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https://github.com/saitohirga/WSJT-X.git
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4362 lines
189 KiB
C++
4362 lines
189 KiB
C++
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// (C) Copyright John Maddock 2006.
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// Use, modification and distribution are subject to the
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// Boost Software License, Version 1.0. (See accompanying file
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// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
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#include <boost/math/constants/constants.hpp>
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#include <boost/math/tools/precision.hpp>
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#include <boost/math/tools/polynomial.hpp>
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#include <boost/numeric/ublas/matrix.hpp>
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#include <boost/numeric/ublas/io.hpp>
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#include <boost/numeric/ublas/operation.hpp>
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#include <boost/cstdint.hpp>
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#include <boost/bind.hpp>
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#include <boost/math/special_functions/log1p.hpp>
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#include <boost/lexical_cast.hpp>
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#include <boost/array.hpp>
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#include "mp_t.hpp"
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//
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// this is a sort of recursive include, since this file
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// is used to create the contents of gamma.hpp.
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// However, we will be using the "generic" version of gamma
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// which doesn't use a lanczos approximation.
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//
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#include <boost/math/special_functions/gamma.hpp>
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#include <iostream>
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#include <iomanip>
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#include <vector>
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using boost::numeric::ublas::matrix;
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using boost::numeric::ublas::vector;
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struct lanczos_spot_data
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{
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int N;
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double g;
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double err;
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};
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lanczos_spot_data sweet_spots[] = {
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// a few selected spots from Pugh's thesis:
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3, 2.03209, 6.4e-7,
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4, 3.655180, 8.5e-8,
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9, 8.406094, 6.9e-15,
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10, 9.656578, 2.1e-16,
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11, 10.900511, 6.1e-18,
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12, 12.066012, 1.1e-19,
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13, 13.144565, 5.2e-21,
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14, 13.726821, 4.0e-22,
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15, 14.977863, 1.2e-23,
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20, 20.298892, 7.8e-31,
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21, 21.508926, 2.1e-32,
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22, 22.618910, 1.8e-34,
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23, 23.118012, 5.2e-35,
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// some more we've found, these are all the points where the first
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// negleted term from the Lanczos series changes sign, there is one
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// point just above that point, and one just below:
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3, 0.58613894134759903, 0.00036580426080686315,
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3, 0.58613894879817963, 0.00036580416320668711,
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3, 1.249951496720314, 4.2305536798953929e-005,
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3, 1.2499515041708946, 4.2305534015367749e-005,
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3, 1.8384982049465179, 1.164778449938813e-005,
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3, 1.8384982123970985, 1.1647786529958048e-005,
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3, 2.3276706635951996, 5.3732582230468998e-006,
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3, 2.3276706710457802, 5.3732600294516588e-006,
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3, 2.6038404405117035, 7.4060771171390379e-007,
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3, 2.6038404479622841, 7.406058543248471e-007,
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4, 0.62704569846391678, 0.00012800533557793425,
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4, 0.62704570591449738, 0.00012800531551557324,
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4, 1.3344274088740349, 9.0166140418663663e-006,
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4, 1.3344274163246155, 9.0166116338646694e-006,
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4, 1.9880444705486298, 1.4955557778725745e-006,
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4, 1.9880444779992104, 1.4955557079994109e-006,
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4, 2.5835030898451805, 4.5877676859600527e-007,
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4, 2.5835030972957611, 4.5877667876325684e-007,
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4, 3.1086832508444786, 2.1441725587707906e-007,
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4, 3.1086832582950592, 2.1441723688775765e-007,
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4, 3.5476611852645874, 1.0813760990515544e-007,
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4, 3.547661192715168, 1.0813766234070327e-007,
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4, 3.6492579728364944, 1.047642251124206e-007,
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4, 3.649257980287075, 1.0476415796074403e-007,
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5, 0.65435918420553207, 5.5274429026631278e-005,
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5, 0.65435919165611267, 5.527441185903385e-005,
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5, 1.389187254011631, 2.6307833416554052e-006,
