/usr/lib/python2.7/dist-packages/mpmath/libmp/gammazeta.py is in python-mpmath 0.19-3.
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-----------------------------------------------------------------------
This module implements gamma- and zeta-related functions:
* Bernoulli numbers
* Factorials
* The gamma function
* Polygamma functions
* Harmonic numbers
* The Riemann zeta function
* Constants related to these functions
-----------------------------------------------------------------------
"""
import math
from .backend import xrange
from .backend import MPZ, MPZ_ZERO, MPZ_ONE, MPZ_THREE, gmpy
from .libintmath import list_primes, ifac, ifac2, moebius
from .libmpf import (\
round_floor, round_ceiling, round_down, round_up,
round_nearest, round_fast,
lshift, sqrt_fixed, isqrt_fast,
fzero, fone, fnone, fhalf, ftwo, finf, fninf, fnan,
from_int, to_int, to_fixed, from_man_exp, from_rational,
mpf_pos, mpf_neg, mpf_abs, mpf_add, mpf_sub,
mpf_mul, mpf_mul_int, mpf_div, mpf_sqrt, mpf_pow_int,
mpf_rdiv_int,
mpf_perturb, mpf_le, mpf_lt, mpf_gt, mpf_shift,
negative_rnd, reciprocal_rnd,
bitcount, to_float, mpf_floor, mpf_sign, ComplexResult
)
from .libelefun import (\
constant_memo,
def_mpf_constant,
mpf_pi, pi_fixed, ln2_fixed, log_int_fixed, mpf_ln2,
mpf_exp, mpf_log, mpf_pow, mpf_cosh,
mpf_cos_sin, mpf_cosh_sinh, mpf_cos_sin_pi, mpf_cos_pi, mpf_sin_pi,
ln_sqrt2pi_fixed, mpf_ln_sqrt2pi, sqrtpi_fixed, mpf_sqrtpi,
cos_sin_fixed, exp_fixed
)
from .libmpc import (\
mpc_zero, mpc_one, mpc_half, mpc_two,
mpc_abs, mpc_shift, mpc_pos, mpc_neg,
mpc_add, mpc_sub, mpc_mul, mpc_div,
mpc_add_mpf, mpc_mul_mpf, mpc_div_mpf, mpc_mpf_div,
mpc_mul_int, mpc_pow_int,
mpc_log, mpc_exp, mpc_pow,
mpc_cos_pi, mpc_sin_pi,
mpc_reciprocal, mpc_square,
mpc_sub_mpf
)
# Catalan's constant is computed using Lupas's rapidly convergent series
# (listed on http://mathworld.wolfram.com/CatalansConstant.html)
# oo
# ___ n-1 8n 2 3 2
# 1 \ (-1) 2 (40n - 24n + 3) [(2n)!] (n!)
# K = --- ) -----------------------------------------
# 64 /___ 3 2
# n (2n-1) [(4n)!]
# n = 1
@constant_memo
def catalan_fixed(prec):
prec = prec + 20
a = one = MPZ_ONE << prec
s, t, n = 0, 1, 1
while t:
a *= 32 * n**3 * (2*n-1)
a //= (3-16*n+16*n**2)**2
t = a * (-1)**(n-1) * (40*n**2-24*n+3) // (n**3 * (2*n-1))
s += t
n += 1
return s >> (20 + 6)
# Khinchin's constant is relatively difficult to compute. Here
# we use the rational zeta series
# oo 2*n-1
# ___ ___
# \ ` zeta(2*n)-1 \ ` (-1)^(k+1)
# log(K)*log(2) = ) ------------ ) ----------
# /___. n /___. k
# n = 1 k = 1
# which adds half a digit per term. The essential trick for achieving
# reasonable efficiency is to recycle both the values of the zeta
# function (essentially Bernoulli numbers) and the partial terms of
# the inner sum.
# An alternative might be to use K = 2*exp[1/log(2) X] where
# / 1 1 [ pi*x*(1-x^2) ]
# X = | ------ log [ ------------ ].
# / 0 x(1+x) [ sin(pi*x) ]
# and integrate numerically. In practice, this seems to be slightly
# slower than the zeta series at high precision.
@constant_memo
def khinchin_fixed(prec):
wp = int(prec + prec**0.5 + 15)
s = MPZ_ZERO
fac = from_int(4)
t = ONE = MPZ_ONE << wp
pi = mpf_pi(wp)
pipow = twopi2 = mpf_shift(mpf_mul(pi, pi, wp), 2)
n = 1
while 1:
zeta2n = mpf_abs(mpf_bernoulli(2*n, wp))
zeta2n = mpf_mul(zeta2n, pipow, wp)
zeta2n = mpf_div(zeta2n, fac, wp)
zeta2n = to_fixed(zeta2n, wp)
term = (((zeta2n - ONE) * t) // n) >> wp
if term < 100:
break
#if not n % 10:
# print n, math.log(int(abs(term)))
s += term
t += ONE//(2*n+1) - ONE//(2*n)
n += 1
fac = mpf_mul_int(fac, (2*n)*(2*n-1), wp)
pipow = mpf_mul(pipow, twopi2, wp)
s = (s << wp) // ln2_fixed(wp)
K = mpf_exp(from_man_exp(s, -wp), wp)
K = to_fixed(K, prec)
return K
# Glaisher's constant is defined as A = exp(1/2 - zeta'(-1)).
# One way to compute it would be to perform direct numerical
# differentiation, but computing arbitrary Riemann zeta function
# values at high precision is expensive. We instead use the formula
# A = exp((6 (-zeta'(2))/pi^2 + log 2 pi + gamma)/12)
# and compute zeta'(2) from the series representation
# oo
# ___
# \ log k
# -zeta'(2) = ) -----
# /___ 2
# k
# k = 2
# This series converges exceptionally slowly, but can be accelerated
# using Euler-Maclaurin formula. The important insight is that the
# E-M integral can be done in closed form and that the high order
# are given by
# n / \
# d | log x | a + b log x
# --- | ----- | = -----------
# n | 2 | 2 + n
# dx \ x / x
# where a and b are integers given by a simple recurrence. Note
# that just one logarithm is needed. However, lots of integer
# logarithms are required for the initial summation.
# This algorithm could possibly be turned into a faster algorithm
# for general evaluation of zeta(s) or zeta'(s); this should be
# looked into.
@constant_memo
def glaisher_fixed(prec):
wp = prec + 30
# Number of direct terms to sum before applying the Euler-Maclaurin
# formula to the tail. TODO: choose more intelligently
N = int(0.33*prec + 5)
ONE = MPZ_ONE << wp
# Euler-Maclaurin, step 1: sum log(k)/k**2 for k from 2 to N-1
s = MPZ_ZERO
for k in range(2, N):
#print k, N
s += log_int_fixed(k, wp) // k**2
logN = log_int_fixed(N, wp)
#logN = to_fixed(mpf_log(from_int(N), wp+20), wp)
# E-M step 2: integral of log(x)/x**2 from N to inf
s += (ONE + logN) // N
# E-M step 3: endpoint correction term f(N)/2
s += logN // (N**2 * 2)
# E-M step 4: the series of derivatives
pN = N**3
a = 1
b = -2
j = 3
fac = from_int(2)
k = 1
while 1:
# D(2*k-1) * B(2*k) / fac(2*k) [D(n) = nth derivative]
D = ((a << wp) + b*logN) // pN
D = from_man_exp(D, -wp)
B = mpf_bernoulli(2*k, wp)
term = mpf_mul(B, D, wp)
term = mpf_div(term, fac, wp)
term = to_fixed(term, wp)
if abs(term) < 100:
break
#if not k % 10:
# print k, math.log(int(abs(term)), 10)
s -= term
# Advance derivative twice
a, b, pN, j = b-a*j, -j*b, pN*N, j+1
a, b, pN, j = b-a*j, -j*b, pN*N, j+1
k += 1
fac = mpf_mul_int(fac, (2*k)*(2*k-1), wp)
# A = exp((6*s/pi**2 + log(2*pi) + euler)/12)
pi = pi_fixed(wp)
s *= 6
s = (s << wp) // (pi**2 >> wp)
s += euler_fixed(wp)
s += to_fixed(mpf_log(from_man_exp(2*pi, -wp), wp), wp)
s //= 12
A = mpf_exp(from_man_exp(s, -wp), wp)
return to_fixed(A, prec)
# Apery's constant can be computed using the very rapidly convergent
# series
# oo
# ___ 2 10
# \ n 205 n + 250 n + 77 (n!)
# zeta(3) = ) (-1) ------------------- ----------
# /___ 64 5
# n = 0 ((2n+1)!)
@constant_memo
def apery_fixed(prec):
prec += 20
d = MPZ_ONE << prec
term = MPZ(77) << prec
n = 1
s = MPZ_ZERO
while term:
s += term
d *= (n**10)
d //= (((2*n+1)**5) * (2*n)**5)
term = (-1)**n * (205*(n**2) + 250*n + 77) * d
n += 1
return s >> (20 + 6)
"""
Euler's constant (gamma) is computed using the Brent-McMillan formula,
gamma ~= I(n)/J(n) - log(n), where
I(n) = sum_{k=0,1,2,...} (n**k / k!)**2 * H(k)
J(n) = sum_{k=0,1,2,...} (n**k / k!)**2
H(k) = 1 + 1/2 + 1/3 + ... + 1/k
The error is bounded by O(exp(-4n)). Choosing n to be a power
of two, 2**p, the logarithm becomes particularly easy to calculate.[1]
We use the formulation of Algorithm 3.9 in [2] to make the summation
more efficient.
