117 lines
2.4 KiB
Python
117 lines
2.4 KiB
Python
"""
|
|
Precompute coefficients of Temme's asymptotic expansion for gammainc.
|
|
|
|
This takes about 8 hours to run on a 2.3 GHz Macbook Pro with 4GB ram.
|
|
|
|
Sources:
|
|
[1] NIST, "Digital Library of Mathematical Functions",
|
|
https://dlmf.nist.gov/
|
|
|
|
"""
|
|
import os
|
|
from scipy.special._precompute.utils import lagrange_inversion
|
|
|
|
try:
|
|
import mpmath as mp
|
|
except ImportError:
|
|
pass
|
|
|
|
|
|
def compute_a(n):
|
|
"""a_k from DLMF 5.11.6"""
|
|
a = [mp.sqrt(2)/2]
|
|
for k in range(1, n):
|
|
ak = a[-1]/k
|
|
for j in range(1, len(a)):
|
|
ak -= a[j]*a[-j]/(j + 1)
|
|
ak /= a[0]*(1 + mp.mpf(1)/(k + 1))
|
|
a.append(ak)
|
|
return a
|
|
|
|
|
|
def compute_g(n):
|
|
"""g_k from DLMF 5.11.3/5.11.5"""
|
|
a = compute_a(2*n)
|
|
g = [mp.sqrt(2)*mp.rf(0.5, k)*a[2*k] for k in range(n)]
|
|
return g
|
|
|
|
|
|
def eta(lam):
|
|
"""Function from DLMF 8.12.1 shifted to be centered at 0."""
|
|
if lam > 0:
|
|
return mp.sqrt(2*(lam - mp.log(lam + 1)))
|
|
elif lam < 0:
|
|
return -mp.sqrt(2*(lam - mp.log(lam + 1)))
|
|
else:
|
|
return 0
|
|
|
|
|
|
def compute_alpha(n):
|
|
"""alpha_n from DLMF 8.12.13"""
|
|
coeffs = mp.taylor(eta, 0, n - 1)
|
|
return lagrange_inversion(coeffs)
|
|
|
|
|
|
def compute_d(K, N):
|
|
"""d_{k, n} from DLMF 8.12.12"""
|
|
M = N + 2*K
|
|
d0 = [-mp.mpf(1)/3]
|
|
alpha = compute_alpha(M + 2)
|
|
for n in range(1, M):
|
|
d0.append((n + 2)*alpha[n+2])
|
|
d = [d0]
|
|
g = compute_g(K)
|
|
for k in range(1, K):
|
|
dk = []
|
|
for n in range(M - 2*k):
|
|
dk.append((-1)**k*g[k]*d[0][n] + (n + 2)*d[k-1][n+2])
|
|
d.append(dk)
|
|
for k in range(K):
|
|
d[k] = d[k][:N]
|
|
return d
|
|
|
|
|
|
header = \
|
|
r"""/* This file was automatically generated by _precomp/gammainc.py.
|
|
* Do not edit it manually!
|
|
*/
|
|
|
|
#ifndef IGAM_H
|
|
#define IGAM_H
|
|
|
|
#define K {}
|
|
#define N {}
|
|
|
|
static const double d[K][N] =
|
|
{{"""
|
|
|
|
footer = \
|
|
r"""
|
|
#endif
|
|
"""
|
|
|
|
|
|
def main():
|
|
print(__doc__)
|
|
K = 25
|
|
N = 25
|
|
with mp.workdps(50):
|
|
d = compute_d(K, N)
|
|
fn = os.path.join(os.path.dirname(__file__), '..', 'cephes', 'igam.h')
|
|
with open(fn + '.new', 'w') as f:
|
|
f.write(header.format(K, N))
|
|
for k, row in enumerate(d):
|
|
row = [mp.nstr(x, 17, min_fixed=0, max_fixed=0) for x in row]
|
|
f.write('{')
|
|
f.write(", ".join(row))
|
|
if k < K - 1:
|
|
f.write('},\n')
|
|
else:
|
|
f.write('}};\n')
|
|
f.write(footer)
|
|
os.rename(fn + '.new', fn)
|
|
|
|
|
|
if __name__ == "__main__":
|
|
main()
|