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5, 1.3891872614622116, 2.6307826816540255e-006,
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5, 2.0801975876092911, 2.9014694764756466e-007,
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5, 2.0801975950598717, 2.9014696142909125e-007,
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5, 2.7272208333015442, 5.9176544536206063e-008,
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5, 2.7272208407521248, 5.9176528019009275e-008,
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5, 3.3269794136285782, 1.9379184595472579e-008,
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5, 3.3269794210791588, 1.9379189057179851e-008,
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5, 3.8722873777151108, 9.0123694745032944e-009,
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5, 3.8722873851656914, 9.0123661007867372e-009,
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5, 4.3516943231225014, 4.9816097608842378e-009,
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5, 4.351694330573082, 4.9816087871575021e-009,
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6, 0.67425807565450668, 2.7371903086326379e-005,
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6, 0.67425808310508728, 2.7371893712684938e-005,
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6, 1.4284561350941658, 9.4138428977455083e-007,
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6, 1.4284561425447464, 9.4138417896969612e-007,
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6, 2.1447814926505089, 7.4109049995920916e-008,
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6, 2.1447815001010895, 7.4109044064461768e-008,
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6, 2.8244904428720474, 1.0708333024058614e-008,
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6, 2.824490450322628, 1.0708333661155651e-008,
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6, 3.4669468700885773, 2.4940116660002578e-009,
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6, 3.4669468775391579, 2.4940118560592828e-009,
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6, 4.0696377158164978, 8.5267622527507512e-010,
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6, 4.0696377232670784, 8.5267624048985822e-010,
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6, 4.6278460025787354, 3.9323405231792876e-010,
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6, 4.6278460100293159, 3.9323393550021793e-010,
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6, 5.1345816180109978, 2.1943462473419721e-010,
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|
6, 5.1345816254615784, 2.1943458105218707e-010,
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6, 5.5917193591594696, 1.242916896389048e-010,
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6, 5.5917193666100502, 1.2429175679293363e-010,
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|
7, 0.68959406018257141, 1.4934177419305624e-005,
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7, 0.68959406763315201, 1.4934171696393048e-005,
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|
7, 1.4584139510989189, 3.8891482410396904e-007,
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7, 1.4584139585494995, 3.8891475436265016e-007,
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|
7, 2.1934122517704964, 2.2977989848570384e-008,
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7, 2.193412259221077, 2.2977982819591052e-008,
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7, 2.8964230641722679, 2.4742239281355561e-009,
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|
7, 2.8964230716228485, 2.4742234478616864e-009,
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7, 3.5678729563951492, 4.2816138965285379e-010,
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7, 3.5678729638457298, 4.2816128067858218e-010,
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7, 4.2069913893938065, 1.0931003472338519e-010,
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7, 4.2069913968443871, 1.0931004659777642e-010,
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7, 4.8118086308240891, 3.8518213120095777e-011,
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7, 4.8118086382746696, 3.8518212172324096e-011,
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7, 5.3790060579776764, 1.7613602961109952e-011,
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7, 5.379006065428257, 1.7613603275796266e-011,
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|
7, 5.9039439857006073, 9.7098281187573355e-012,
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|
7, 5.9039439931511879, 9.7098271522398387e-012,
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|
7, 6.3838147521018982, 5.7878801570305524e-012,
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|
7, 6.3838147595524788, 5.7878771973838702e-012,
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7, 6.7833591029047966, 2.829092368392454e-012,
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7, 6.7833591103553772, 2.8290901754246269e-012,
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|
8, 0.70188850909471512, 8.7610752024780369e-006,
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8, 0.70188851654529572, 8.7610704217796693e-006,
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8, 1.4822541922330856, 1.7881701068328982e-007,
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8, 1.4822541996836662, 1.7881694190606741e-007,
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8, 2.231784276664257, 8.2272386690167009e-009,
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8, 2.2317842841148376, 8.2272367814664617e-009,
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8, 2.9525627419352531, 6.8591536474980462e-010,
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8, 2.9525627493858337, 6.8591515664364383e-010,
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8, 3.6455128863453865, 9.1549235855884194e-011,
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|
8, 3.6455128937959671, 9.1549217088498416e-011,
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|
8, 4.3106606528162956, 1.8015466799942756e-011,
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|
8, 4.3106606602668762, 1.8015467001952356e-011,
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|
8, 4.9472371935844421, 4.918149362493385e-012,
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|
8, 4.9472372010350227, 4.9181495178774722e-012,
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|
8, 5.553666315972805, 1.7720036810370511e-012,
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|
8, 5.5536663234233856, 1.7720035779693502e-012,
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|
8, 6.1274929493665695, 8.0403304333886761e-013,
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|
8, 6.1274929568171501, 8.0403274526493467e-013,
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|
8, 6.6654317826032639, 4.3551924163930176e-013,
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|
8, 6.6654317900538445, 4.3551918580885179e-013,
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|
8, 7.1646013185381889, 2.6142608803934048e-013,
|
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|
8, 7.1646013259887695, 2.6142610866873453e-013,
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|