Reference:
[1] Xavier Gourdon & Pascal Sebah, The Euler constant: gamma
http://numbers.computation.free.fr/Constants/Gamma/gamma.pdf
[2] Jonathan Borwein & David Bailey, Mathematics by Experiment,
A K Peters, 2003
"""
@constant_memo
def euler_fixed(prec):
extra = 30
prec += extra
# choose p such that exp(-4*(2**p)) < 2**-n
p = int(math.log((prec/4) * math.log(2), 2)) + 1
n = 2**p
A = U = -p*ln2_fixed(prec)
B = V = MPZ_ONE << prec
k = 1
while 1:
B = B*n**2//k**2
A = (A*n**2//k + B)//k
U += A
V += B
if max(abs(A), abs(B)) < 100:
break
k += 1
return (U<<(prec-extra))//V
# Use zeta accelerated formulas for the Mertens and twin
# prime constants; see
# http://mathworld.wolfram.com/MertensConstant.html
# http://mathworld.wolfram.com/TwinPrimesConstant.html
@constant_memo
def mertens_fixed(prec):
wp = prec + 20
m = 2
s = mpf_euler(wp)
while 1:
t = mpf_zeta_int(m, wp)
if t == fone:
break
t = mpf_log(t, wp)
t = mpf_mul_int(t, moebius(m), wp)
t = mpf_div(t, from_int(m), wp)
s = mpf_add(s, t)
m += 1
return to_fixed(s, prec)
@constant_memo
def twinprime_fixed(prec):
def I(n):
return sum(moebius(d)<<(n//d) for d in xrange(1,n+1) if not n%d)//n
wp = 2*prec + 30
res = fone
primes = [from_rational(1,p,wp) for p in [2,3,5,7]]
ppowers = [mpf_mul(p,p,wp) for p in primes]
n = 2
while 1:
a = mpf_zeta_int(n, wp)
for i in range(4):
a = mpf_mul(a, mpf_sub(fone, ppowers[i]), wp)
ppowers[i] = mpf_mul(ppowers[i], primes[i], wp)
a = mpf_pow_int(a, -I(n), wp)
if mpf_pos(a, prec+10, 'n') == fone:
break
#from libmpf import to_str
#print n, to_str(mpf_sub(fone, a), 6)
res = mpf_mul(res, a, wp)
n += 1
res = mpf_mul(res, from_int(3*15*35), wp)
res = mpf_div(res, from_int(4*16*36), wp)
return to_fixed(res, prec)
mpf_euler = def_mpf_constant(euler_fixed)
mpf_apery = def_mpf_constant(apery_fixed)
mpf_khinchin = def_mpf_constant(khinchin_fixed)
mpf_glaisher = def_mpf_constant(glaisher_fixed)
mpf_catalan = def_mpf_constant(catalan_fixed)
mpf_mertens = def_mpf_constant(mertens_fixed)
mpf_twinprime = def_mpf_constant(twinprime_fixed)
#-----------------------------------------------------------------------#
# #
# Bernoulli numbers #
# #
#-----------------------------------------------------------------------#
MAX_BERNOULLI_CACHE = 3000
"""
Small Bernoulli numbers and factorials are used in numerous summations,
so it is critical for speed that sequential computation is fast and that
values are cached up to a fairly high threshold.
On the other hand, we also want to support fast computation of isolated
large numbers. Currently, no such acceleration is provided for integer
factorials (though it is for large floating-point factorials, which are
computed via gamma if the precision is low enough).
For sequential computation of Bernoulli numbers, we use Ramanujan's formula
/ n + 3 \
B = (A(n) - S(n)) / | |
n \ n /
where A(n) = (n+3)/3 when n = 0 or 2 (mod 6), A(n) = -(n+3)/6
when n = 4 (mod 6), and
[n/6]
___
\ / n + 3 \
S(n) = ) | | * B
/___ \ n - 6*k / n-6*k
k = 1
For isolated large Bernoulli numbers, we use the Riemann zeta function
to calculate a numerical value for B_n. The von Staudt-Clausen theorem
can then be used to optionally find the exact value of the
numerator and denominator.
"""
bernoulli_cache = {}
f3 = from_int(3)
f6 = from_int(6)
def bernoulli_size(n):
"""Accurately estimate the size of B_n (even n > 2 only)"""
lgn = math.log(n,2)
return int(2.326 + 0.5*lgn + n*(lgn - 4.094))
BERNOULLI_PREC_CUTOFF = bernoulli_size(MAX_BERNOULLI_CACHE)
def mpf_bernoulli(n, prec, rnd=None):
"""Computation of Bernoulli numbers (numerically)"""
if n < 2:
if n < 0:
raise ValueError("Bernoulli numbers only defined for n >= 0")
if n == 0:
return fone
if n == 1:
return mpf_neg(fhalf)
# For odd n > 1, the Bernoulli numbers are zero
if n & 1:
return fzero
# If precision is extremely high, we can save time by computing
# the Bernoulli number at a lower precision that is sufficient to
# obtain the exact fraction, round to the exact fraction, and
# convert the fraction back to an mpf value at the original precision
if prec > BERNOULLI_PREC_CUTOFF and prec > bernoulli_size(n)*1.1 + 1000:
p, q = bernfrac(n)
return from_rational(p, q, prec, rnd or round_floor)
if n > MAX_BERNOULLI_CACHE:
return mpf_bernoulli_huge(n, prec, rnd)
wp = prec + 30
# Reuse nearby precisions
wp += 32 - (prec & 31)
cached = bernoulli_cache.get(wp)
if cached:
numbers, state = cached
if n in numbers:
if not rnd:
return numbers[n]
return mpf_pos(numbers[n], prec, rnd)
m, bin, bin1 = state
if n - m > 10:
return mpf_bernoulli_huge(n, prec, rnd)
else:
if n > 10:
return mpf_bernoulli_huge(n, prec, rnd)
numbers = {0:fone}
m, bin, bin1 = state = [2, MPZ(10), MPZ_ONE]
bernoulli_cache[wp] = (numbers, state)
while m <= n:
#print m
case = m % 6
# Accurately estimate size of B_m so we can use
# fixed point math without using too much precision
szbm = bernoulli_size(m)
s = 0
sexp = max(0, szbm) - wp
if m < 6:
a = MPZ_ZERO
else:
a = bin1
for j in xrange(1, m//6+1):
usign, uman, uexp, ubc = u = numbers[m-6*j]
if usign:
uman = -uman
s += lshift(a*uman, uexp-sexp)
# Update inner binomial coefficient
j6 = 6*j
a *= ((m-5-j6)*(m-4-j6)*(m-3-j6)*(m-2-j6)*(m-1-j6)*(m-j6))
a //= ((4+j6)*(5+j6)*(6+j6)*(7+j6)*(8+j6)*(9+j6))
if case == 0: b = mpf_rdiv_int(m+3, f3, wp)
if case == 2: b = mpf_rdiv_int(m+3, f3, wp)
if case == 4: b = mpf_rdiv_int(-m-3, f6, wp)
s = from_man_exp(s, sexp, wp)
b = mpf_div(mpf_sub(b, s, wp), from_int(bin), wp)
numbers[m] = b
m += 2
# Update outer binomial coefficient
bin = bin * ((m+2)*(m+3)) // (m*(m-1))
if m > 6:
bin1 = bin1 * ((2+m)*(3+m)) // ((m-7)*(m-6))
state[:] = [m, bin, bin1]
return numbers[n]
def mpf_bernoulli_huge(n, prec, rnd=None):
wp = prec + 10
piprec = wp + int(math.log(n,2))
v = mpf_gamma_int(n+1, wp)
v = mpf_mul(v, mpf_zeta_int(n, wp), wp)
v = mpf_mul(v, mpf_pow_int(mpf_pi(piprec), -n, wp))
v = mpf_shift(v, 1-n)
if not n & 3:
v = mpf_neg(v)
return mpf_pos(v, prec, rnd or round_fast)
def bernfrac(n):
r"""
Returns a tuple of integers `(p, q)` such that `p/q = B_n` exactly,
where `B_n` denotes the `n`-th Bernoulli number. The fraction is
always reduced to lowest terms. Note that for `n > 1` and `n` odd,
`B_n = 0`, and `(0, 1)` is returned.
**Examples**
The first few Bernoulli numbers are exactly::
>>> from mpmath import *
>>> for n in range(15):
... p, q = bernfrac(n)
... print("%s %s/%s" % (n, p, q))
...
0 1/1
1 -1/2
2 1/6
3 0/1
4 -1/30
5 0/1
6 1/42
7 0/1
8 -1/30
9 0/1
10 5/66
11 0/1
12 -691/2730
13 0/1
14 7/6
This function works for arbitrarily large `n`::
>>> p, q = bernfrac(10**4)
>>> print(q)
2338224387510
>>> print(len(str(p)))
27692
>>> mp.dps = 15
>>> print(mpf(p) / q)
-9.04942396360948e+27677
>>> print(bernoulli(10**4))
-9.04942396360948e+27677
.. note ::
:func:`~mpmath.bernoulli` computes a floating-point approximation
directly, without computing the exact fraction first.
This is much faster for large `n`.
**Algorithm**
:func:`~mpmath.bernfrac` works by computing the value of `B_n` numerically
and then using the von Staudt-Clausen theorem [1] to reconstruct
the exact fraction. For large `n`, this is significantly faster than
computing `B_1, B_2, \ldots, B_2` recursively with exact arithmetic.
The implementation has been tested for `n = 10^m` up to `m = 6`.
In practice, :func:`~mpmath.bernfrac` appears to be about three times
slower than the specialized program calcbn.exe [2]
**References**
1. MathWorld, von Staudt-Clausen Theorem:
http://mathworld.wolfram.com/vonStaudt-ClausenTheorem.html
2. The Bernoulli Number Page:
http://www.bernoulli.org/
"""
n = int(n)
if n < 3:
return [(1, 1), (-1, 2), (1, 6)][n]
if n & 1:
return (0, 1)
q = 1
for k in list_primes(n+1):
if not (n % (k-1)):
q *= k
prec = bernoulli_size(n) + int(math.log(q,2)) + 20
b = mpf_bernoulli(n, prec)
p = mpf_mul(b, from_int(q))
pint = to_int(p, round_nearest)
return (pint, q)
#-----------------------------------------------------------------------#
# #
# The gamma function (OLD IMPLEMENTATION) #
# #
#-----------------------------------------------------------------------#
"""
We compute the real factorial / gamma function using Spouge's approximation
x! = (x+a)**(x+1/2) * exp(-x-a) * [c_0 + S(x) + eps]
where S(x) is the sum of c_k/(x+k) from k = 1 to a-1 and the coefficients
are given by
c_0 = sqrt(2*pi)
(-1)**(k-1)
c_k = ----------- (a-k)**(k-1/2) exp(-k+a), k = 1,2,...,a-1
(k - 1)!
As proved by Spouge, if we choose a = log(2)/log(2*pi)*n = 0.38*n, the
relative error eps is less than 2^(-n) for any x in the right complex
half-plane (assuming a > 2). In practice, it seems that a can be chosen
quite a bit lower still (30-50%); this possibility should be investigated.
For negative x, we use the reflection formula.
References:
-----------
John L. Spouge, "Computation of the gamma, digamma, and trigamma
functions", SIAM Journal on Numerical Analysis 31 (1994), no. 3, 931-944.
