boost/math/special_functions/detail/hypergeometric_1F1_negative_b_regions.hpp
/////////////////////////////////////////////////////////////////////////////// // Copyright 2019 John Maddock // Distributed under the Boost // Software License, Version 1.0. (See accompanying file // LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) #ifndef BOOST_MATH_DETIAL_1F1_MAP_NEG_B_HPP #define BOOST_MATH_DETIAL_1F1_MAP_NEG_B_HPP namespace boost { namespace math { namespace detail { // // hypergeometric_1F1_negative_b_recurrence_region maps out the domains over which // forward and backwards recurrences are stable when b < 0. // Returns -1, 0, or 1: // -1: Backwards recurrence is stable. // +1: Forwards recurrence is stable. // 0: Neither recurrence is stable. // template <class T> int hypergeometric_1F1_negative_b_recurrence_region(const T& a, const T& b, const T& z) { BOOST_MATH_STD_USING static const double domain[][4] = { { 1e-300, -1000000.1, 278609.88983674475, 465687}, { 1e-300, -400000.03999999998, 111530.83622048119, 111544}, { 1e-300, -160000.016, 44699.113858309654, 44712}, { 1e-300, -64000.006399999998, 17966.035866477272, 17980}, { 1e-300, -25600.002560000001, 7273.793013139204, 7287}, { 1e-300, -10240.001024000001, 2998.0791015625, 3012}, { 1e-300, -4096.0004096000002, 1291.0461647727273, 1306}, { 1e-300, -1638.40016384, 619.421875, 636}, { 1e-300, -655.36006553599998, 374.62926136363637, 397}, { 1e-300, -262.14402621440001, 335.29958677685943, 370}, { 1e-300, -104.85761048576001, 412.83057851239664, 462}, { 1e-300, -41.943044194304001, 513.00603693181824, 570}, { 1e-300, -16.777217677721602, 584.14449896694214, 644}, { 1e-300, -6.7108870710886404, 623.30417097107443, 684}, { 1e-300, -2.6843548284354561, 642.31960227272725, 704}, { 1e-300, -1.0737419313741825, 650.20738636363626, 711}, { 1.0000000000000001e-275, -1000000.1, 278597.3203069513, 448872}, { 1.0000000000000001e-275, -400000.03999999998, 111518.14741654831, 111531}, { 1.0000000000000001e-275, -160000.016, 44686.621781782669, 44700}, { 1.0000000000000001e-275, -64000.006399999998, 17953.944740988994, 17967}, { 1.0000000000000001e-275, -25600.002560000001, 7261.1431107954559, 7274}, { 1.0000000000000001e-275, -10240.001024000001, 2985.014559659091, 2999}, { 1.0000000000000001e-275, -4096.0004096000002, 1277.4573863636363, 1292}, { 1.0000000000000001e-275, -1638.40016384, 603.88778409090912, 620}, { 1.0000000000000001e-275, -655.36006553599998, 354.10653409090912, 376}, { 1.0000000000000001e-275, -262.14402621440001, 303.91367510330571, 337}, { 1.0000000000000001e-275, -104.85761048576001, 367.0454545454545, 416}, { 1.0000000000000001e-275, -41.943044194304001, 460.06028861053716, 516}, { 1.0000000000000001e-275, -16.777217677721602, 528.36389462809916, 588}, { 1.0000000000000001e-275, -6.7108870710886404, 566.33555010330565, 627}, { 1.0000000000000001e-275, -2.6843548284354561, 585.32541322314046, 646}, { 1.0000000000000001e-275, -1.0737419313741825, 593.0082967458676, 654}, { 1.0000000000000002e-250, -1000000.1, 278584.75303113955, 431059}, { 1.0000000000000002e-250, -400000.03999999998, 111505.53022349965, 111519}, { 1.0000000000000002e-250, -160000.016, 44674.060190374199, 44687}, { 1.0000000000000002e-250, -64000.006399999998, 17941.25452769886, 17954}, { 1.0000000000000002e-250, -25600.002560000001, 7248.2002840909081, 7262}, { 1.0000000000000002e-250, -10240.001024000001, 2972.017267400568, 2985}, { 1.0000000000000002e-250, -4096.0004096000002, 1263.6363636363637, 1278}, { 1.0000000000000002e-250, -1638.40016384, 588.35369318181824, 604}, { 1.0000000000000002e-250, -655.36006553599998, 334.18039772727275, 355}, { 1.0000000000000002e-250, -262.14402621440001, 273.7357954545455, 306}, { 1.0000000000000002e-250, -104.85761048576001, 323.05010330578506, 370}, { 1.0000000000000002e-250, -41.943044194304001, 