8, 7.6312204822897911, 1.5767278184484503e-013,
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|
8, 7.6312204897403717, 1.576728292094547e-013,
|
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|
8, 7.8779716044664383, 6.0579161144149095e-014,
|
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|
8, 7.8779716119170189, 6.0579145495868576e-014,
|
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|
9, 0.7120363786816597, 5.4383680639900953e-006,
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|
9, 0.7120363861322403, 5.4383647430909934e-006,
|
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|
9, 1.5018193200230598, 8.9331552269793478e-008,
|
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|
9, 1.5018193274736404, 8.9331529217486569e-008,
|
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|
9, 2.2630845010280609, 3.2922940247769628e-009,
|
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|
9, 2.2630845084786415, 3.2922936681951672e-009,
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|
9, 2.9980199560523033, 2.1887825488935961e-010,
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|
9, 2.9980199635028839, 2.1887818854849753e-010,
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|
9, 3.7078000977635384, 2.3213138957857667e-011,
|
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|
9, 3.707800105214119, 2.3213132917173771e-011,
|
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|
9, 4.3928612694144249, 3.6226509291513855e-012,
|
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|
9, 4.3928612768650055, 3.6226507116549233e-012,
|
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|
9, 5.0530400574207306, 7.8472620106742312e-013,
|
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|
9, 5.0530400648713112, 7.8472599177745781e-013,
|
||
|
9, 5.6876183524727821, 2.2542091218064693e-013,
|
||
|
9, 5.6876183599233627, 2.2542085472082343e-013,
|
||
|
9, 6.2953110262751579, 8.261026286619692e-014,
|
||
|
9, 6.2953110337257385, 8.2610267677539022e-014,
|
||
|
9, 6.8742315545678139, 3.7226229880661886e-014,
|
||
|
9, 6.8742315620183945, 3.722621661830273e-014,
|
||
|
9, 7.4219388365745544, 1.9804198141916339e-014,
|
||
|
9, 7.421938844025135, 1.9804197841080089e-014,
|
||
|
9, 7.936030775308609, 1.1800366216622659e-014,
|
||
|
9, 7.9360307827591896, 1.1800362381207757e-014,
|
||
|
9, 8.4175376817584038, 7.341772297716249e-015,
|
||
|
9, 8.4175376892089844, 7.3417744970482286e-015,
|
||
|
10, 0.72060277312994003, 3.5324438420494327e-006,
|
||
|
10, 0.72060278058052063, 3.5324425213780505e-006,
|
||
|
10, 1.5182576552033424, 4.7690979892410133e-008,
|
||
|
10, 1.518257662653923, 4.7690968887170632e-008,
|
||
|
10, 2.2892609387636185, 1.4396365431590603e-009,
|
||
|
10, 2.2892609462141991, 1.4396363283470758e-009,
|
||
|
10, 3.0358340814709663, 7.8118059457005561e-011,
|
||
|
10, 3.0358340889215469, 7.8118048721726738e-011,
|
||
|
10, 3.7592824399471283, 6.7417155809106958e-012,
|
||
|
10, 3.7592824473977089, 6.7417147580480395e-012,
|
||
|
10, 4.4602697342634201, 8.5434024476488055e-013,
|
||
|
10, 4.4602697417140007, 8.5434000006256964e-013,
|
||
|
10, 5.1389683187007904, 1.5015237157428729e-013,
|
||
|
10, 5.138968326151371, 1.5015232503714518e-013,
|
||
|
10, 5.795127771794796, 3.5035278065891386e-014,
|
||
|
10, 5.7951277792453766, 3.5035266685032581e-014,
|
||
|
10, 6.4280949085950851, 1.0474518584578706e-014,
|
||
|
10, 6.4280949160456657, 1.047451900091486e-014,
|
||
|
10, 7.0368044599890709, 3.8902615249132166e-015,
|
||
|
10, 7.0368044674396515, 3.8902605386615012e-015,
|
||
|
10, 7.6197631657123566, 1.742442940023884e-015,
|
||
|
10, 7.6197631731629372, 1.7424426824540998e-015,
|
||
|
10, 8.1750892177224159, 9.1144555287546294e-016,
|
||
|
10, 8.1750892251729965, 9.1144560389833963e-016,
|
||
|
10, 8.7008367404341698, 5.3541847847332351e-016,
|
||
|
10, 8.7008367478847504, 5.3541844723127514e-016,
|
||
|
10, 9.1965094059705734, 3.3560389199894529e-016,
|
||
|
10, 9.196509413421154, 3.3560370421140035e-016,
|
||
|
10, 9.6678015515208244, 2.1059571242602905e-016,
|
||
|
10, 9.667801558971405, 2.1059579862533338e-016,
|
||
|
11, 0.72796426713466644, 2.3816013324880167e-006,
|
||
|
11, 0.72796427458524704, 2.3816003939892356e-006,
|
||
|
11, 1.5323273167014122, 2.6884539057733283e-008,
|
||
|
11, 1.5323273241519928, 2.6884532967611841e-008,
|
||
|
11, 2.3115825653076172, 6.7682186491374862e-010,
|
||
|
11, 2.3115825727581978, 6.768217495558193e-010,
|
||
|
11, 3.0679489448666573, 3.0544688315708309e-011,
|
||
|
11, 3.0679489523172379, 3.054468452617125e-011,
|
||
|
11, 3.8027990683913231, 2.186949692679853e-012,
|
||
|
11, 3.8027990758419037, 2.1869496552307528e-012,
|
||
|
11, 4.5169280469417572, 2.2947351040888336e-013,
|
||
|
11, 4.5169280543923378, 2.2947348773371414e-013,
|
||
|
11, 5.2107065990567207, 3.3353640359739479e-014,
|
||
|
11, 5.2107066065073013, 3.335363154144172e-014,
|
||
|
11, 5.8841593414545059, 6.4351159890193404e-015,
|
||
|
11, 5.8841593489050865, 6.435115249418266e-015,
|
||
|
11, 6.5370001494884491, 1.5930548620019077e-015,
|
||
|
11, 6.5370001569390297, 1.59305482454352e-015,
|
||
|
11, 7.1686412021517754, 4.9183710858836462e-016,
|
||
|
11, 7.168641209602356, 4.9183713116224844e-016,
|
||
|
11, 7.7781866490840912, 1.8464804238726119e-016,
|
||
|
11, 7.7781866565346718, 1.8464801709489452e-016,
|
||
|
11, 8.3644260391592979, 8.2263320775918765e-017,
|
||
|
11, 8.3644260466098785, 8.2263297787354186e-017,
|
||
|
11, 8.9258662387728691, 4.2375346461043962e-017,
|
||
|
11, 8.9258662462234497, 4.2375329267920249e-017,
|
||
|
11, 9.4609249979257584, 2.4482616150935944e-017,
|
||
|
11, 9.460925005376339, 2.4482619419420328e-017,
|
||
|
11, 9.968697115778923, 1.5254006021629585e-017,
|
||
|
11, 9.9686971232295036, 1.5254009931381535e-017,
|
||
|
11, 10.451380550861359, 9.7234265908062383e-018,
|
||
|
11, 10.451380558311939, 9.7234241296957717e-018,
|
||
|
11, 10.909898526966572, 6.3147967493977272e-018,
|
||
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|
||
|
47, 38.614104360342026, 8.7068430193326876e-66,
|
||
|
47, 38.614104367792606, 8.7068402136754839e-66,
|
||
|
47, 39.186051674187183, 4.7496069713042888e-66,
|
||
|
47, 39.186051681637764, 4.7496053262899038e-66,
|
||
|
47, 39.752189837396145, 2.6711638237868345e-66,
|
||
|
47, 39.752189844846725, 2.6711631630388347e-66,
|
||
|
47, 40.312457516789436, 1.5462743542785204e-66,
|
||
|
47, 40.312457524240017, 1.5462749942360846e-66,
|
||
|
47, 40.86679895222187, 9.197576438140017e-67,
|
||
|
47, 40.866798959672451, 9.1975795377859577e-67,
|
||
|
47, 41.415166474878788, 5.6155851602094119e-67,
|
||
|
47, 41.415166482329369, 5.6155838713502694e-67,
|
||
|
47, 41.957523517310619, 3.5168645107743576e-67,
|
||
|
47, 41.9575235247612, 3.5168655127769924e-67,
|
||
|
47, 42.493848152458668, 2.2493284118817205e-67,
|
||
|
47, 42.493848159909248, 2.2493291438227212e-67,
|
||
|
47, 43.024137258529663, 1.4660650820917181e-67,
|
||
|
47, 43.024137265980244, 1.4660656316867027e-67,
|
||
|
47, 43.548411078751087, 9.7156719582995046e-68,
|
||
|
47, 43.548411086201668, 9.7156694474500461e-68,
|
||
|
47, 44.066718079149723, 6.5310259797325952e-68,
|
||
|
47, 44.066718086600304, 6.5310260879476081e-68,
|
||
|
47, 44.579139620065689, 4.442438020547419e-68,
|
||
|
47, 44.57913962751627, 4.4424398622724323e-68,
|
||
|