"""
spouge_cache = {}
def calc_spouge_coefficients(a, prec):
wp = prec + int(a*1.4)
c = [0] * a
# b = exp(a-1)
b = mpf_exp(from_int(a-1), wp)
# e = exp(1)
e = mpf_exp(fone, wp)
# sqrt(2*pi)
sq2pi = mpf_sqrt(mpf_shift(mpf_pi(wp), 1), wp)
c[0] = to_fixed(sq2pi, prec)
for k in xrange(1, a):
# c[k] = ((-1)**(k-1) * (a-k)**k) * b / sqrt(a-k)
term = mpf_mul_int(b, ((-1)**(k-1) * (a-k)**k), wp)
term = mpf_div(term, mpf_sqrt(from_int(a-k), wp), wp)
c[k] = to_fixed(term, prec)
# b = b / (e * k)
b = mpf_div(b, mpf_mul(e, from_int(k), wp), wp)
return c
# Cached lookup of coefficients
def get_spouge_coefficients(prec):
# This exact precision has been used before
if prec in spouge_cache:
return spouge_cache[prec]
for p in spouge_cache:
if 0.8 <= prec/float(p) < 1:
return spouge_cache[p]
# Here we estimate the value of a based on Spouge's inequality for
# the relative error
a = max(3, int(0.38*prec)) # 0.38 = log(2)/log(2*pi), ~= 1.26*n
coefs = calc_spouge_coefficients(a, prec)
spouge_cache[prec] = (prec, a, coefs)
return spouge_cache[prec]
def spouge_sum_real(x, prec, a, c):
x = to_fixed(x, prec)
s = c[0]
for k in xrange(1, a):
s += (c[k] << prec) // (x + (k << prec))
return from_man_exp(s, -prec, prec, round_floor)
# Unused: for fast computation of gamma(p/q)
def spouge_sum_rational(p, q, prec, a, c):
s = c[0]
for k in xrange(1, a):
s += c[k] * q // (p+q*k)
return from_man_exp(s, -prec, prec, round_floor)
# For a complex number a + b*I, we have
#
# c_k (a+k)*c_k b * c_k
# ------------- = --------- - ------- * I
# (a + b*I) + k M M
#
# 2 2 2 2 2
# where M = (a+k) + b = (a + b ) + (2*a*k + k )
def spouge_sum_complex(re, im, prec, a, c):
re = to_fixed(re, prec)
im = to_fixed(im, prec)
sre, sim = c[0], 0
mag = ((re**2)>>prec) + ((im**2)>>prec)
for k in xrange(1, a):
M = mag + re*(2*k) + ((k**2) << prec)
sre += (c[k] * (re + (k << prec))) // M
sim -= (c[k] * im) // M
re = from_man_exp(sre, -prec, prec, round_floor)
im = from_man_exp(sim, -prec, prec, round_floor)
return re, im
def mpf_gamma_int_old(n, prec, rounding=round_fast):
if n < 1000:
return from_int(ifac(n-1), prec, rounding)
# XXX: choose the cutoff less arbitrarily
size = int(n*math.log(n,2))
if prec > size/20.0:
return from_int(ifac(n-1), prec, rounding)
return mpf_gamma(from_int(n), prec, rounding)
def mpf_factorial_old(x, prec, rounding=round_fast):
return mpf_gamma_old(x, prec, rounding, p1=0)
def mpc_factorial_old(x, prec, rounding=round_fast):
return mpc_gamma_old(x, prec, rounding, p1=0)
def mpf_gamma_old(x, prec, rounding=round_fast, p1=1):
"""
Computes the gamma function of a real floating-point argument.
With p1=0, computes a factorial instead.
"""
sign, man, exp, bc = x
if not man:
if x == finf:
return finf
if x == fninf or x == fnan:
return fnan
# More precision is needed for enormous x. TODO:
# use Stirling's formula + Euler-Maclaurin summation
size = exp + bc
if size > 5:
size = int(size * math.log(size,2))
wp = prec + max(0, size) + 15
if exp >= 0:
if sign or (p1 and not man):
raise ValueError("gamma function pole")
# A direct factorial is fastest
if exp + bc <= 10:
return from_int(ifac((man<<exp)-p1), prec, rounding)
reflect = sign or exp+bc < -1
if p1:
# Should be done exactly!
x = mpf_sub(x, fone)
# x < 0.25
if reflect:
# gamma = pi / (sin(pi*x) * gamma(1-x))
wp += 15
pix = mpf_mul(x, mpf_pi(wp), wp)
t = mpf_sin_pi(x, wp)
g = mpf_gamma_old(mpf_sub(fone, x), wp)
return mpf_div(pix, mpf_mul(t, g, wp), prec, rounding)
sprec, a, c = get_spouge_coefficients(wp)
s = spouge_sum_real(x, sprec, a, c)
# gamma = exp(log(x+a)*(x+0.5) - xpa) * s
xpa = mpf_add(x, from_int(a), wp)
logxpa = mpf_log(xpa, wp)
xph = mpf_add(x, fhalf, wp)
t = mpf_sub(mpf_mul(logxpa, xph, wp), xpa, wp)
t = mpf_mul(mpf_exp(t, wp), s, prec, rounding)
return t
def mpc_gamma_old(x, prec, rounding=round_fast, p1=1):
re, im = x
if im == fzero:
return mpf_gamma_old(re, prec, rounding, p1), fzero
# More precision is needed for enormous x.
sign, man, exp, bc = re
isign, iman, iexp, ibc = im
if re == fzero:
size = iexp+ibc
else:
size = max(exp+bc, iexp+ibc)
if size > 5:
size = int(size * math.log(size,2))
reflect = sign or (exp+bc < -1)
wp = prec + max(0, size) + 25
# Near x = 0 pole (TODO: other poles)
if p1:
if size < -prec-5:
return mpc_add_mpf(mpc_div(mpc_one, x, 2*prec+10), \
mpf_neg(mpf_euler(2*prec+10)), prec, rounding)
elif size < -5:
wp += (-2*size)
if p1:
# Should be done exactly!
re_orig = re
re = mpf_sub(re, fone, bc+abs(exp)+2)
x = re, im
if reflect:
# Reflection formula
wp += 15
pi = mpf_pi(wp), fzero
pix = mpc_mul(x, pi, wp)
t = mpc_sin_pi(x, wp)
u = mpc_sub(mpc_one, x, wp)
g = mpc_gamma_old(u, wp)
w = mpc_mul(t, g, wp)
return mpc_div(pix, w, wp)
# Extremely close to the real line?
# XXX: reflection formula
if iexp+ibc < -wp:
a = mpf_gamma_old(re_orig, wp)
b = mpf_psi0(re_orig, wp)
gamma_diff = mpf_div(a, b, wp)
return mpf_pos(a, prec, rounding), mpf_mul(gamma_diff, im, prec, rounding)
sprec, a, c = get_spouge_coefficients(wp)
s = spouge_sum_complex(re, im, sprec, a, c)
# gamma = exp(log(x+a)*(x+0.5) - xpa) * s
repa = mpf_add(re, from_int(a), wp)
logxpa = mpc_log((repa, im), wp)
reph = mpf_add(re, fhalf, wp)
t = mpc_sub(mpc_mul(logxpa, (reph, im), wp), (repa, im), wp)
t = mpc_mul(mpc_exp(t, wp), s, prec, rounding)
return t
#-----------------------------------------------------------------------#
# #
# Polygamma functions #
# #
#-----------------------------------------------------------------------#
"""
For all polygamma (psi) functions, we use the Euler-Maclaurin summation
formula. It looks slightly different in the m = 0 and m > 0 cases.
For m = 0, we have
oo
___ B
(0) 1 \ 2 k -2 k
psi (z) ~ log z + --- - ) ------ z
2 z /___ (2 k)!
k = 1
Experiment shows that the minimum term of the asymptotic series
reaches 2^(-p) when Re(z) > 0.11*p. So we simply use the recurrence
for psi (equivalent, in fact, to summing to the first few terms
directly before applying E-M) to obtain z large enough.
Since, very crudely, log z ~= 1 for Re(z) > 1, we can use
fixed-point arithmetic (if z is extremely large, log(z) itself
is a sufficient approximation, so we can stop there already).
For Re(z) << 0, we could use recurrence, but this is of course
inefficient for large negative z, so there we use the
reflection formula instead.
For m > 0, we have
N - 1
___
~~~(m) [ \ 1 ] 1 1
psi (z) ~ [ ) -------- ] + ---------- + -------- +
[ /___ m+1 ] m+1 m
k = 1 (z+k) ] 2 (z+N) m (z+N)
oo
___ B
\ 2 k (m+1) (m+2) ... (m+2k-1)
+ ) ------ ------------------------
/___ (2 k)! m + 2 k
k = 1 (z+N)
where ~~~ denotes the function rescaled by 1/((-1)^(m+1) m!).
Here again N is chosen to make z+N large enough for the minimum
term in the last series to become smaller than eps.
TODO: the current estimation of N for m > 0 is *very suboptimal*.
TODO: implement the reflection formula for m > 0, Re(z) << 0.
It is generally a combination of multiple cotangents. Need to
figure out a reasonably simple way to generate these formulas
on the fly.
TODO: maybe use exact algorithms to compute psi for integral
and certain rational arguments, as this can be much more
efficient. (On the other hand, the availability of these
special values provides a convenient way to test the general
algorithm.)
"""
# Harmonic numbers are just shifted digamma functions
# We should calculate these exactly when x is an integer
# and when doing so is faster.
def mpf_harmonic(x, prec, rnd):
if x in (fzero, fnan, finf):
return x
a = mpf_psi0(mpf_add(fone, x, prec+5), prec)
return mpf_add(a, mpf_euler(prec+5, rnd), prec, rnd)
def mpc_harmonic(z, prec, rnd):
if z[1] == fzero:
return (mpf_harmonic(z[0], prec, rnd), fzero)
a = mpc_psi0(mpc_add_mpf(z, fone, prec+5), prec)
return mpc_add_mpf(a, mpf_euler(prec+5, rnd), prec, rnd)
def mpf_psi0(x, prec, rnd=round_fast):
"""
Computation of the digamma function (psi function of order 0)
of a real argument.
"""
sign, man, exp, bc = x
wp = prec + 10
if not man:
if x == finf: return x
if x == fninf or x == fnan: return fnan
if x == fzero or (exp >= 0 and sign):
raise ValueError("polygamma pole")
# Reflection formula
if sign and exp+bc > 3:
c, s = mpf_cos_sin_pi(x, wp)
q = mpf_mul(mpf_div(c, s, wp), mpf_pi(wp), wp)
p = mpf_psi0(mpf_sub(fone, x, wp), wp)
return mpf_sub(p, q, prec, rnd)
# The logarithmic term is accurate enough
if (not sign) and bc + exp > wp:
return mpf_log(mpf_sub(x, fone, wp), prec, rnd)
# Initial recurrence to obtain a large enough x
m = to_int(x)
n = int(0.11*wp) + 2
s = MPZ_ZERO
x = to_fixed(x, wp)
one = MPZ_ONE << wp
if m < n:
for k in xrange(m, n):
s -= (one << wp) // x
x += one
x -= one
# Logarithmic term
s += to_fixed(mpf_log(from_man_exp(x, -wp, wp), wp), wp)
# Endpoint term in Euler-Maclaurin expansion
s += (one << wp) // (2*x)
# Euler-Maclaurin remainder sum
x2 = (x*x) >> wp
t = one
prev = 0
k = 1
while 1:
t = (t*x2) >> wp
bsign, bman, bexp, bbc = mpf_bernoulli(2*k, wp)
offset = (bexp + 2*wp)
if offset >= 0: term = (bman << offset) // (t*(2*k))
else: term = (bman >> (-offset)) // (t*(2*k))
if k & 1: s -= term
else: s += term
if k > 2 and term >= prev:
break
prev = term
k += 1
return from_man_exp(s, -wp, wp, rnd)
def mpc_psi0(z, prec, rnd=round_fast):
"""
Computation of the digamma function (psi function of order 0)
of a complex argument.