407.41541838842977, 463}, { 1.0000000000000002e-250, -16.777217677721602, 472.78376807851242, 532}, { 1.0000000000000002e-250, -6.7108870710886404, 509.47443181818176, 570}, { 1.0000000000000002e-250, -2.6843548284354561, 528.11466942148763, 589}, { 1.0000000000000002e-250, -1.0737419313741825, 535.52815082644634, 596}, { 1.0000000000000003e-225, -1000000.1, 278572.12184906006, 278585}, { 1.0000000000000003e-225, -400000.03999999998, 111493.21068649805, 551165}, { 1.0000000000000003e-225, -160000.016, 44661.126509232956, 44674}, { 1.0000000000000003e-225, -64000.006399999998, 17928.069602272728, 17942}, { 1.0000000000000003e-225, -25600.002560000001, 7235.550426136364, 7249}, { 1.0000000000000003e-225, -10240.001024000001, 2959.153053977273, 2973}, { 1.0000000000000003e-225, -4096.0004096000002, 1250.2649147727275, 1264}, { 1.0000000000000003e-225, -1638.40016384, 573.37073863636363, 589}, { 1.0000000000000003e-225, -655.36006553599998, 315.21946022727275, 335}, { 1.0000000000000003e-225, -262.14402621440001, 245.11137654958674, 275}, { 1.0000000000000003e-225, -104.85761048576001, 280.71022727272725, 326}, { 1.0000000000000003e-225, -41.943044194304001, 355.45519111570246, 411}, { 1.0000000000000003e-225, -16.777217677721602, 417.41670971074382, 476}, { 1.0000000000000003e-225, -6.7108870710886404, 452.71984762396687, 513}, { 1.0000000000000003e-225, -2.6843548284354561, 470.45648243801656, 532}, { 1.0000000000000003e-225, -1.0737419313741825, 478.06438855888422, 539}, { 1.0000000000000004e-200, -1000000.1, 278559.54284667969, 278573}, { 1.0000000000000004e-200, -400000.03999999998, 111480.15247691762, 111493}, { 1.0000000000000004e-200, -160000.016, 44648.944635564636, 44662}, { 1.0000000000000004e-200, -64000.006399999998, 17916.076071999294, 17929}, { 1.0000000000000004e-200, -25600.002560000001, 7223.06005859375, 7236}, { 1.0000000000000004e-200, -10240.001024000001, 2946.222301136364, 2960}, { 1.0000000000000004e-200, -4096.0004096000002, 1236.0916193181818, 1251}, { 1.0000000000000004e-200, -1638.40016384, 558.40909090909076, 574}, { 1.0000000000000004e-200, -655.36006553599998, 297.05220170454544, 316}, { 1.0000000000000004e-200, -262.14402621440001, 218.16438533057845, 247}, { 1.0000000000000004e-200, -104.85761048576001, 239.76949896694211, 283}, { 1.0000000000000004e-200, -41.943044194304001, 304.40728305785115, 359}, { 1.0000000000000004e-200, -16.777217677721602, 362.01349431818181, 421}, { 1.0000000000000004e-200, -6.7108870710886404, 396.18123708677683, 456}, { 1.0000000000000004e-200, -2.6843548284354561, 413.36518595041321, 474}, { 1.0000000000000004e-200, -1.0737419313741825, 420.31249999999994, 482}, { 1.0000000000000006e-175, -1000000.1, 278547.12638697342, 465687}, { 1.0000000000000006e-175, -400000.03999999998, 111468.15407492899, 111481}, { 1.0000000000000006e-175, -160000.016, 44636.134832208802, 44649}, { 1.0000000000000006e-175, -64000.006399999998, 17903.091796875, 17917}, { 1.0000000000000006e-175, -25600.002560000001, 7210.247247869318, 7223}, { 1.0000000000000006e-175, -10240.001024000001, 2933.2915482954545, 2947}, { 1.0000000000000006e-175, -4096.0004096000002, 1223.600497159091, 1237}, { 1.0000000000000006e-175, -1638.40016384, 544.06640625, 559}, { 1.0000000000000006e-175, -655.36006553599998, 279.11931818181813, 298}, { 1.0000000000000006e-175, -262.14402621440001, 192.94921875, 220}, { 1.0000000000000006e-175, -104.85761048576001, 200.78641528925618, 242}, { 1.0000000000000006e-175, -41.943044194304001, 254.19679752066114, 308}, { 1.0000000000000006e-175, -16.777217677721602, 307.10227272727263, 366}, { 1.0000000000000006e-175, -6.7108870710886404, 339.36918904958679, 400}, { 1.0000000000000006e-175, -2.6843548284354561, 356.1554106404958, 417}, { 1.0000000000000006e-175, -1.0737419313741825, 