47, 45.085793986916542, 3.0499884991862987e-68,
|
||
|
47, 45.085793994367123, 3.0499874345652321e-68,
|
||
|
47, 45.586839035153389, 2.1083600642969428e-68,
|
||
|
47, 45.58683904260397, 2.1083609447567937e-68,
|
||
|
47, 46.082472607493401, 1.4938147144274907e-68,
|
||
|
47, 46.082472614943981, 1.4938153354270103e-68,
|
||
|
47, 46.572930358350277, 1.0602813294127947e-68,
|
||
|
47, 46.572930365800858, 1.0602819311490823e-68,
|
||
|
47, 47.058480642735958, 7.5239566886279539e-69,
|
||
|
47, 47.058480650186539, 7.5239599111949498e-69,
|
||
|
47, 47.539416998624802, 5.3260856872391961e-69,
|
||
|
47, 47.539417006075382, 5.3260856592438819e-69,
|
||
|
48, 0.8081108033657074, 3.790537216779154e-09,
|
||
|
48, 0.80811081081628799, 3.7905242694972794e-09,
|
||
|
48, 1.6817938983440399, 2.2504531682948115e-12,
|
||
|
48, 1.6817939057946205, 2.2504483820670804e-12,
|
||
|
48, 2.5442163795232773, 2.9637275510465843e-15,
|
||
|
48, 2.5442163869738579, 2.9637227676420457e-15,
|
||
|
48, 3.3967880085110664, 6.9290264628280876e-18,
|
||
|
48, 3.396788015961647, 6.9290183337540871e-18,
|
||
|
48, 4.2404954805970192, 2.5428118362391363e-20,
|
||
|
48, 4.2404954880475998, 2.5428091511397717e-20,
|
||
|
48, 5.0760592594742775, 1.3535154982948461e-22,
|
||
|
48, 5.0760592669248581, 1.3535145215948875e-22,
|
||
|
48, 5.9040295109152794, 9.8885231815640295e-25,
|
||
|
48, 5.90402951836586, 9.8885168237279911e-25,
|
||
|
48, 6.7248413637280464, 9.518302995514812e-27,
|
||
|
48, 6.724841371178627, 9.5183002220780383e-27,
|
||
|
48, 7.5388482883572578, 1.1696457368664985e-28,
|
||
|
48, 7.5388482958078384, 1.169645374012626e-28,
|
||
|
48, 8.346343457698822, 1.7894833131442889e-30,
|
||
|
48, 8.3463434651494026, 1.7894824081088118e-30,
|
||
|
48, 9.1475740000605583, 3.3397293242857729e-32,
|
||
|
48, 9.1475740075111389, 3.339728176140613e-32,
|
||
|
48, 9.9427508860826492, 7.4755270996096026e-34,
|
||
|
48, 9.9427508935332298, 7.4755256245678417e-34,
|
||
|
48, 10.732056066393852, 1.9783655047619916e-35,
|
||
|
48, 10.732056073844433, 1.9783652198251722e-35,
|
||
|
48, 11.51564771682024, 6.1149507579983915e-37,
|
||
|
48, 11.515647724270821, 6.1149494622707338e-37,
|
||
|
48, 12.293664202094078, 2.1842708433101863e-38,
|
||
|
48, 12.293664209544659, 2.1842700354917262e-38,
|
||
|
48, 13.066227130591869, 8.9337538377953074e-40,
|
||
|
48, 13.06622713804245, 8.9337502859897068e-40,
|
||
|
48, 13.833443738520145, 4.149924620485562e-41,
|
||
|
48, 13.833443745970726, 4.1499237221656357e-41,
|
||
|
48, 14.595408782362938, 2.1736394745132589e-42,
|
||
|
48, 14.595408789813519, 2.173638813907354e-42,
|
||
|
48, 15.352206096053123, 1.2754745467294774e-43,
|
||
|
48, 15.352206103503704, 1.2754741693102798e-43,
|
||
|
48, 16.103909812867641, 8.3362836739294546e-45,
|
||
|
48, 16.103909820318222, 8.3362820544674689e-45,
|
||
|
48, 16.850585393607616, 6.0368656344880979e-46,
|
||
|
48, 16.850585401058197, 6.0368647910276049e-46,
|
||
|
48, 17.592290475964546, 4.8208082646747087e-47,
|
||
|
48, 17.592290483415127, 4.8208065336171195e-47,
|
||
|
48, 18.32907559722662, 4.2267802866741478e-48,
|
||
|
48, 18.3290756046772, 4.2267788247523572e-48,
|
||
|
48, 19.060984767973423, 4.0527677000262975e-49,
|
||
|
48, 19.060984775424004, 4.0527678696098182e-49,
|
||
|
48, 19.788055963814259, 4.2340523931993972e-50,
|
||
|
48, 19.788055971264839, 4.2340524276091812e-50,
|
||
|
48, 20.510321587324142, 4.8034783455880191e-51,
|
||
|
48, 20.510321594774723, 4.8034780348731884e-51,
|
||
|
48, 21.227808803319931, 5.899186088148869e-52,
|
||
|
48, 21.227808810770512, 5.8991860762611689e-52,
|
||
|
48, 21.940539844334126, 7.8199420923371399e-53,
|
||
|
48, 21.940539851784706, 7.8199427514005375e-53,
|
||
|
48, 22.648532271385193, 1.1158743726808565e-53,
|
||
|
48, 22.648532278835773, 1.1158741101131104e-53,
|
||
|
48, 23.351799219846725, 1.7097382876171394e-54,
|
||
|
48, 23.351799227297306, 1.7097378990809491e-54,
|
||
|
48, 24.050349555909634, 2.8062000362026087e-55,
|
||
|
48, 24.050349563360214, 2.8062001977088843e-55,
|
||
|
48, 24.744188040494919, 4.9228561141945363e-56,
|
||
|
48, 24.744188047945499, 4.9228560221850344e-56,
|
||
|
48, 25.433315485715866, 9.2112000380037501e-57,
|
||
|
48, 25.433315493166447, 9.2111973817050768e-57,
|
||
|
48, 26.117728859186172, 1.8346687861945116e-57,
|
||
|
48, 26.117728866636753, 1.8346682350681618e-57,
|
||
|
48, 26.797421365976334, 3.8826606576023508e-58,
|
||
|
48, 26.797421373426914, 3.8826600315338496e-58,
|
||
|
48, 27.47238253057003, 8.7148703279250707e-59,
|
||
|
48, 27.472382538020611, 8.7148674108098657e-59,
|
||
|
48, 28.142598286271095, 2.0711829791398647e-59,
|
||
|
48, 28.142598293721676, 2.0711832322860257e-59,
|
||
|
48, 28.808050975203514, 5.203584010105859e-60,
|
||
|
48, 28.808050982654095, 5.2035828616190226e-60,
|
||
|
48, 29.468719460070133, 1.3798854862844653e-60,
|
||
|
48, 29.468719467520714, 1.3798850951568444e-60,
|
||
|
48, 30.124579109251499, 3.8565307240880389e-61,
|
||
|
48, 30.12457911670208, 3.8565299743759719e-61,
|
||
|
48, 30.775601848959923, 1.134342736876812e-61,
|
||
|
48, 30.775601856410503, 1.1343429076941469e-61,
|
||
|
48, 31.42175617814064, 3.5066060548219019e-62,
|
||
|
48, 31.421756185591221, 3.5066056252289294e-62,
|
||
|
48, 32.063007235527039, 1.1377415805787006e-62,
|
||
|
48, 32.063007242977619, 1.1377415420726448e-62,
|
||
|
48, 32.699316792190075, 3.8694437991207317e-63,
|
||
|
48, 32.699316799640656, 3.8694430678757391e-63,
|
||
|
48, 33.330643333494663, 1.3776788233453704e-63,
|
||
|
48, 33.330643340945244, 1.3776791661638305e-63,
|
||
|
48, 33.956942074000835, 5.1285952783952753e-64,
|
||
|
48, 33.956942081451416, 5.1285947602355625e-64,
|
||
|
48, 34.578165091574192, 1.9937127927134763e-64,
|
||
|
48, 34.578165099024773, 1.9937126593845751e-64,
|
||
|
48, 35.194261386990547, 8.0836531292607215e-65,
|
||
|
48, 35.194261394441128, 8.0836543290621667e-65,
|
||
|
48, 35.805177040398121, 3.4143034915050112e-65,
|
||
|
48, 35.805177047848701, 3.4143040521491458e-65,
|
||
|
48, 36.410855405032635, 1.5004193222265229e-65,
|
||
|
48, 36.410855412483215, 1.5004189889736636e-65,
|
||
|
48, 37.011237367987633, 6.8526723104762059e-66,
|
||
|
48, 37.011237375438213, 6.8526715926507402e-66,
|
||
|
};
|
||
|
|
||
|
//
|
||
|
// A bunch of helper functions used for calculating the coefficients:
|
||
|
//
|
||
|
mp_t factorial(unsigned n)
|
||
|
{
|
||
|
static boost::array<boost::array<mp_t, 1>, 201> result;
|
||
|
static bool init = false;
|
||
|
if(!init)
|
||
|
{
|
||
|
unsigned k = 1;
|
||
|
mp_t fact = 1;
|
||
|
do{
|
||
|
result[k-1][0] = fact;
|
||
|
fact *= k++;
|
||
|
}
|
||
|
while(k < result.size());
|
||
|
init = true;
|
||
|
}
|
||
|
//static const int array_size = result.size();
|
||
|
if(n < result.size())
|
||
|
return result[n][0];
|
||
|
|
||
|
unsigned i = (unsigned)result.size()-1;
|
||
|
mp_t r = result[i][0];
|
||
|
while(i < n)
|
||
|
r *= ++i;
|
||
|
return r;
|
||
|
}
|
||
|
|
||
|
template <class T>
|
||
|
T binomial(int n, int k, T)
|
||
|
{
|
||
|
T result;
|
||
|
if(k < 0)
|
||
|
{
|
||
|
result = 0;
|
||
|
}
|
||
|
else if(k > n)
|
||
|
{
|
||
|
result = 1;
|
||
|
}
|
||
|
else
|
||
|
{
|
||
|
result = factorial(n);
|
||
|
result /= factorial(k);
|
||
|
result /= factorial(n-k);
|
||
|
}
|
||
|
return result;
|
||
|
}
|
||
|
//
|
||
|
// Functions for creating the matrices that generate the coefficents.