"""
re, im = z
# Fall back to the real case
if im == fzero:
return (mpf_psi0(re, prec, rnd), fzero)
wp = prec + 20
sign, man, exp, bc = re
# Reflection formula
if sign and exp+bc > 3:
c = mpc_cos_pi(z, wp)
s = mpc_sin_pi(z, wp)
q = mpc_mul_mpf(mpc_div(c, s, wp), mpf_pi(wp), wp)
p = mpc_psi0(mpc_sub(mpc_one, z, wp), wp)
return mpc_sub(p, q, prec, rnd)
# Just the logarithmic term
if (not sign) and bc + exp > wp:
return mpc_log(mpc_sub(z, mpc_one, wp), prec, rnd)
# Initial recurrence to obtain a large enough z
w = to_int(re)
n = int(0.11*wp) + 2
s = mpc_zero
if w < n:
for k in xrange(w, n):
s = mpc_sub(s, mpc_reciprocal(z, wp), wp)
z = mpc_add_mpf(z, fone, wp)
z = mpc_sub(z, mpc_one, wp)
# Logarithmic and endpoint term
s = mpc_add(s, mpc_log(z, wp), wp)
s = mpc_add(s, mpc_div(mpc_half, z, wp), wp)
# Euler-Maclaurin remainder sum
z2 = mpc_square(z, wp)
t = mpc_one
prev = mpc_zero
k = 1
eps = mpf_shift(fone, -wp+2)
while 1:
t = mpc_mul(t, z2, wp)
bern = mpf_bernoulli(2*k, wp)
term = mpc_mpf_div(bern, mpc_mul_int(t, 2*k, wp), wp)
s = mpc_sub(s, term, wp)
szterm = mpc_abs(term, 10)
if k > 2 and mpf_le(szterm, eps):
break
prev = term
k += 1
return s
# Currently unoptimized
def mpf_psi(m, x, prec, rnd=round_fast):
"""
Computation of the polygamma function of arbitrary integer order
m >= 0, for a real argument x.
"""
if m == 0:
return mpf_psi0(x, prec, rnd=round_fast)
return mpc_psi(m, (x, fzero), prec, rnd)[0]
def mpc_psi(m, z, prec, rnd=round_fast):
"""
Computation of the polygamma function of arbitrary integer order
m >= 0, for a complex argument z.
"""
if m == 0:
return mpc_psi0(z, prec, rnd)
re, im = z
wp = prec + 20
sign, man, exp, bc = re
if not im[1]:
if im in (finf, fninf, fnan):
return (fnan, fnan)
if not man:
if re == finf and im == fzero:
return (fzero, fzero)
if re == fnan:
return (fnan, fnan)
# Recurrence
w = to_int(re)
n = int(0.4*wp + 4*m)
s = mpc_zero
if w < n:
for k in xrange(w, n):
t = mpc_pow_int(z, -m-1, wp)
s = mpc_add(s, t, wp)
z = mpc_add_mpf(z, fone, wp)
zm = mpc_pow_int(z, -m, wp)
z2 = mpc_pow_int(z, -2, wp)
# 1/m*(z+N)^m
integral_term = mpc_div_mpf(zm, from_int(m), wp)
s = mpc_add(s, integral_term, wp)
# 1/2*(z+N)^(-(m+1))
s = mpc_add(s, mpc_mul_mpf(mpc_div(zm, z, wp), fhalf, wp), wp)
a = m + 1
b = 2
k = 1
# Important: we want to sum up to the *relative* error,
# not the absolute error, because psi^(m)(z) might be tiny
magn = mpc_abs(s, 10)
magn = magn[2]+magn[3]
eps = mpf_shift(fone, magn-wp+2)
while 1:
zm = mpc_mul(zm, z2, wp)
bern = mpf_bernoulli(2*k, wp)
scal = mpf_mul_int(bern, a, wp)
scal = mpf_div(scal, from_int(b), wp)
term = mpc_mul_mpf(zm, scal, wp)
s = mpc_add(s, term, wp)
szterm = mpc_abs(term, 10)
if k > 2 and mpf_le(szterm, eps):
break
#print k, to_str(szterm, 10), to_str(eps, 10)
a *= (m+2*k)*(m+2*k+1)
b *= (2*k+1)*(2*k+2)
k += 1
# Scale and sign factor
v = mpc_mul_mpf(s, mpf_gamma(from_int(m+1), wp), prec, rnd)
if not (m & 1):
v = mpf_neg(v[0]), mpf_neg(v[1])
return v
#-----------------------------------------------------------------------#
# #
# Riemann zeta function #
# #
#-----------------------------------------------------------------------#
"""
We use zeta(s) = eta(s) / (1 - 2**(1-s)) and Borwein's approximation
n-1
___ k
-1 \ (-1) (d_k - d_n)
eta(s) ~= ---- ) ------------------
d_n /___ s
k = 0 (k + 1)
where
k
___ i
\ (n + i - 1)! 4
d_k = n ) ---------------.
/___ (n - i)! (2i)!
i = 0
If s = a + b*I, the absolute error for eta(s) is bounded by
3 (1 + 2|b|)
------------ * exp(|b| pi/2)
n
(3+sqrt(8))
Disregarding the linear term, we have approximately,
log(err) ~= log(exp(1.58*|b|)) - log(5.8**n)
log(err) ~= 1.58*|b| - log(5.8)*n
log(err) ~= 1.58*|b| - 1.76*n
log2(err) ~= 2.28*|b| - 2.54*n
So for p bits, we should choose n > (p + 2.28*|b|) / 2.54.
References:
-----------
Peter Borwein, "An Efficient Algorithm for the Riemann Zeta Function"
http://www.cecm.sfu.ca/personal/pborwein/PAPERS/P117.ps
http://en.wikipedia.org/wiki/Dirichlet_eta_function
"""
borwein_cache = {}
def borwein_coefficients(n):
if n in borwein_cache:
return borwein_cache[n]
ds = [MPZ_ZERO] * (n+1)
d = MPZ_ONE
s = ds[0] = MPZ_ONE
for i in range(1, n+1):
d = d * 4 * (n+i-1) * (n-i+1)
d //= ((2*i) * ((2*i)-1))
s += d
ds[i] = s
borwein_cache[n] = ds
return ds
ZETA_INT_CACHE_MAX_PREC = 1000
zeta_int_cache = {}
def mpf_zeta_int(s, prec, rnd=round_fast):
"""
Optimized computation of zeta(s) for an integer s.
"""
wp = prec + 20
s = int(s)
if s in zeta_int_cache and zeta_int_cache[s][0] >= wp:
return mpf_pos(zeta_int_cache[s][1], prec, rnd)
if s < 2:
if s == 1:
raise ValueError("zeta(1) pole")
if not s:
return mpf_neg(fhalf)
return mpf_div(mpf_bernoulli(-s+1, wp), from_int(s-1), prec, rnd)
# 2^-s term vanishes?
if s >= wp:
return mpf_perturb(fone, 0, prec, rnd)
# 5^-s term vanishes?
elif s >= wp*0.431:
t = one = 1 << wp
t += 1 << (wp - s)
t += one // (MPZ_THREE ** s)
t += 1 << max(0, wp - s*2)
return from_man_exp(t, -wp, prec, rnd)
else:
# Fast enough to sum directly?
# Even better, we use the Euler product (idea stolen from pari)
m = (float(wp)/(s-1) + 1)
if m < 30:
needed_terms = int(2.0**m + 1)
if needed_terms < int(wp/2.54 + 5) / 10:
t = fone
for k in list_primes(needed_terms):
#print k, needed_terms
powprec = int(wp - s*math.log(k,2))
if powprec < 2:
break
a = mpf_sub(fone, mpf_pow_int(from_int(k), -s, powprec), wp)
t = mpf_mul(t, a, wp)
return mpf_div(fone, t, wp)
# Use Borwein's algorithm
n = int(wp/2.54 + 5)
d = borwein_coefficients(n)
t = MPZ_ZERO
s = MPZ(s)
for k in xrange(n):
t += (((-1)**k * (d[k] - d[n])) << wp) // (k+1)**s
t = (t << wp) // (-d[n])
t = (t << wp) // ((1 << wp) - (1 << (wp+1-s)))
if (s in zeta_int_cache and zeta_int_cache[s][0] < wp) or (s not in zeta_int_cache):
zeta_int_cache[s] = (wp, from_man_exp(t, -wp-wp))
return from_man_exp(t, -wp-wp, prec, rnd)
def mpf_zeta(s, prec, rnd=round_fast, alt=0):
sign, man, exp, bc = s
if not man:
if s == fzero:
if alt:
return fhalf
else:
return mpf_neg(fhalf)
if s == finf:
return fone
return fnan
wp = prec + 20
# First term vanishes?
if (not sign) and (exp + bc > (math.log(wp,2) + 2)):
return mpf_perturb(fone, alt, prec, rnd)
# Optimize for integer arguments
elif exp >= 0:
if alt:
if s == fone:
return mpf_ln2(prec, rnd)
z = mpf_zeta_int(to_int(s), wp, negative_rnd[rnd])
q = mpf_sub(fone, mpf_pow(ftwo, mpf_sub(fone, s, wp), wp), wp)
return mpf_mul(z, q, prec, rnd)
else:
return mpf_zeta_int(to_int(s), prec, rnd)
