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/* (C) Copyright John Maddock 2006.
* Use, modification and distribution are subject to the
* Boost Software License, Version 1.0. (See accompanying file
* LICENSE_1_0.txt or copy at https://www.boost.org/LICENSE_1_0.txt)
*/
/* Both lanczos.h and lanczos.c were formed from Boost's lanczos.hpp
*
* Scipy changes:
* - 06-22-2016: Removed all code not related to double precision and
* ported to c for use in Cephes. Note that the order of the
* coefficients is reversed to match the behavior of polevl.
*/
/*
* Optimal values for G for each N are taken from
* https://web.viu.ca/pughg/phdThesis/phdThesis.pdf,
* as are the theoretical error bounds.
*
* Constants calculated using the method described by Godfrey
* https://my.fit.edu/~gabdo/gamma.txt and elaborated by Toth at
* https://www.rskey.org/gamma.htm using NTL::RR at 1000 bit precision.
*/
/*
* Lanczos Coefficients for N=13 G=6.024680040776729583740234375
* Max experimental error (with arbitrary precision arithmetic) 1.196214e-17
* Generated with compiler: Microsoft Visual C++ version 8.0 on Win32 at Mar 23 2006
*
* Use for double precision.
*/
#pragma once
#include "../config.h"
#include "polevl.h"
namespace xsf {
namespace cephes {
namespace detail {
constexpr double lanczos_num[] = {
2.506628274631000270164908177133837338626, 210.8242777515793458725097339207133627117,
8071.672002365816210638002902272250613822, 186056.2653952234950402949897160456992822,
2876370.628935372441225409051620849613599, 31426415.58540019438061423162831820536287,
248874557.8620541565114603864132294232163, 1439720407.311721673663223072794912393972,
6039542586.35202800506429164430729792107, 17921034426.03720969991975575445893111267,
35711959237.35566804944018545154716670596, 42919803642.64909876895789904700198885093,
23531376880.41075968857200767445163675473};
constexpr double lanczos_denom[] = {1, 66, 1925, 32670, 357423, 2637558, 13339535,
45995730, 105258076, 150917976, 120543840, 39916800, 0};
constexpr double lanczos_sum_expg_scaled_num[] = {
0.006061842346248906525783753964555936883222, 0.5098416655656676188125178644804694509993,
19.51992788247617482847860966235652136208, 449.9445569063168119446858607650988409623,
6955.999602515376140356310115515198987526, 75999.29304014542649875303443598909137092,
601859.6171681098786670226533699352302507, 3481712.15498064590882071018964774556468,
14605578.08768506808414169982791359218571, 43338889.32467613834773723740590533316085,
86363131.28813859145546927288977868422342, 103794043.1163445451906271053616070238554,
56906521.91347156388090791033559122686859};
constexpr double lanczos_sum_expg_scaled_denom[] = {
1, 66, 1925, 32670, 357423, 2637558, 13339535, 45995730, 105258076, 150917976, 120543840, 39916800, 0};
constexpr double lanczos_sum_near_1_d[] = {
0.3394643171893132535170101292240837927725e-9, -0.2499505151487868335680273909354071938387e-8,
0.8690926181038057039526127422002498960172e-8, -0.1933117898880828348692541394841204288047e-7,
0.3075580174791348492737947340039992829546e-7, -0.2752907702903126466004207345038327818713e-7,
-0.1515973019871092388943437623825208095123e-5, 0.004785200610085071473880915854204301886437,
-0.1993758927614728757314233026257810172008, 1.483082862367253753040442933770164111678,
-3.327150580651624233553677113928873034916, 2.208709979316623790862569924861841433016};
constexpr double lanczos_sum_near_2_d[] = {
0.1009141566987569892221439918230042368112e-8, -0.7430396708998719707642735577238449585822e-8,
0.2583592566524439230844378948704262291927e-7, -0.5746670642147041587497159649318454348117e-7,
0.9142922068165324132060550591210267992072e-7, -0.8183698410724358930823737982119474130069e-7,
-0.4506604409707170077136555010018549819192e-5, 0.01422519127192419234315002746252160965831,
-0.5926941084905061794445733628891024027949, 4.408830289125943377923077727900630927902,
-9.8907772644920670589288081640128194231, 6.565936202082889535528455955485877361223};
XSF_HOST_DEVICE double lanczos_sum(double x) { return ratevl(x, lanczos_num, 12, lanczos_denom, 12); }
XSF_HOST_DEVICE double lanczos_sum_near_1(double dx) {
double result = 0;
unsigned k;
for (k = 1; k <= 12; ++k) {
result += (-lanczos_sum_near_1_d[k - 1] * dx) / (k * dx + k * k);
}
return result;
}
XSF_HOST_DEVICE double lanczos_sum_near_2(double dx) {
double result = 0;
double x = dx + 2;
unsigned k;
for (k = 1; k <= 12; ++k) {
result += (-lanczos_sum_near_2_d[k - 1] * dx) / (x + k * x + k * k - 1);
}
return result;
}
} // namespace detail
constexpr double lanczos_g = 6.024680040776729583740234375;
XSF_HOST_DEVICE double lanczos_sum_expg_scaled(double x) {
return ratevl(x, detail::lanczos_sum_expg_scaled_num, 12, detail::lanczos_sum_expg_scaled_denom, 12);
}
} // namespace cephes
} // namespace xsf
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