|
||
|
// See http://my.fit.edu/~gabdo/gamma.txt and http://www.rskey.org/gamma.htm
|
||
|
//
|
||
|
template <class T>
|
||
|
matrix<T> make_B(unsigned n, T)
|
||
|
{
|
||
|
matrix<T> result(n, n);
|
||
|
for(unsigned i = 0; i < n; ++i)
|
||
|
{
|
||
|
for(unsigned j = 0; j < n; ++j)
|
||
|
{
|
||
|
if(i == 0)
|
||
|
result(i, j) = 1;
|
||
|
else if(j >= i)
|
||
|
{
|
||
|
T r = binomial(i+j-1, j-i, T());
|
||
|
if((j-i) %2)
|
||
|
r = -r;
|
||
|
result(i, j) = r;
|
||
|
}
|
||
|
else
|
||
|
result(i, j) = 0;
|
||
|
}
|
||
|
}
|
||
|
return result;
|
||
|
}
|
||
|
|
||
|
template <class T>
|
||
|
matrix<T> make_C(unsigned n, T)
|
||
|
{
|
||
|
matrix<T> result(n, n);
|
||
|
for(unsigned i = 0; i < n; ++i)
|
||
|
{
|
||
|
for(unsigned j = 0; j < n; ++j)
|
||
|
{
|
||
|
if((i==0) && (j == 0))
|
||
|
result(i, j) = 0.5;
|
||
|
else if(j > i)
|
||
|
result(i, j) = 0;
|
||
|
else
|
||
|
{
|
||
|
T r = 0;
|
||
|
for(unsigned k = 0; k <= i; ++k)
|
||
|
{
|
||
|
r += binomial(2*i, 2*k, T()) * binomial(k, k+j-i, T());
|
||
|
}
|
||
|
if((i-j)%2)
|
||
|
r = -r;
|
||
|
result(i, j) = r;
|
||
|
}
|
||
|
}
|
||
|
}
|
||
|
return result;
|
||
|
}
|
||
|
|
||
|
template <class T>
|
||
|
matrix<T> make_D(unsigned n, T)
|
||
|
{
|
||
|
matrix<T> result(n, n);
|
||
|
for(unsigned i = 0; i < n; ++i)
|
||
|
{
|
||
|
for(unsigned j = 0; j < n; ++j)
|
||
|
{
|
||
|
if(i != j)
|
||
|
result(i, j) = 0;
|
||
|
else if(i == 0)
|
||
|
result(i, j) = 1;
|
||
|
else if(i == 1)
|
||
|
result(i, j) = -1;
|
||
|
else
|
||
|
{
|
||
|
T r = result(i-1, i-1) * (2 * (2*i - 1));
|
||
|
r /= (i-1);
|
||
|
result(i, j) = r;
|
||
|
}
|
||
|
}
|
||
|
}
|
||
|
return result;
|
||
|
}
|
||
|
|
||
|
template <class T>
|
||
|
matrix<T> make_F(unsigned n, T g)
|
||
|
{
|
||
|
using namespace std;
|
||
|
matrix<T> result(n, 1);
|
||
|
for(unsigned i = 0; i < n; ++i)
|
||
|
{
|
||
|
T r = factorial(2 * i);
|
||
|
r /= factorial(i);
|
||
|
r *= exp(double(i) + g + 0.5);
|
||
|
r /= ldexp(1.0, (2*i)-1);
|
||
|
r /= pow(g + i + 0.5, (long)i);
|
||
|
r /= sqrt(g + i + 0.5);
|
||
|
result(i, 0) = r;
|
||
|
}
|
||
|
return result;
|
||
|
}
|
||
|
|
||
|
template <class T>
|
||
|
struct lanczos_info
|
||
|
{
|
||
|
int n; // number of coefficients
|
||
|
T r; // the arbitrary parameter
|
||
|
std::vector<T> c; // the coefficients
|
||
|
T err; // error found
|
||
|
};
|
||
|
//
|
||
|
// Create the coefficients, caching previous matrix values as appropriate:
|
||
|
//
|
||
|
template <class T>
|
||
|
lanczos_info<T> generate_lanczos(unsigned n, T g)
|
||
|
{
|
||
|
static unsigned last_n = 0;
|
||
|
static matrix<T> last_prefix;
|
||
|
|
||
|
matrix<T> B, C, D;
|
||
|
matrix<T> F, P, E, T1, T2;
|
||
|
|
||
|
if(n == last_n)
|
||
|
{
|
||
|
F = make_F(n, g);
|
||
|
P = prod(last_prefix, F);
|
||
|
}
|
||
|
else
|
||
|
{
|
||
|
B = make_B(n, g);
|
||
|
C = make_C(n, g);
|
||
|
D = make_D(n, g);
|
||
|
F = make_F(n, g);
|
||
|
|
||
|
T1 = prod(D, B);
|
||
|
T2 = prod(T1, C);
|
||
|
P = prod(T2, F);
|
||
|
|
||
|
last_n = n;
|
||
|
last_prefix = T2;
|
||
|
}
|
||
|
|
||
|
lanczos_info<T> result;
|
||
|
result.n = n;
|
||
|
result.r = g;
|
||
|
for(unsigned i = 0; i < n; ++i)
|
||
|
result.c.push_back(P(i, 0));
|
||
|
|
||
|
return result;
|
||
|
}
|
||
|
//
|
||
|
// Returns the factorials and half factorials:
|
||
|
//
|
||
|
template <class T>
|
||
|
std::vector<std::vector<T> > const & get_test_data()
|
||
|
{
|
||
|
// returns factorials and half factorials in the range [0, 100]:
|
||
|
static std::vector<std::vector<T> > data;
|
||
|
if(data.empty())
|
||
|
{
|
||
|
T fact = 1;
|
||
|
int k = 1;
|
||
|
std::vector<T> item;
|
||
|
do
|
||
|
{
|
||
|
data.push_back(item);
|
||
|
data.back().push_back(k-1);
|
||
|
data.back().push_back(fact);
|
||
|
data.back().push_back(log(fact));
|
||
|
fact = fact * T(k++);
|
||
|
}while(k < 100);
|
||
|
|
||
|
fact = 0.5;
|
||
|
T srpi = sqrt(boost::math::constants::pi<T>());
|
||
|
T mul = 1.5;
|
||
|
do{
|
||
|
data.push_back(item);
|
||
|
data.back().push_back(mul-1);
|
||
|
data.back().push_back(fact*srpi);
|
||
|
data.back().push_back(log(fact*srpi));
|
||
|
fact *= mul;
|
||
|
mul += 1;
|
||
|
}while(mul < 100);
|
||
|
}
|
||
|
|
||
|
return data;
|
||
|
}
|
||
|
|
||
|
struct sort_on_0
|
||
|
{
|
||
|
template<class T>
|
||
|
bool operator ()(const T& a, const T& b)
|
||
|
{
|
||
|
return ((a[0]) < (b[0]));
|
||
|
}
|
||
|
};
|
||
|
//
|
||
|
// Get test data for points near 0, 1, 2 etc.