363.13694473140492, 424}, { 1.0000000000000007e-150, -1000000.1, 278534.48812689661, 448872}, { 1.0000000000000007e-150, -400000.03999999998, 111455.15580610794, 111468}, { 1.0000000000000007e-150, -160000.016, 44623.643454811798, 44637}, { 1.0000000000000007e-150, -64000.006399999998, 17891.00373493541, 17904}, { 1.0000000000000007e-150, -25600.002560000001, 7197.631392045454, 7211}, { 1.0000000000000007e-150, -10240.001024000001, 2920.3607954545455, 2934}, { 1.0000000000000007e-150, -4096.0004096000002, 1210.3267045454545, 1224}, { 1.0000000000000007e-150, -1638.40016384, 530.0033735795455, 545}, { 1.0000000000000007e-150, -655.36006553599998, 262.37215909090907, 280}, { 1.0000000000000007e-150, -262.14402621440001, 169.07024793388427, 194}, { 1.0000000000000007e-150, -104.85761048576001, 164.12706611570246, 203}, { 1.0000000000000007e-150, -41.943044194304001, 206.02079028925615, 258}, { 1.0000000000000007e-150, -16.777217677721602, 253.0191115702479, 311}, { 1.0000000000000007e-150, -6.7108870710886404, 283.23056559917353, 343}, { 1.0000000000000007e-150, -2.6843548284354561, 299.29041838842977, 360}, { 1.0000000000000007e-150, -1.0737419313741825, 305.7925490702479, 367}, { 1.0000000000000008e-125, -1000000.1, 278522.04145396838, 285024}, { 1.0000000000000008e-125, -400000.03999999998, 111443.00279374557, 111456}, { 1.0000000000000008e-125, -160000.016, 44611.143155184654, 44624}, { 1.0000000000000008e-125, -64000.006399999998, 17878.306263316765, 17891}, { 1.0000000000000008e-125, -25600.002560000001, 7185.1384943181802, 7198}, { 1.0000000000000008e-125, -10240.001024000001, 2907.430042613636, 2921}, { 1.0000000000000008e-125, -4096.0004096000002, 1197.0621448863635, 1211}, { 1.0000000000000008e-125, -1638.40016384, 516.05983664772725, 531}, { 1.0000000000000008e-125, -655.36006553599998, 246.17897727272725, 263}, { 1.0000000000000008e-125, -262.14402621440001, 147.2197830578512, 171}, { 1.0000000000000008e-125, -104.85761048576001, 130.599173553719, 166}, { 1.0000000000000008e-125, -41.943044194304001, 159.12190082644628, 209}, { 1.0000000000000008e-125, -16.777217677721602, 199.5635330578512, 256}, { 1.0000000000000008e-125, -6.7108870710886404, 227.1807205578512, 287}, { 1.0000000000000008e-125, -2.6843548284354561, 242.41670971074376, 303}, { 1.0000000000000008e-125, -1.0737419313741825, 248.39036673553716, 309}, { 1.0000000000000009e-100, -1000000.1, 278509.36086203833, 278546}, { 1.0000000000000009e-100, -400000.03999999998, 111430.10303574696, 551165}, { 1.0000000000000009e-100, -160000.016, 44598.652055220169, 44612}, { 1.0000000000000009e-100, -64000.006399999998, 17865.125532670456, 17879}, { 1.0000000000000009e-100, -25600.002560000001, 7172.046519886364, 7186}, { 1.0000000000000009e-100, -10240.001024000001, 2895.015980113636, 2908}, { 1.0000000000000009e-100, -4096.0004096000002, 1184.2329545454545, 1198}, { 1.0000000000000009e-100, -1638.40016384, 502.04403409090901, 517}, { 1.0000000000000009e-100, -655.36006553599998, 231.13849431818178, 247}, { 1.0000000000000009e-100, -262.14402621440001, 128.02290482954544, 149}, { 1.0000000000000009e-100, -104.85761048576001, 100.76188016528923, 133}, { 1.0000000000000009e-100, -41.943044194304001, 115.10072314049584, 162}, { 1.0000000000000009e-100, -16.777217677721602, 147.05255681818178, 203}, { 1.0000000000000009e-100, -6.7108870710886404, 171.19963842975204, 231}, { 1.0000000000000009e-100, -2.6843548284354561, 185.58367768595039, 246}, { 1.0000000000000009e-100, -1.0737419313741825, 191.0575929752066, 252}, { 1.0000000000000009e-75, -1000000.1, 278497.00512279174, 278510}, { 1.0000000000000009e-75, -400000.03999999998, 111417.96751369131, 111431}, { 1.0000000000000009e-75, -160000.016, 44586.151478160507, 44599}, { 1.0000000000000009e-75, -64000.006399999998, 17853.128506747154, 17866}, { 