lanczos.h

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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.
 */
 
#ifndef LANCZOS_H
#define LANCZOS_H
 
 
static const double lanczos_num[13] = {
    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
};
 
static const double lanczos_denom[13] = {
    1,
    66,
    1925,
    32670,
    357423,
    2637558,
    13339535,
    45995730,
    105258076,
    150917976,
    120543840,
    39916800,
    0
};
 
static const double lanczos_sum_expg_scaled_num[13] = {
    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
};
 
static const double lanczos_sum_expg_scaled_denom[13] = {
    1,
    66,
    1925,
    32670,
    357423,
    2637558,
    13339535,
    45995730,
    105258076,
    150917976,
    120543840,
    39916800,
    0
};
 
static const double lanczos_sum_near_1_d[12] = {
    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
};
 
static const double lanczos_sum_near_2_d[12] = {
    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
};
 
static const double lanczos_g = 6.024680040776729583740234375;
 
#endif