|
||
|
template <class T>
|
||
|
std::vector<std::vector<T> > const & get_test_data_near_x(T x)
|
||
|
{
|
||
|
//
|
||
|
// this *must* be called with a *very* high precision
|
||
|
// type T in order to stand any chance of long double
|
||
|
// precision results:
|
||
|
//
|
||
|
using namespace std;
|
||
|
using namespace boost::math;
|
||
|
static std::vector<std::vector<T> > data;
|
||
|
static T last_x(-10000);
|
||
|
if(last_x != x)
|
||
|
data.clear();
|
||
|
if(!data.size())
|
||
|
{
|
||
|
last_x = x;
|
||
|
std::vector<T> item;
|
||
|
for(int i = -4; i > -20; --i)
|
||
|
{
|
||
|
double val = ldexp(2.0, i);
|
||
|
data.push_back(item);
|
||
|
data.back().push_back(val+x);
|
||
|
data.back().push_back(tgamma(T(val)+x));
|
||
|
data.back().push_back(lgamma(T(val)+x));
|
||
|
data.push_back(item);
|
||
|
data.back().push_back(-val+x);
|
||
|
data.back().push_back(tgamma(T(-val)+x));
|
||
|
data.back().push_back(lgamma(T(-val)+x));
|
||
|
}
|
||
|
T v = x-0.5;
|
||
|
T interval = T(1)/8;
|
||
|
while(v <= x+0.5)
|
||
|
{
|
||
|
if((floor(v) != v) || (v > 0))
|
||
|
{
|
||
|
data.push_back(item);
|
||
|
data.back().push_back(v);
|
||
|
data.back().push_back(tgamma(v));
|
||
|
data.back().push_back(lgamma(v));
|
||
|
}
|
||
|
v += interval;
|
||
|
}
|
||
|
//
|
||
|
// sort data:
|
||
|
//
|
||
|
std::sort(data.begin(), data.end(), sort_on_0());
|
||
|
}
|
||
|
return data;
|
||
|
}
|
||
|
|
||
|
template <class T>
|
||
|
std::vector<std::vector<T> > const & get_test_data_near_1()
|
||
|
{
|
||
|
return get_test_data_near_x(T(1));
|
||
|
}
|
||
|
|
||
|
template <class T>
|
||
|
std::vector<std::vector<T> > const & get_test_data_near_2()
|
||
|
{
|
||
|
return get_test_data_near_x(T(2));
|
||
|
}
|
||
|
|
||
|
//
|
||
|
// Converts Lanczos approximation into rational form via
|
||
|
// polynomial arithmetic:
|
||
|
//
|
||
|
template <class T>
|
||
|
struct lanczos_rational
|
||
|
{
|
||
|
std::vector<T> num, denom;
|
||
|
int N;
|
||
|
T g;
|
||
|
|
||
|
T gamma(T z)
|
||
|
{
|
||
|
using namespace std;
|
||
|
BOOST_ASSERT(num.size() == denom.size());
|
||
|
BOOST_ASSERT(num.size() == N);
|
||
|
T zgh = z + g - T(0.5);
|
||
|
T prefix = pow(zgh, T(z - 0.5)) / exp(zgh);
|
||
|
T s1, s2;
|
||
|
if(z < 1)
|
||
|
{
|
||
|
s1 = num[N-1];
|
||
|
s2 = denom[N-1];
|
||
|
for(unsigned i = N-2; i >= 0; --i)
|
||
|
{
|
||
|
s1 *= z;
|
||
|
s2 *= z;
|
||
|
s1 += num[i];
|
||
|
s2 += num[i];
|
||
|
}
|
||
|
s1 /= s2;
|
||
|
return prefix * s1;
|
||
|
}
|
||
|
else
|
||
|
{
|
||
|
z = 1/z;
|
||
|
s1 = num[0];
|
||
|
s2 = denom[0];
|
||
|
for(int i = 1; i < N; ++i)
|
||
|
{
|
||
|
s1 *= z;
|
||
|
s2 *= z;
|
||
|
s1 += num[i];
|
||
|
s2 += denom[i];
|
||
|
}
|
||
|
s1 /= s2;
|
||
|
return prefix * s1;
|
||
|
}
|
||
|
}
|
||
|
T factorial(T z)
|
||
|
{
|
||
|
return this->gamma(z+1);
|
||
|
}
|
||
|
template <class U>
|
||
|
lanczos_rational(const lanczos_info<U>& info)
|
||
|
{
|
||
|
U l_denom_coef[2] = { 0, 1 };
|
||
|
boost::math::tools::polynomial<U> l_denom(l_denom_coef, 1), l_num(info.c[1]);
|
||
|
for(unsigned i = 2; i < info.c.size(); ++i)
|
||
|
{
|
||
|
l_denom_coef[0] = i - 1;
|
||
|
boost::math::tools::polynomial<U> bot(l_denom_coef, 1), top(info.c[i]);
|
||
|
l_num *= bot;
|
||
|
top *= l_denom;
|
||
|
l_denom *= bot;
|
||
|
l_num += top;
|
||
|
}
|
||
|
l_num += l_denom * info.c[0];
|
||
|
for(unsigned i = 0; i < info.c.size(); ++i)
|
||
|
{
|
||
|
num.push_back(boost::math::tools::real_cast<T>(l_num[i]));
|
||
|
denom.push_back(boost::math::tools::real_cast<T>(l_denom[i]));
|
||
|
}
|
||
|
//std::cout << l_num << std::endl;
|
||
|
//std::cout << l_denom << std::endl;
|
||
|
BOOST_ASSERT(num.size() == l_num.degree()+1);
|
||
|
BOOST_ASSERT(denom.size() == l_denom.degree()+1);
|
||
|
g = boost::math::tools::real_cast<T>(info.r);
|
||
|
N = info.n;
|
||
|
/*
|
||
|
for(unsigned i = 0; i < num.size(); ++i)
|
||
|
std::cout << num[i] << " ";
|
||
|
std::cout << std::endl;
|
||
|
for(unsigned i = 0; i < denom.size(); ++i)
|
||
|
std::cout << denom[i] << " ";
|
||
|
std::cout << std::endl;
|
||
|
*/
|
||
|
}
|
||
|
};
|
||
|
//
|
||
|
// Code to test an approximation against the factorials and
|
||
|
// half factorials, returns the max error found:
|
||
|
//
|
||
|
template <class T, class R>
|
||
|
T get_max_error(const lanczos_info<T>& dat, R)
|
||
|
{
|
||
|
T max_error = 0;
|
||
|
std::vector<std::vector<T> > const & tests = get_test_data<T>();
|
||
|
lanczos_rational<R> rational(dat);
|
||
|
|
||
|
for(unsigned i = 0; i < tests.size(); ++i)
|
||
|
{
|
||
|
R input = boost::math::tools::real_cast<R>(tests[i][0]);
|
||
|
if(std::numeric_limits<R>::is_specialized && (boost::math::tools::real_cast<R>(tests[i][1]) > (std::numeric_limits<R>::max)()))
|
||
|
continue;
|
||
|
|
||
|
R gamr = rational.factorial(input);
|
||
|
if(std::numeric_limits<R>::is_specialized && (gamr > (std::numeric_limits<R>::max)()))
|
||
|
continue;
|
||
|
T gam = gamr;
|
||
|
T exp = tests[i][1];
|
||
|
T err = relative_error(gam, exp);
|
||
|
if(err > max_error)
|
||
|
max_error = err;
|
||
|
}
|
||
|
|
||
|
return max_error;