# Negative: use the reflection formula
# Borwein only proves the accuracy bound for x >= 1/2. However, based on
# tests, the accuracy without reflection is quite good even some distance
# to the left of 1/2. XXX: verify this.
if sign:
# XXX: could use the separate refl. formula for Dirichlet eta
if alt:
q = mpf_sub(fone, mpf_pow(ftwo, mpf_sub(fone, s, wp), wp), wp)
return mpf_mul(mpf_zeta(s, wp), q, prec, rnd)
# XXX: -1 should be done exactly
y = mpf_sub(fone, s, 10*wp)
a = mpf_gamma(y, wp)
b = mpf_zeta(y, wp)
c = mpf_sin_pi(mpf_shift(s, -1), wp)
wp2 = wp + max(0,exp+bc)
pi = mpf_pi(wp+wp2)
d = mpf_div(mpf_pow(mpf_shift(pi, 1), s, wp2), pi, wp2)
return mpf_mul(a,mpf_mul(b,mpf_mul(c,d,wp),wp),prec,rnd)
# Near pole
r = mpf_sub(fone, s, wp)
asign, aman, aexp, abc = mpf_abs(r)
pole_dist = -2*(aexp+abc)
if pole_dist > wp:
if alt:
return mpf_ln2(prec, rnd)
else:
q = mpf_neg(mpf_div(fone, r, wp))
return mpf_add(q, mpf_euler(wp), prec, rnd)
else:
wp += max(0, pole_dist)
t = MPZ_ZERO
#wp += 16 - (prec & 15)
# Use Borwein's algorithm
n = int(wp/2.54 + 5)
d = borwein_coefficients(n)
t = MPZ_ZERO
sf = to_fixed(s, wp)
ln2 = ln2_fixed(wp)
for k in xrange(n):
u = (-sf*log_int_fixed(k+1, wp, ln2)) >> wp
#esign, eman, eexp, ebc = mpf_exp(u, wp)
#offset = eexp + wp
#if offset >= 0:
# w = ((d[k] - d[n]) * eman) << offset
#else:
# w = ((d[k] - d[n]) * eman) >> (-offset)
eman = exp_fixed(u, wp, ln2)
w = (d[k] - d[n]) * eman
if k & 1:
t -= w
else:
t += w
t = t // (-d[n])
t = from_man_exp(t, -wp, wp)
if alt:
return mpf_pos(t, prec, rnd)
else:
q = mpf_sub(fone, mpf_pow(ftwo, mpf_sub(fone, s, wp), wp), wp)
return mpf_div(t, q, prec, rnd)
def mpc_zeta(s, prec, rnd=round_fast, alt=0, force=False):
re, im = s
if im == fzero:
return mpf_zeta(re, prec, rnd, alt), fzero
# slow for large s
if (not force) and mpf_gt(mpc_abs(s, 10), from_int(prec)):
raise NotImplementedError
wp = prec + 20
# Near pole
r = mpc_sub(mpc_one, s, wp)
asign, aman, aexp, abc = mpc_abs(r, 10)
pole_dist = -2*(aexp+abc)
if pole_dist > wp:
if alt:
q = mpf_ln2(wp)
y = mpf_mul(q, mpf_euler(wp), wp)
g = mpf_shift(mpf_mul(q, q, wp), -1)
g = mpf_sub(y, g)
z = mpc_mul_mpf(r, mpf_neg(g), wp)
z = mpc_add_mpf(z, q, wp)
return mpc_pos(z, prec, rnd)
else:
q = mpc_neg(mpc_div(mpc_one, r, wp))
q = mpc_add_mpf(q, mpf_euler(wp), wp)
return mpc_pos(q, prec, rnd)
else:
wp += max(0, pole_dist)
# Reflection formula. To be rigorous, we should reflect to the left of
# re = 1/2 (see comments for mpf_zeta), but this leads to unnecessary
# slowdown for interesting values of s
if mpf_lt(re, fzero):
# XXX: could use the separate refl. formula for Dirichlet eta
if alt:
q = mpc_sub(mpc_one, mpc_pow(mpc_two, mpc_sub(mpc_one, s, wp),
wp), wp)
return mpc_mul(mpc_zeta(s, wp), q, prec, rnd)
# XXX: -1 should be done exactly
y = mpc_sub(mpc_one, s, 10*wp)
a = mpc_gamma(y, wp)
b = mpc_zeta(y, wp)
c = mpc_sin_pi(mpc_shift(s, -1), wp)
rsign, rman, rexp, rbc = re
isign, iman, iexp, ibc = im
mag = max(rexp+rbc, iexp+ibc)
wp2 = wp + max(0, mag)
pi = mpf_pi(wp+wp2)
pi2 = (mpf_shift(pi, 1), fzero)
d = mpc_div_mpf(mpc_pow(pi2, s, wp2), pi, wp2)
return mpc_mul(a,mpc_mul(b,mpc_mul(c,d,wp),wp),prec,rnd)
n = int(wp/2.54 + 5)
n += int(0.9*abs(to_int(im)))
d = borwein_coefficients(n)
ref = to_fixed(re, wp)
imf = to_fixed(im, wp)
tre = MPZ_ZERO
tim = MPZ_ZERO
one = MPZ_ONE << wp
one_2wp = MPZ_ONE << (2*wp)
critical_line = re == fhalf
ln2 = ln2_fixed(wp)
pi2 = pi_fixed(wp-1)
wp2 = wp+wp
for k in xrange(n):
log = log_int_fixed(k+1, wp, ln2)
# A square root is much cheaper than an exp
if critical_line:
w = one_2wp // isqrt_fast((k+1) << wp2)
else:
w = exp_fixed((-ref*log) >> wp, wp)
if k & 1:
w *= (d[n] - d[k])
else:
w *= (d[k] - d[n])
wre, wim = cos_sin_fixed((-imf*log)>>wp, wp, pi2)
tre += (w * wre) >> wp
tim += (w * wim) >> wp
tre //= (-d[n])
tim //= (-d[n])
tre = from_man_exp(tre, -wp, wp)
tim = from_man_exp(tim, -wp, wp)
if alt:
return mpc_pos((tre, tim), prec, rnd)
else:
q = mpc_sub(mpc_one, mpc_pow(mpc_two, r, wp), wp)
return mpc_div((tre, tim), q, prec, rnd)
def mpf_altzeta(s, prec, rnd=round_fast):
return mpf_zeta(s, prec, rnd, 1)
def mpc_altzeta(s, prec, rnd=round_fast):
return mpc_zeta(s, prec, rnd, 1)
# Not optimized currently
mpf_zetasum = None
def pow_fixed(x, n, wp):
if n == 1:
return x
y = MPZ_ONE << wp
while n:
if n & 1:
y = (y*x) >> wp
n -= 1
x = (x*x) >> wp
n //= 2
return y
# TODO: optimize / cleanup interface / unify with list_primes
sieve_cache = []
primes_cache = []
mult_cache = []
def primesieve(n):
global sieve_cache, primes_cache, mult_cache
if n < len(sieve_cache):
sieve = sieve_cache#[:n+1]
primes = primes_cache[:primes_cache.index(max(sieve))+1]
mult = mult_cache#[:n+1]
return sieve, primes, mult
sieve = [0] * (n+1)
mult = [0] * (n+1)
primes = list_primes(n)
for p in primes:
#sieve[p::p] = p
for k in xrange(p,n+1,p):
sieve[k] = p
for i, p in enumerate(sieve):
if i >= 2:
m = 1
n = i // p
while not n % p:
n //= p
m += 1
mult[i] = m
sieve_cache = sieve
primes_cache = primes
mult_cache = mult
return sieve, primes, mult
def zetasum_sieved(critical_line, sre, sim, a, n, wp):
if a < 1:
raise ValueError("a cannot be less than 1")
sieve, primes, mult = primesieve(a+n)
basic_powers = {}
one = MPZ_ONE << wp
one_2wp = MPZ_ONE << (2*wp)
wp2 = wp+wp
ln2 = ln2_fixed(wp)
pi2 = pi_fixed(wp-1)
for p in primes:
if p*2 > a+n:
break
log = log_int_fixed(p, wp, ln2)
cos, sin = cos_sin_fixed((-sim*log)>>wp, wp, pi2)
if critical_line:
u = one_2wp // isqrt_fast(p<<wp2)
else:
u = exp_fixed((-sre*log)>>wp, wp)
pre = (u*cos) >> wp
pim = (u*sin) >> wp
basic_powers[p] = [(pre, pim)]
tre, tim = pre, pim
for m in range(1,int(math.log(a+n,p)+0.01)+1):
tre, tim = ((pre*tre-pim*tim)>>wp), ((pim*tre+pre*tim)>>wp)
basic_powers[p].append((tre,tim))
xre = MPZ_ZERO
xim = MPZ_ZERO
if a == 1:
xre += one
aa = max(a,2)
for k in xrange(aa, a+n+1):
p = sieve[k]
if p in basic_powers:
m = mult[k]
tre, tim = basic_powers[p][m-1]
while 1:
k //= p**m
if k == 1:
break
p = sieve[k]
m = mult[k]
pre, pim = basic_powers[p][m-1]
tre, tim = ((pre*tre-pim*tim)>>wp), ((pim*tre+pre*tim)>>wp)
else:
log = log_int_fixed(k, wp, ln2)
cos, sin = cos_sin_fixed((-sim*log)>>wp, wp, pi2)
if critical_line:
u = one_2wp // isqrt_fast(k<<wp2)
else:
u = exp_fixed((-sre*log)>>wp, wp)
tre = (u*cos) >> wp
tim = (u*sin) >> wp
xre += tre
xim += tim
return xre, xim
# Set to something large to disable
ZETASUM_SIEVE_CUTOFF = 10
def mpc_zetasum(s, a, n, derivatives, reflect, prec):
"""
Fast version of mp._zetasum, assuming s = complex, a = integer.
"""
wp = prec + 10
have_derivatives = derivatives != [0]
have_one_derivative = len(derivatives) == 1
# parse s
sre, sim = s
critical_line = (sre == fhalf)
sre = to_fixed(sre, wp)
sim = to_fixed(sim, wp)
if a > 0 and n > ZETASUM_SIEVE_CUTOFF and not have_derivatives and not reflect:
re, im = zetasum_sieved(critical_line, sre, sim, a, n, wp)
xs = [(from_man_exp(re, -wp, prec, 'n'), from_man_exp(im, -wp, prec, 'n'))]
return xs, []
maxd = max(derivatives)
if not have_one_derivative:
derivatives = range(maxd+1)
# x_d = 0, y_d = 0
xre = [MPZ_ZERO for d in derivatives]
xim = [MPZ_ZERO for d in derivatives]
if reflect:
yre = [MPZ_ZERO for d in derivatives]
yim = [MPZ_ZERO for d in derivatives]
else:
yre = yim = []
one = MPZ_ONE << wp
one_2wp = MPZ_ONE << (2*wp)
ln2 = ln2_fixed(wp)
pi2 = pi_fixed(wp-1)
wp2 = wp+wp
for w in xrange(a, a+n+1):
log = log_int_fixed(w, wp, ln2)
cos, sin = cos_sin_fixed((-sim*log)>>wp, wp, pi2)
if critical_line:
u = one_2wp // isqrt_fast(w<<wp2)
else:
u = exp_fixed((-sre*log)>>wp, wp)
xterm_re = (u * cos) >> wp
xterm_im = (u * sin) >> wp
if reflect:
reciprocal = (one_2wp // (u*w))
yterm_re = (reciprocal * cos) >> wp
yterm_im = (reciprocal * sin) >> wp
if have_derivatives:
if have_one_derivative:
log = pow_fixed(log, maxd, wp)
xre[0] += (xterm_re * log) >> wp
xim[0] += (xterm_im * log) >> wp
if reflect:
yre[0] += (yterm_re * log) >> wp
yim[0] += (yterm_im * log) >> wp
else:
t = MPZ_ONE << wp
for d in derivatives:
xre[d] += (xterm_re * t) >> wp
xim[d] += (xterm_im * t) >> wp
if reflect:
yre[d] += (yterm_re * t) >> wp
yim[d] += (yterm_im * t) >> wp
t = (t * log) >> wp
else:
xre[0] += xterm_re
xim[0] += xterm_im
if reflect:
yre[0] += yterm_re
yim[0] += yterm_im
if have_derivatives:
if have_one_derivative:
if maxd % 2:
xre[0] = -xre[0]
xim[0] = -xim[0]
if reflect:
yre[0] = -yre[0]
yim[0] = -yim[0]
else:
xre = [(-1)**d * xre[d] for d in derivatives]
xim = [(-1)**d * xim[d] for d in derivatives]
if reflect:
yre = [(-1)**d * yre[d] for d in derivatives]
yim = [(-1)**d * yim[d] for d in derivatives]
xs = [(from_man_exp(xa, -wp, prec, 'n'), from_man_exp(xb, -wp, prec, 'n'))
for (xa, xb) in zip(xre, xim)]
ys = [(from_man_exp(ya, -wp, prec, 'n'), from_man_exp(yb, -wp, prec, 'n'))
for (ya, yb) in zip(yre, yim)]
