR formula."""
if j == 1:
return fone
s = fzero
pi = pi_fixed(prec)
for h in xrange(1, j):
if igcd(h,j) != 1:
continue
# & with mask to compute fractional part of fixed-point number
one = 1 << prec
onemask = one - 1
half = one >> 1
g = 0
if j >= 3:
for k in xrange(1, j):
t = h*k*one//j
if t > 0: frac = t & onemask
else: frac = -((-t) & onemask)
g += k*(frac - half)
g = ((g - 2*h*n*one)*pi//j) >> prec
s = mpf_add(s, mpf_cos(from_man_exp(g, -prec), prec), prec)
return s
def _as_mpf_val(self, prec):
return mlib.from_man_exp(mpmath.gammazeta.catalan_fixed(prec+10), -prec-10)
def _as_mpf_val(self, prec):
return mlib.from_man_exp(phi_fixed(prec+10), -prec-10)
def summand(k, _term=[term]):
if k:
k = int(k)
_term[0] *= MP_BASE(func1(k-1))
_term[0] //= MP_BASE(func2(k-1))
return make_mpf(from_man_exp(_term[0], -prec2))
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 Richardson extrapolation
# Need to use at least quad precision because a lot of cancellation
# might occur in the extrapolation process
prec2 = 4*prec
one = MP_BASE(1) << prec2
term = expr.subs(n, 0)
term = (MP_BASE(term.p) << prec2) // term.q
def summand(k, _term=[term]):
if k:
k = int(k)
_term[0] *= MP_BASE(func1(k-1))
_term[0] //= MP_BASE(func2(k-1))
return make_mpf(from_man_exp(_term[0], -prec2))
orig = mp.prec
try:
mp.prec = prec
v = nsum(summand, [0, mpmath_inf], method='richardson')
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_
def _as_mpf_val(self, prec):
return mlib.from_man_exp(mpmath.gammazeta.catalan_fixed(prec+10), -prec-10)
def _as_mpf_val(self, prec):
return mlib.from_man_exp(phi_fixed(prec+10), -prec-10)
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_