|
||
|
}
|
||
|
//
|
||
|
// This is what prints the "best" approximation out as code:
|
||
|
//
|
||
|
template <class T>
|
||
|
void print_code(const lanczos_info<T>& l, const char* name)
|
||
|
{
|
||
|
std::cout << std::scientific << std::setprecision(std::numeric_limits<T>::digits10 + 3);
|
||
|
using namespace std;
|
||
|
lanczos_info<T> l2(l);
|
||
|
T factor = exp(l.r);
|
||
|
for(unsigned k = 0; k < l2.c.size(); ++k)
|
||
|
l2.c[k] /= factor;
|
||
|
|
||
|
lanczos_rational<T> rat(l);
|
||
|
T max_term = 0;
|
||
|
for(unsigned i = 0; i < rat.denom.size(); ++i)
|
||
|
{
|
||
|
if(rat.denom[i] > max_term)
|
||
|
max_term = rat.denom[i];
|
||
|
}
|
||
|
const char* denom_type;
|
||
|
const char* cast_type;
|
||
|
const char* suffix_type;
|
||
|
if(max_term < (std::numeric_limits<boost::uint16_t>::max)())
|
||
|
{
|
||
|
denom_type = "boost::uint16_t";
|
||
|
cast_type = "static_cast<boost::uint16_t>";
|
||
|
suffix_type = "u";
|
||
|
}
|
||
|
else if(max_term < (std::numeric_limits<boost::uint32_t>::max)())
|
||
|
{
|
||
|
denom_type = "boost::uint32_t";
|
||
|
cast_type = "static_cast<boost::uint32_t>";
|
||
|
suffix_type = "u";
|
||
|
}
|
||
|
#ifdef BOOST_HAS_LONG_LONG
|
||
|
else if(max_term < (std::numeric_limits<boost::uint64_t>::max)())
|
||
|
{
|
||
|
denom_type = "boost::uint64_t";
|
||
|
cast_type = "";
|
||
|
suffix_type = "uLL";
|
||
|
}
|
||
|
#endif
|
||
|
else
|
||
|
{
|
||
|
denom_type = "T";
|
||
|
cast_type = "static_cast<T>";
|
||
|
suffix_type = "L";
|
||
|
}
|
||
|
|
||
|
std::cout <<
|
||
|
"//\n"
|
||
|
"// Lanczos Coefficients for N=" << l.n << " G=" << l.r << "\n"
|
||
|
"// Max experimental error (with ";
|
||
|
if(std::strlen(name) == 0)
|
||
|
std::cout << "arbitary";
|
||
|
else
|
||
|
std::cout << name;
|
||
|
std::cout << " precision arithmetic) " << l.err <<
|
||
|
"\n// Generated with compiler: " << BOOST_COMPILER << " on " << BOOST_PLATFORM << " at " << __DATE__ << "\n"
|
||
|
"//\n"
|
||
|
"struct lanczos" << l.n << name << "\n"
|
||
|
"{\n"
|
||
|
" template <class T>\n"
|
||
|
" static T lanczos_sum(const T& z)\n"
|
||
|
" {\n"
|
||
|
" static const T num[" << rat.num.size() << "] = {\n";
|
||
|
|
||
|
for(unsigned i = 0; i < rat.num.size(); ++i)
|
||
|
{
|
||
|
std::cout << " static_cast<T>(" << rat.num[i] << "L)";
|
||
|
if(i != rat.num.size() - 1)
|
||
|
std::cout << ",";
|
||
|
std::cout << "\n";
|
||
|
}
|
||
|
std::cout <<
|
||
|
" };\n"
|
||
|
" static const " << denom_type << " denom[" << rat.denom.size() << "] = {\n";
|
||
|
for(unsigned i = 0; i < rat.denom.size(); ++i)
|
||
|
{
|
||
|
std::cout << " " << cast_type << "(" << rat.denom[i] << suffix_type << ")";
|
||
|
if(i != rat.denom.size() - 1)
|
||
|
std::cout << ",";
|
||
|
std::cout << "\n";
|
||
|
}
|
||
|
std::cout <<
|
||
|
" };\n"
|
||
|
" return boost::math::tools::evaluate_rational(num, denom, z);\n"
|
||
|
" }\n\n"
|
||
|
" template <class T>\n"
|
||
|
" static T lanczos_sum_expG_scaled(const T& z)\n"
|
||
|
" {\n"
|
||
|
" static const T num[" << rat.num.size() << "] = {\n";
|
||
|
|
||
|
for(unsigned i = 0; i < rat.num.size(); ++i)
|
||
|
{
|
||
|
std::cout << " static_cast<T>(" << (rat.num[i]/factor) << "L)";
|
||
|
if(i != rat.num.size() - 1)
|
||
|
std::cout << ",";
|
||
|
std::cout << "\n";
|
||
|
}
|
||
|
std::cout <<
|
||
|
" };\n"
|
||
|
" static const " << denom_type << " denom[" << rat.denom.size() << "] = {\n";
|
||
|
for(unsigned i = 0; i < rat.denom.size(); ++i)
|
||
|
{
|
||
|
std::cout << " " << cast_type << "(" << rat.denom[i] << suffix_type << ")";
|
||
|
if(i != rat.denom.size() - 1)
|
||
|
std::cout << ",";
|
||
|
std::cout << "\n";
|
||
|
}
|
||
|
std::cout <<
|
||
|
" };\n"
|
||
|
" return boost::math::tools::evaluate_rational(num, denom, z);\n"
|
||
|
" }\n\n";
|
||
|
|
||
|
std::cout <<
|
||
|
"\n template<class T>\n"
|
||
|
" static T lanczos_sum_near_1(const T& dz)\n"
|
||
|
" {\n"
|
||
|
" static const T d[" << l2.n-1 << "] = {\n";
|
||
|
|
||
|
factor = sqrt((l.r + 0.5)/boost::math::constants::e<T>()) / exp(l.r);
|
||
|
|
||
|
for(int i = 1; i < l.n; ++i)
|
||
|
{
|
||
|
std::cout << " static_cast<T>(" << l.c[i]*factor << "L)";
|
||
|
if(i != l.n - 1)
|
||
|
std::cout << ",";
|
||
|
std::cout << "\n";
|
||
|
}
|
||
|
std::cout <<
|
||
|
" };\n"
|
||
|
" T result = 0;\n"
|
||
|
" for(unsigned k = 1; k <= sizeof(d)/sizeof(d[0]); ++k)\n"
|
||
|
" {\n"
|
||
|
" result += (-d[k-1]*dz)/(k*dz + k*k);\n"
|
||
|
" }\n"
|
||
|
" return result;\n"
|
||
|
" }\n";
|
||
|
|
||
|
std::cout <<
|
||
|
"\n template<class T>\n"
|
||
|
" static T lanczos_sum_near_2(const T& dz)\n"
|
||
|
" {\n"
|
||
|
" static const T d[" << l2.n-1 << "] = {\n";
|
||
|
|
||
|
factor = pow(boost::math::constants::e<T>()/(l.r+1.5), T(1.5)) * exp(l.r);
|
||
|
//pow((l.r + 1.5)/boost::math::constants::e<T>(), 1.5) / exp(l.r);
|
||
|
|
||
|
for(int i = 1; i < l.n; ++i)
|
||
|
{
|
||
|
std::cout << " static_cast<T>(" << l.c[i]/factor << "L),\n";
|
||
|
}
|
||
|
std::cout <<
|
||
|
" };\n"
|
||
|
" T result = 0;\n"
|
||
|
" T z = dz + 2;\n"
|
||
|
" for(unsigned k = 1; k <= sizeof(d)/sizeof(d[0]); ++k)\n"