return xs, ys
#-----------------------------------------------------------------------#
# #
# The gamma function (NEW IMPLEMENTATION) #
# #
#-----------------------------------------------------------------------#
# Higher means faster, but more precomputation time
MAX_GAMMA_TAYLOR_PREC = 5000
# Need to derive higher bounds for Taylor series to go higher
assert MAX_GAMMA_TAYLOR_PREC < 15000
# Use Stirling's series if abs(x) > beta*prec
# Important: must be large enough for convergence!
GAMMA_STIRLING_BETA = 0.2
SMALL_FACTORIAL_CACHE_SIZE = 150
gamma_taylor_cache = {}
gamma_stirling_cache = {}
small_factorial_cache = [from_int(ifac(n)) for \
n in range(SMALL_FACTORIAL_CACHE_SIZE+1)]
def zeta_array(N, prec):
"""
zeta(n) = A * pi**n / n! + B
where A is a rational number (A = Bernoulli number
for n even) and B is an infinite sum over powers of exp(2*pi).
(B = 0 for n even).
TODO: this is currently only used for gamma, but could
be very useful elsewhere.
"""
extra = 30
wp = prec+extra
zeta_values = [MPZ_ZERO] * (N+2)
pi = pi_fixed(wp)
# STEP 1:
one = MPZ_ONE << wp
zeta_values[0] = -one//2
f_2pi = mpf_shift(mpf_pi(wp),1)
exp_2pi_k = exp_2pi = mpf_exp(f_2pi, wp)
# Compute exponential series
# Store values of 1/(exp(2*pi*k)-1),
# exp(2*pi*k)/(exp(2*pi*k)-1)**2, 1/(exp(2*pi*k)-1)**2
# pi*k*exp(2*pi*k)/(exp(2*pi*k)-1)**2
exps3 = []
k = 1
while 1:
tp = wp - 9*k
if tp < 1:
break
# 1/(exp(2*pi*k-1)
q1 = mpf_div(fone, mpf_sub(exp_2pi_k, fone, tp), tp)
# pi*k*exp(2*pi*k)/(exp(2*pi*k)-1)**2
q2 = mpf_mul(exp_2pi_k, mpf_mul(q1,q1,tp), tp)
q1 = to_fixed(q1, wp)
q2 = to_fixed(q2, wp)
q2 = (k * q2 * pi) >> wp
exps3.append((q1, q2))
# Multiply for next round
exp_2pi_k = mpf_mul(exp_2pi_k, exp_2pi, wp)
k += 1
# Exponential sum
for n in xrange(3, N+1, 2):
s = MPZ_ZERO
k = 1
for e1, e2 in exps3:
if n%4 == 3:
t = e1 // k**n
else:
U = (n-1)//4
t = (e1 + e2//U) // k**n
if not t:
break
s += t
k += 1
zeta_values[n] = -2*s
# Even zeta values
B = [mpf_abs(mpf_bernoulli(k,wp)) for k in xrange(N+2)]
pi_pow = fpi = mpf_pow_int(mpf_shift(mpf_pi(wp), 1), 2, wp)
pi_pow = mpf_div(pi_pow, from_int(4), wp)
for n in xrange(2,N+2,2):
z = mpf_mul(B[n], pi_pow, wp)
zeta_values[n] = to_fixed(z, wp)
pi_pow = mpf_mul(pi_pow, fpi, wp)
pi_pow = mpf_div(pi_pow, from_int((n+1)*(n+2)), wp)
# Zeta sum
reciprocal_pi = (one << wp) // pi
for n in xrange(3, N+1, 4):
U = (n-3)//4
s = zeta_values[4*U+4]*(4*U+7)//4
for k in xrange(1, U+1):
s -= (zeta_values[4*k] * zeta_values[4*U+4-4*k]) >> wp
zeta_values[n] += (2*s*reciprocal_pi) >> wp
for n in xrange(5, N+1, 4):
U = (n-1)//4
s = zeta_values[4*U+2]*(2*U+1)
for k in xrange(1, 2*U+1):
s += ((-1)**k*2*k* zeta_values[2*k] * zeta_values[4*U+2-2*k])>>wp
zeta_values[n] += ((s*reciprocal_pi)>>wp)//(2*U)
return [x>>extra for x in zeta_values]
def gamma_taylor_coefficients(inprec):
"""
Gives the Taylor coefficients of 1/gamma(1+x) as
a list of fixed-point numbers. Enough coefficients are returned
to ensure that the series converges to the given precision
when x is in [0.5, 1.5].
"""
# Reuse nearby cache values (small case)
if inprec < 400:
prec = inprec + (10-(inprec%10))
elif inprec < 1000:
prec = inprec + (30-(inprec%30))
else:
prec = inprec
if prec in gamma_taylor_cache:
return gamma_taylor_cache[prec], prec
# Experimentally determined bounds
if prec < 1000:
N = int(prec**0.76 + 2)
else:
# Valid to at least 15000 bits
N = int(prec**0.787 + 2)
# Reuse higher precision values
for cprec in gamma_taylor_cache:
if cprec > prec:
coeffs = [x>>(cprec-prec) for x in gamma_taylor_cache[cprec][-N:]]
if inprec < 1000:
gamma_taylor_cache[prec] = coeffs
return coeffs, prec
# Cache at a higher precision (large case)
if prec > 1000:
prec = int(prec * 1.2)
wp = prec + 20
A = [0] * N
A[0] = MPZ_ZERO
A[1] = MPZ_ONE << wp
A[2] = euler_fixed(wp)
# SLOW, reference implementation
#zeta_values = [0,0]+[to_fixed(mpf_zeta_int(k,wp),wp) for k in xrange(2,N)]
zeta_values = zeta_array(N, wp)
for k in xrange(3, N):
a = (-A[2]*A[k-1])>>wp
for j in xrange(2,k):
a += ((-1)**j * zeta_values[j] * A[k-j]) >> wp
a //= (1-k)
A[k] = a
A = [a>>20 for a in A]
A = A[::-1]
A = A[:-1]
gamma_taylor_cache[prec] = A
#return A, prec
return gamma_taylor_coefficients(inprec)
def gamma_fixed_taylor(xmpf, x, wp, prec, rnd, type):
# Determine nearest multiple of N/2
#n = int(x >> (wp-1))
#steps = (n-1)>>1
nearest_int = ((x >> (wp-1)) + MPZ_ONE) >> 1
one = MPZ_ONE << wp
coeffs, cwp = gamma_taylor_coefficients(wp)
if nearest_int > 0:
r = one
for i in xrange(nearest_int-1):
x -= one
r = (r*x) >> wp
x -= one
p = MPZ_ZERO
for c in coeffs:
p = c + ((x*p)>>wp)
p >>= (cwp-wp)
if type == 0:
return from_man_exp((r<<wp)//p, -wp, prec, rnd)
if type == 2:
return mpf_shift(from_rational(p, (r<<wp), prec, rnd), wp)
if type == 3:
return mpf_log(mpf_abs(from_man_exp((r<<wp)//p, -wp)), prec, rnd)
else:
r = one
for i in xrange(-nearest_int):
r = (r*x) >> wp
x += one
p = MPZ_ZERO
for c in coeffs:
p = c + ((x*p)>>wp)
p >>= (cwp-wp)
if wp - bitcount(abs(x)) > 10:
# pass very close to 0, so do floating-point multiply
g = mpf_add(xmpf, from_int(-nearest_int)) # exact
r = from_man_exp(p*r,-wp-wp)
r = mpf_mul(r, g, wp)
if type == 0:
return mpf_div(fone, r, prec, rnd)
if type == 2:
return mpf_pos(r, prec, rnd)
if type == 3:
return mpf_log(mpf_abs(mpf_div(fone, r, wp)), prec, rnd)
else:
r = from_man_exp(x*p*r,-3*wp)
if type == 0: return mpf_div(fone, r, prec, rnd)
if type == 2: return mpf_pos(r, prec, rnd)
if type == 3: return mpf_neg(mpf_log(mpf_abs(r), prec, rnd))
def stirling_coefficient(n):
if n in gamma_stirling_cache:
return gamma_stirling_cache[n]
p, q = bernfrac(n)
q *= MPZ(n*(n-1))
gamma_stirling_cache[n] = p, q, bitcount(abs(p)), bitcount(q)
return gamma_stirling_cache[n]
def real_stirling_series(x, prec):
"""
Sums the rational part of Stirling's expansion,
log(sqrt(2*pi)) - z + 1/(12*z) - 1/(360*z^3) + ...