|
||
|
" {\n"
|
||
|
" result += (-d[k-1]*dz)/(z + k*z + k*k - 1);\n"
|
||
|
" }\n"
|
||
|
" return result;\n"
|
||
|
" }\n\n"
|
||
|
" static double g(){ return " << l.r << "; }\n"
|
||
|
"};\n\n";
|
||
|
}
|
||
|
//
|
||
|
// Print out the test values:
|
||
|
//
|
||
|
void print_test_values(const std::vector<std::vector<mp_t> >& v, const char* name, int offset = 1)
|
||
|
{
|
||
|
std::cout << "#define SC_(x) static_cast<T>(BOOST_JOIN(x, L))\n";
|
||
|
std::cout <<
|
||
|
" static const boost::array<boost::array<T, 3>, " << v.size() << "> " << name << " = {\n";
|
||
|
for(unsigned i = 0; i < v.size(); ++i)
|
||
|
{
|
||
|
std::cout << " SC_(" << (v[i][0] + offset) << "), SC_(" << v[i][1] << "), SC_(" << v[i][2] << "),\n";
|
||
|
}
|
||
|
std::cout << " };\n#undef SC_\n\n";
|
||
|
}
|
||
|
//
|
||
|
// Get the error for a specific approximation, and print out it's code:
|
||
|
//
|
||
|
void calculate_lanczos_spot(int n, mp_t r, const char* suffix = "")
|
||
|
{
|
||
|
lanczos_info<mp_t> info = generate_lanczos(n, r);
|
||
|
// note error is calculated at high precision:
|
||
|
info.err = get_max_error(info, r);
|
||
|
print_code(info, suffix);
|
||
|
}
|
||
|
//
|
||
|
// Scan the sweet-spots for the best approximation:
|
||
|
//
|
||
|
template <class T>
|
||
|
void find_best_lanczos(const char* name, T eps, int max_scan = 100)
|
||
|
{
|
||
|
using namespace std;
|
||
|
|
||
|
lanczos_info<mp_t> best;
|
||
|
best.err = 100; // best case had better be better than this!
|
||
|
for(int i = 0; i < sizeof(sweet_spots)/sizeof(sweet_spots[0]); ++i)
|
||
|
{
|
||
|
if((sweet_spots[i].err < eps*10) && (sweet_spots[i].N < max_scan))
|
||
|
{
|
||
|
lanczos_info<mp_t> info = generate_lanczos(sweet_spots[i].N, mp_t(sweet_spots[i].g));
|
||
|
mp_t err = get_max_error(info, eps);
|
||
|
if(err/eps < 1000)
|
||
|
{
|
||
|
std::cout << sweet_spots[i].N << " " << sweet_spots[i].g << " " << err/eps << std::endl;
|
||
|
}
|
||
|
if(err < best.err)
|
||
|
{
|
||
|
best = info;
|
||
|
best.err = err;
|
||
|
}
|
||
|
}
|
||
|
}
|
||
|
std::cout << std::endl;
|
||
|
|
||
|
print_code(best, name);
|
||
|
}
|
||
|
int main(int argc, char*argv [])
|
||
|
{
|
||
|
bool test_double(false), test_long(false), test_float(false), test_quad(false), spots(false), test_data(false);
|
||
|
|
||
|
if(argc < 2)
|
||
|
{
|
||
|
std::cout <<
|
||
|
"Useage:\n"
|
||
|
" -float test type float for the best approximation\n"
|
||
|
" -double test type double for the best approximation\n"
|
||
|
" -long-double test type long double for the best approximation\n"
|
||
|
" -spots print out the best cases found in previous runs\n"
|
||
|
" -data print out the test data\n" << std::flush;
|
||
|
return 1;
|
||
|
}
|
||
|
|
||
|
for(int i = 1; i < argc; ++i)
|
||
|
{
|
||
|
if(std::strcmp(argv[i], "-float") == 0)
|
||
|
test_float = true;
|
||
|
else if(std::strcmp(argv[i], "-double") == 0)
|
||
|
test_double = true;
|
||
|
else if(std::strcmp(argv[i], "-long-double") == 0)
|
||
|
test_long = true;
|
||
|
else if(std::strcmp(argv[i], "-quad-float") == 0)
|
||
|
test_quad = true;
|
||
|
else if(std::strcmp(argv[i], "-spots") == 0)
|
||
|
spots = true;
|
||
|
else if(std::strcmp(argv[i], "-data") == 0)
|
||
|
test_data = true;
|
||
|
}
|
||
|
|
||
|
if(spots)
|
||
|
{
|
||
|
// these are optimal N and R from Pugh:
|
||
|
calculate_lanczos_spot(61, 63.192152);
|
||
|
calculate_lanczos_spot(31, 32.080670);
|
||
|
calculate_lanczos_spot(6, 5.581);
|
||
|
calculate_lanczos_spot(11, 10.900511);
|
||
|
calculate_lanczos_spot(13, 13.144565);
|
||
|
calculate_lanczos_spot(22, 22.61891);
|
||
|
// these are the best cases we've already determined
|
||
|
// from previous runs:
|
||
|
calculate_lanczos_spot(6, 1.428456135094165802001953125);
|
||
|
calculate_lanczos_spot(13, 6.024680040776729583740234375);
|
||
|
calculate_lanczos_spot(17, 12.2252227365970611572265625);
|
||
|
calculate_lanczos_spot(24, 20.3209821879863739013671875);
|
||
|
}
|
||
|
if(test_float)
|
||
|
{
|
||
|
find_best_lanczos("float", std::numeric_limits<float>::epsilon());
|
||
|
}
|
||
|
if(test_double)
|
||
|
{
|
||
|
find_best_lanczos("double", std::numeric_limits<double>::epsilon());
|
||
|
}
|
||
|
|
||
|
if(test_long)
|
||
|
{
|
||
|
find_best_lanczos("long_double", std::numeric_limits<long double>::epsilon());
|
||
|
}
|
||
|
if(test_quad)
|
||
|
{
|
||
|
//find_best_lanczos("quad_float", pow(NTL::quad_float(2), NTL::quad_float(-105)));
|
||
|
}
|
||
|
if(test_data)
|
||
|
{
|
||
|
std::cout << "Test Data follows:\n\n";
|
||
|
|
||
|
std::vector<std::vector<mp_t> > const & tests = get_test_data<mp_t>();
|
||
|
print_test_values(tests, "factorials");
|
||
|
print_test_values(get_test_data_near_1<mp_t>(), "near_1", 0);
|
||
|
print_test_values(get_test_data_near_2<mp_t>(), "near_2", 0);
|
||
|
print_test_values(get_test_data_near_x<mp_t>(mp_t(0)), "near_0", 0);
|
||
|
print_test_values(get_test_data_near_x<mp_t>(mp_t(-10)), "near_m10", 0);
|
||
|
print_test_values(get_test_data_near_x<mp_t>(mp_t(-55)), "near_m55", 0);
|
||
|
}
|
||
|
return 0;
|
||
|
}
|
||
|
|
||
|
|
||
|
|