"""
t = (MPZ_ONE<<(prec+prec)) // x # t = 1/x
u = (t*t)>>prec # u = 1/x**2
s = ln_sqrt2pi_fixed(prec) - x
# Add initial terms of Stirling's series
s += t//12; t = (t*u)>>prec
s -= t//360; t = (t*u)>>prec
s += t//1260; t = (t*u)>>prec
s -= t//1680; t = (t*u)>>prec
if not t: return s
s += t//1188; t = (t*u)>>prec
s -= 691*t//360360; t = (t*u)>>prec
s += t//156; t = (t*u)>>prec
if not t: return s
s -= 3617*t//122400; t = (t*u)>>prec
s += 43867*t//244188; t = (t*u)>>prec
s -= 174611*t//125400; t = (t*u)>>prec
if not t: return s
k = 22
# From here on, the coefficients are growing, so we
# have to keep t at a roughly constant size
usize = bitcount(abs(u))
tsize = bitcount(abs(t))
texp = 0
while 1:
p, q, pb, qb = stirling_coefficient(k)
term_mag = tsize + pb + texp
shift = -texp
m = pb - term_mag
if m > 0 and shift < m:
p >>= m
shift -= m
m = tsize - term_mag
if m > 0 and shift < m:
w = t >> m
shift -= m
else:
w = t
term = (t*p//q) >> shift
if not term:
break
s += term
t = (t*u) >> usize
texp -= (prec - usize)
k += 2
return s
def complex_stirling_series(x, y, prec):
# t = 1/z
_m = (x*x + y*y) >> prec
tre = (x << prec) // _m
tim = (-y << prec) // _m
# u = 1/z**2
ure = (tre*tre - tim*tim) >> prec
uim = tim*tre >> (prec-1)
# s = log(sqrt(2*pi)) - z
sre = ln_sqrt2pi_fixed(prec) - x
sim = -y
# Add initial terms of Stirling's series
sre += tre//12; sim += tim//12;
tre, tim = ((tre*ure-tim*uim)>>prec), ((tre*uim+tim*ure)>>prec)
sre -= tre//360; sim -= tim//360;
tre, tim = ((tre*ure-tim*uim)>>prec), ((tre*uim+tim*ure)>>prec)
sre += tre//1260; sim += tim//1260;
tre, tim = ((tre*ure-tim*uim)>>prec), ((tre*uim+tim*ure)>>prec)
sre -= tre//1680; sim -= tim//1680;
tre, tim = ((tre*ure-tim*uim)>>prec), ((tre*uim+tim*ure)>>prec)
if abs(tre) + abs(tim) < 5: return sre, sim
sre += tre//1188; sim += tim//1188;
tre, tim = ((tre*ure-tim*uim)>>prec), ((tre*uim+tim*ure)>>prec)
sre -= 691*tre//360360; sim -= 691*tim//360360;
tre, tim = ((tre*ure-tim*uim)>>prec), ((tre*uim+tim*ure)>>prec)
sre += tre//156; sim += tim//156;
tre, tim = ((tre*ure-tim*uim)>>prec), ((tre*uim+tim*ure)>>prec)
if abs(tre) + abs(tim) < 5: return sre, sim
sre -= 3617*tre//122400; sim -= 3617*tim//122400;
tre, tim = ((tre*ure-tim*uim)>>prec), ((tre*uim+tim*ure)>>prec)
sre += 43867*tre//244188; sim += 43867*tim//244188;
tre, tim = ((tre*ure-tim*uim)>>prec), ((tre*uim+tim*ure)>>prec)
sre -= 174611*tre//125400; sim -= 174611*tim//125400;
tre, tim = ((tre*ure-tim*uim)>>prec), ((tre*uim+tim*ure)>>prec)
if abs(tre) + abs(tim) < 5: return sre, sim
k = 22
# From here on, the coefficients are growing, so we
# have to keep t at a roughly constant size
usize = bitcount(max(abs(ure), abs(uim)))
tsize = bitcount(max(abs(tre), abs(tim)))
texp = 0
while 1:
p, q, pb, qb = stirling_coefficient(k)
term_mag = tsize + pb + texp
shift = -texp
m = pb - term_mag
if m > 0 and shift < m:
p >>= m
shift -= m
m = tsize - term_mag
if m > 0 and shift < m:
wre = tre >> m
wim = tim >> m
shift -= m
else:
wre = tre
wim = tim
termre = (tre*p//q) >> shift
termim = (tim*p//q) >> shift
if abs(termre) + abs(termim) < 5:
break
sre += termre
sim += termim
tre, tim = ((tre*ure - tim*uim)>>usize), \
((tre*uim + tim*ure)>>usize)
texp -= (prec - usize)
k += 2
return sre, sim
def mpf_gamma(x, prec, rnd='d', type=0):
"""
This function implements multipurpose evaluation of the gamma
function, G(x), as well as the following versions of the same:
type = 0 -- G(x) [standard gamma function]
type = 1 -- G(x+1) = x*G(x+1) = x! [factorial]
type = 2 -- 1/G(x) [reciprocal gamma function]
type = 3 -- log(|G(x)|) [log-gamma function, real part]
"""
# Specal values
sign, man, exp, bc = x
if not man:
if x == fzero:
if type == 1: return fone
if type == 2: return fzero
raise ValueError("gamma function pole")
if x == finf:
if type == 2: return fzero
return finf
return fnan
# First of all, for log gamma, numbers can be well beyond the fixed-point
# range, so we must take care of huge numbers before e.g. trying
# to convert x to the nearest integer
if type == 3:
wp = prec+20
if exp+bc > wp and not sign:
return mpf_sub(mpf_mul(x, mpf_log(x, wp), wp), x, prec, rnd)
# We strongly want to special-case small integers
is_integer = exp >= 0
if is_integer:
# Poles
if sign:
if type == 2:
return fzero
raise ValueError("gamma function pole")
# n = x
n = man << exp
if n < SMALL_FACTORIAL_CACHE_SIZE:
if type == 0:
return mpf_pos(small_factorial_cache[n-1], prec, rnd)
if type == 1:
return mpf_pos(small_factorial_cache[n], prec, rnd)
if type == 2:
return mpf_div(fone, small_factorial_cache[n-1], prec, rnd)
if type == 3:
return mpf_log(small_factorial_cache[n-1], prec, rnd)
else:
# floor(abs(x))
n = int(man >> (-exp))
# Estimate size and precision
# Estimate log(gamma(|x|),2) as x*log(x,2)
mag = exp + bc
gamma_size = n*mag
if type == 3:
wp = prec + 20
else:
wp = prec + bitcount(gamma_size) + 20
# Very close to 0, pole
if mag < -wp:
if type == 0:
return mpf_sub(mpf_div(fone,x, wp),mpf_shift(fone,-wp),prec,rnd)
if type == 1: return mpf_sub(fone, x, prec, rnd)
if type == 2: return mpf_add(x, mpf_shift(fone,mag-wp), prec, rnd)
if type == 3: return mpf_neg(mpf_log(mpf_abs(x), prec, rnd))
# From now on, we assume having a gamma function
if type == 1:
return mpf_gamma(mpf_add(x, fone), prec, rnd, 0)
# Special case integers (those not small enough to be caught above,
# but still small enough for an exact factorial to be faster
# than an approximate algorithm), and half-integers
if exp >= -1:
if is_integer:
if gamma_size < 10*wp:
if type == 0:
return from_int(ifac(n-1), prec, rnd)
if type == 2:
return from_rational(MPZ_ONE, ifac(n-1), prec, rnd)
if type == 3:
return mpf_log(from_int(ifac(n-1)), prec, rnd)
# half-integer
if n < 100 or gamma_size < 10*wp:
if sign:
w = sqrtpi_fixed(wp)
if n % 2: f = ifac2(2*n+1)
else: f = -ifac2(2*n+1)
if type == 0:
return mpf_shift(from_rational(w, f, prec, rnd), -wp+n+1)
if type == 2:
return mpf_shift(from_rational(f, w, prec, rnd), wp-n-1)
if type == 3:
return mpf_log(mpf_shift(from_rational(w, abs(f),
prec, rnd), -wp+n+1), prec, rnd)
elif n == 0:
if type == 0: return mpf_sqrtpi(prec, rnd)
if type == 2: return mpf_div(fone, mpf_sqrtpi(wp), prec, rnd)
if type == 3: return mpf_log(mpf_sqrtpi(wp), prec, rnd)
else:
w = sqrtpi_fixed(wp)
w = from_man_exp(w * ifac2(2*n-1), -wp-n)
if type == 0: return mpf_pos(w, prec, rnd)
if type == 2: return mpf_div(fone, w, prec, rnd)
if type == 3: return mpf_log(mpf_abs(w), prec, rnd)
# Convert to fixed point
offset = exp + wp
if offset >= 0: absxman = man << offset
else: absxman = man >> (-offset)
# For log gamma, provide accurate evaluation for x = 1+eps and 2+eps
if type == 3 and not sign:
one = MPZ_ONE << wp
one_dist = abs(absxman-one)
two_dist = abs(absxman-2*one)
cancellation = (wp - bitcount(min(one_dist, two_dist)))
if cancellation > 10:
xsub1 = mpf_sub(fone, x)
xsub2 = mpf_sub(ftwo, x)
xsub1mag = xsub1[2]+xsub1[3]
xsub2mag = xsub2[2]+xsub2[3]
if xsub1mag < -wp:
return mpf_mul(mpf_euler(wp), mpf_sub(fone, x), prec, rnd)
if xsub2mag < -wp:
return mpf_mul(mpf_sub(fone, mpf_euler(wp)),
mpf_sub(x, ftwo), prec, rnd)
# Proceed but increase precision
wp += max(-xsub1mag, -xsub2mag)
offset = exp + wp
if offset >= 0: absxman = man << offset
else: absxman = man >> (-offset)
# Use Taylor series if appropriate
n_for_stirling = int(GAMMA_STIRLING_BETA*wp)
if n < max(100, n_for_stirling) and wp < MAX_GAMMA_TAYLOR_PREC:
if sign:
absxman = -absxman
return gamma_fixed_taylor(x, absxman, wp, prec, rnd, type)
# Use Stirling's series
# First ensure that |x| is large enough for rapid convergence
xorig = x
# Argument reduction
r = 0
if n < n_for_stirling:
r = one = MPZ_ONE << wp
d = n_for_stirling - n
for k in xrange(d):
r = (r * absxman) >> wp
absxman += one
x = xabs = from_man_exp(absxman, -wp)
if sign:
x = mpf_neg(x)
else:
xabs = mpf_abs(x)
# Asymptotic series
y = real_stirling_series(absxman, wp)
u = to_fixed(mpf_log(xabs, wp), wp)
u = ((absxman - (MPZ_ONE<<(wp-1))) * u) >> wp
y += u
w = from_man_exp(y, -wp)
# Compute final value
if sign:
# Reflection formula
A = mpf_mul(mpf_sin_pi(xorig, wp), xorig, wp)
B = mpf_neg(mpf_pi(wp))
if type == 0 or type == 2:
A = mpf_mul(A, mpf_exp(w, wp))
if r:
B = mpf_mul(B, from_man_exp(r, -wp), wp)
if type == 0:
return mpf_div(B, A, prec, rnd)
if type == 2:
return mpf_div(A, B, prec, rnd)
if type == 3:
if r:
B = mpf_mul(B, from_man_exp(r, -wp), wp)
A = mpf_add(mpf_log(mpf_abs(A), wp), w, wp)
return mpf_sub(mpf_log(mpf_abs(B), wp), A, prec, rnd)
else:
if type == 0:
if r:
return mpf_div(mpf_exp(w, wp),
from_man_exp(r, -wp), prec, rnd)
return mpf_exp(w, prec, rnd)
if type == 2:
if r:
return mpf_div(from_man_exp(r, -wp),
mpf_exp(w, wp), prec, rnd)
return mpf_exp(mpf_neg(w), prec, rnd)
if type == 3:
if r:
return mpf_sub(w, mpf_log(from_man_exp(r,-wp), wp), prec, rnd)
return mpf_pos(w, prec, rnd)
def mpc_gamma(z, prec, rnd='d', type=0):
a, b = z
asign, aman, aexp, abc = a
bsign, bman, bexp, bbc = b
if b == fzero:
# Imaginary part on negative half-axis for log-gamma function
if type == 3 and asign:
re = mpf_gamma(a, prec, rnd, 3)
n = (-aman) >> (-aexp)
im = mpf_mul_int(mpf_pi(prec+10), n, prec, rnd)
return re, im
return mpf_gamma(a, prec, rnd, type), fzero
# Some kind of complex inf/nan
if (not aman and aexp) or (not bman and bexp):
return (fnan, fnan)
# Initial working precision
wp = prec + 20
amag = aexp+abc
bmag = bexp+bbc
if aman:
mag = max(amag, bmag)
else:
mag = bmag
# Close to 0
if mag < -8:
if mag < -wp:
# 1/gamma(z) = z + euler*z^2 + O(z^3)
v = mpc_add(z, mpc_mul_mpf(mpc_mul(z,z,wp),mpf_euler(wp),wp), wp)
if type == 0: return mpc_reciprocal(v, prec, rnd)
if type == 1: return mpc_div(z, v, prec, rnd)
if type == 2: return mpc_pos(v, prec, rnd)
if type == 3: return mpc_log(mpc_reciprocal(v, prec), prec, rnd)
elif type != 1:
wp += (-mag)
# Handle huge log-gamma values; must do this before converting to
# a fixed-point value. TODO: determine a precise cutoff of validity
# depending on amag and bmag
if type == 3 and mag > wp and ((not asign) or (bmag >= amag)):
return mpc_sub(mpc_mul(z, mpc_log(z, wp), wp), z, prec, rnd)
# From now on, we assume having a gamma function
if type == 1:
return mpc_gamma((mpf_add(a, fone), b), prec, rnd, 0)
an = abs(to_int(a))
bn = abs(to_int(b))
absn = max(an, bn)
gamma_size = absn*mag
if type == 3:
pass
else:
wp += bitcount(gamma_size)
# Reflect to the right half-plane. Note that Stirling's expansion
# is valid in the left half-plane too, as long as we're not too close
# to the real axis, but in order to use this argument reduction
# in the negative direction must be implemented.
#need_reflection = asign and ((bmag < 0) or (amag-bmag > 4))
need_reflection = asign
zorig = z
if need_reflection:
z = mpc_neg(z)
asign, aman, aexp, abc = a = z[0]
bsign, bman, bexp, bbc = b = z[1]
# Imaginary part very small compared to real one?
yfinal = 0
balance_prec = 0
if bmag < -10:
# Check z ~= 1 and z ~= 2 for loggamma
if type == 3:
zsub1 = mpc_sub_mpf(z, fone)
if zsub1[0] == fzero:
cancel1 = -bmag
else:
cancel1 = -max(zsub1[0][2]+zsub1[0][3], bmag)
if cancel1 > wp:
pi = mpf_pi(wp)
x = mpc_mul_mpf(zsub1, pi, wp)
x = mpc_mul(x, x, wp)
x = mpc_div_mpf(x, from_int(12), wp)
y = mpc_mul_mpf(zsub1, mpf_neg(mpf_euler(wp)), wp)
yfinal = mpc_add(x, y, wp)
if not need_reflection:
return mpc_pos(yfinal, prec, rnd)
elif cancel1 > 0:
wp += cancel1
zsub2 = mpc_sub_mpf(z, ftwo)
if zsub2[0] == fzero:
cancel2 = -bmag
else:
cancel2 = -max(zsub2[0][2]+zsub2[0][3], bmag)
if cancel2 > wp:
pi = mpf_pi(wp)
t = mpf_sub(mpf_mul(pi, pi), from_int(6))
x = mpc_mul_mpf(mpc_mul(zsub2, zsub2, wp), t, wp)
x = mpc_div_mpf(x, from_int(12), wp)
y = mpc_mul_mpf(zsub2, mpf_sub(fone, mpf_euler(wp)), wp)
yfinal = mpc_add(x, y, wp)
if not need_reflection:
return mpc_pos(yfinal, prec, rnd)
elif cancel2 > 0:
wp += cancel2
if bmag < -wp:
# Compute directly from the real gamma function.
pp = 2*(wp+10)
aabs = mpf_abs(a)
eps = mpf_shift(fone, amag-wp)
x1 = mpf_gamma(aabs, pp, type=type)
x2 = mpf_gamma(mpf_add(aabs, eps), pp, type=type)
xprime = mpf_div(mpf_sub(x2, x1, pp), eps, pp)
y = mpf_mul(b, xprime, prec, rnd)
yfinal = (x1, y)
# Note: we still need to use the reflection formula for
# near-poles, and the correct branch of the log-gamma function
if not need_reflection:
return mpc_pos(yfinal, prec, rnd)
else:
balance_prec += (-bmag)
wp += balance_prec
n_for_stirling = int(GAMMA_STIRLING_BETA*wp)
need_reduction = absn < n_for_stirling
afix = to_fixed(a, wp)
bfix = to_fixed(b, wp)
r = 0
if not yfinal:
zprered = z
# Argument reduction
if absn < n_for_stirling:
absn = complex(an, bn)
d = int((1 + n_for_stirling**2 - bn**2)**0.5 - an)
rre = one = MPZ_ONE << wp
rim = MPZ_ZERO
for k in xrange(d):
rre, rim = ((afix*rre-bfix*rim)>>wp), ((afix*rim + bfix*rre)>>wp)
afix += one
r = from_man_exp(rre, -wp), from_man_exp(rim, -wp)
a = from_man_exp(afix, -wp)
z = a, b
yre, yim = complex_stirling_series(afix, bfix, wp)
# (z-1/2)*log(z) + S
lre, lim = mpc_log(z, wp)
lre = to_fixed(lre, wp)
lim = to_fixed(lim, wp)
yre = ((lre*afix - lim*bfix)>>wp) - (lre>>1) + yre
yim = ((lre*bfix + lim*afix)>>wp) - (lim>>1) + yim
y = from_man_exp(yre, -wp), from_man_exp(yim, -wp)
if r and type == 3:
# If re(z) > 0 and abs(z) <= 4, the branches of loggamma(z)
# and log(gamma(z)) coincide. Otherwise, use the zeroth order
# Stirling expansion to compute the correct imaginary part.
y = mpc_sub(y, mpc_log(r, wp), wp)
zfa = to_float(zprered[0])
zfb = to_float(zprered[1])
zfabs = math.hypot(zfa,zfb)
#if not (zfa > 0.0 and zfabs <= 4):
yfb = to_float(y[1])
u = math.atan2(zfb, zfa)
if zfabs <= 0.5:
gi = 0.577216*zfb - u
else:
gi = -zfb - 0.5*u + zfa*u + zfb*math.log(zfabs)
n = int(math.floor((gi-yfb)/(2*math.pi)+0.5))
y = (y[0], mpf_add(y[1], mpf_mul_int(mpf_pi(wp), 2*n, wp), wp))
if need_reflection:
if type == 0 or type == 2:
A = mpc_mul(mpc_sin_pi(zorig, wp), zorig, wp)
B = (mpf_neg(mpf_pi(wp)), fzero)
if yfinal:
if type == 2:
A = mpc_div(A, yfinal, wp)
else:
A = mpc_mul(A, yfinal, wp)
else:
A = mpc_mul(A, mpc_exp(y, wp), wp)
if r:
B = mpc_mul(B, r, wp)
if type == 0: return mpc_div(B, A, prec, rnd)
if type == 2: return mpc_div(A, B, prec, rnd)
# Reflection formula for the log-gamma function with correct branch
# http://functions.wolfram.com/GammaBetaErf/LogGamma/16/01/01/0006/
# LogGamma[z] == -LogGamma[-z] - Log[-z] +
# Sign[Im[z]] Floor[Re[z]] Pi I + Log[Pi] -
# Log[Sin[Pi (z - Floor[Re[z]])]] -
# Pi I (1 - Abs[Sign[Im[z]]]) Abs[Floor[Re[z]]]
if type == 3:
if yfinal:
s1 = mpc_neg(yfinal)
else:
s1 = mpc_neg(y)
# s -= log(-z)
s1 = mpc_sub(s1, mpc_log(mpc_neg(zorig), wp), wp)
# floor(re(z))
rezfloor = mpf_floor(zorig[0])
imzsign = mpf_sign(zorig[1])
pi = mpf_pi(wp)
t = mpf_mul(pi, rezfloor)
t = mpf_mul_int(t, imzsign, wp)
s1 = (s1[0], mpf_add(s1[1], t, wp))
s1 = mpc_add_mpf(s1, mpf_log(pi, wp), wp)
t = mpc_sin_pi(mpc_sub_mpf(zorig, rezfloor), wp)
t = mpc_log(t, wp)
s1 = mpc_sub(s1, t, wp)
# Note: may actually be unused, because we fall back
# to the mpf_ function for real arguments
if not imzsign:
t = mpf_mul(pi, mpf_floor(rezfloor), wp)
s1 = (s1[0], mpf_sub(s1[1], t, wp))
return mpc_pos(s1, prec, rnd)
else:
if type == 0:
if r:
return mpc_div(mpc_exp(y, wp), r, prec, rnd)
return mpc_exp(y, prec, rnd)
if type == 2:
if r:
return mpc_div(r, mpc_exp(y, wp), prec, rnd)
return mpc_exp(mpc_neg(y), prec, rnd)
if type == 3:
return mpc_pos(y, prec, rnd)
def mpf_factorial(x, prec, rnd='d'):
return mpf_gamma(x, prec, rnd, 1)
def mpc_factorial(x, prec, rnd='d'):
return mpc_gamma(x, prec, rnd, 1)
def mpf_rgamma(x, prec, rnd='d'):
return mpf_gamma(x, prec, rnd, 2)
def mpc_rgamma(x, prec, rnd='d'):
return mpc_gamma(x, prec, rnd, 2)
def mpf_loggamma(x, prec, rnd='d'):
sign, man, exp, bc = x
if sign:
raise ComplexResult
return mpf_gamma(x, prec, rnd, 3)
def mpc_loggamma(z, prec, rnd='d'):
a, b = z
asign, aman, aexp, abc = a
bsign, bman, bexp, bbc = b
if b == fzero and asign:
re = mpf_gamma(a, prec, rnd, 3)
n = (-aman) >> (-aexp)
im = mpf_mul_int(mpf_pi(prec+10), n, prec, rnd)
return re, im
return mpc_gamma(z, prec, rnd, 3)
def mpf_gamma_int(n, prec, rnd=round_fast):
if n < SMALL_FACTORIAL_CACHE_SIZE:
return mpf_pos(small_factorial_cache[n-1], prec, rnd)
return mpf_gamma(from_int(n), prec, rnd)
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