1.0000000000000009e-75, -25600.002560000001, 7160.0009918212891, 7173}, { 1.0000000000000009e-75, -10240.001024000001, 2882.563210227273, 2896}, { 1.0000000000000009e-75, -4096.0004096000002, 1171.4037642045455, 1185}, { 1.0000000000000009e-75, -1638.40016384, 489.03941761363637, 503}, { 1.0000000000000009e-75, -655.36006553599998, 216.05113636363637, 232}, { 1.0000000000000009e-75, -262.14402621440001, 110.08296745867767, 129}, { 1.0000000000000009e-75, -104.85761048576001, 74.400826446280988, 102}, { 1.0000000000000009e-75, -41.943044194304001, 75.619834710743788, 118}, { 1.0000000000000009e-75, -16.777217677721602, 96.373966942148741, 150}, { 1.0000000000000009e-75, -6.7108870710886404, 116.33910123966942, 175}, { 1.0000000000000009e-75, -2.6843548284354561, 129.00947507747935, 189}, { 1.0000000000000009e-75, -1.0737419313741825, 134.06379132231399, 195}, { 1.0000000000000011e-50, -1000000.1, 278484.14271457132, 465687}, { 1.0000000000000011e-50, -400000.03999999998, 111405.16246448863, 111418}, { 1.0000000000000011e-50, -160000.016, 44573.165838068184, 44587}, { 1.0000000000000011e-50, -64000.006399999998, 17840.147727272728, 17854}, { 1.0000000000000011e-50, -25600.002560000001, 7147.375887784091, 7160}, { 1.0000000000000011e-50, -10240.001024000001, 2869.888583096591, 2883}, { 1.0000000000000011e-50, -4096.0004096000002, 1158.1576704545455, 1172}, { 1.0000000000000011e-50, -1638.40016384, 476.00257457386363, 490}, { 1.0000000000000011e-50, -655.36006553599998, 202.20170454545453, 217}, { 1.0000000000000011e-50, -262.14402621440001, 94.02682722107437, 111}, { 1.0000000000000011e-50, -104.85761048576001, 53.0810950413223, 76}, { 1.0000000000000011e-50, -41.943044194304001, 42.355371900826441, 78}, { 1.0000000000000011e-50, -16.777217677721602, 50.051652892561975, 100}, { 1.0000000000000011e-50, -6.7108870710886404, 63.174070247933884, 120}, { 1.0000000000000011e-50, -2.6843548284354561, 72.956805268595033, 132}, { 1.0000000000000011e-50, -1.0737419313741825, 77.104855371900811, 138}, { 1.0000000000000012e-25, -1000000.1, 278471.90274041548, 448844}, { 1.0000000000000012e-25, -400000.03999999998, 111392.85056374289, 111423}, { 1.0000000000000012e-25, -160000.016, 44561.0361328125, 44574}, { 1.0000000000000012e-25, -64000.006399999998, 17828.149314186794, 17841}, { 1.0000000000000012e-25, -25600.002560000001, 7134.765625, 7148}, { 1.0000000000000012e-25, -10240.001024000001, 2857.211647727273, 2870}, { 1.0000000000000012e-25, -4096.0004096000002, 1146.1576704545453, 1159}, { 1.0000000000000012e-25, -1638.40016384, 463.0625, 476}, { 1.0000000000000012e-25, -655.36006553599998, 189.27556818181819, 203}, { 1.0000000000000012e-25, -262.14402621440001, 79.163223140495859, 95}, { 1.0000000000000012e-25, -104.85761048576001, 36.15702479338843, 54}, { 1.0000000000000012e-25, -41.943044194304001, 19.008264462809912, 44}, { 1.0000000000000012e-25, -16.777217677721602, 14.220686983471072, 53}, { 1.0000000000000012e-25, -6.7108870710886404, 15.207081141998497, 66}, { 1.0000000000000012e-25, -2.6843548284354561, 19.000046957175051, 76}, { 1.0000000000000012e-25, -1.0737419313741825, 21.00738364892468, 81}, { 1, -1000000.1, 278455, 285026}, { 1, -400000.03999999998, 111376, 111392}, { 1, -160000.016, 44544, 44561}, { 1, -64000.006399999998, 17812, 17828}, { 1, -25600.002560000001, 7119, 7135}, { 1, -10240.001024000001, 2841, 2858}, { 1, -4096.0004096000002, 1130, 1147}, { 1, -1638.40016384, 447, 464}, { 1, -655.36006553599998, 174, 190}, { 1, -262.14402621440001, 64, 81}, { 1, -104.85761048576001, 21, 37}, { 1, -41.943044194304001, 5, 20}, { 1, -16.777217677721602, 0, 16}, { 1, -6.7108870710886404, 0, 17}, { 1, -2.6843548284354561, 0, 20}, { 1, -1.0737419313741825, 0, 21}, { 2.5, -1000000.1, 278450, 278466}, { 2.5, -400000.03999999998, 111372, 111388}, { 2.5, -160000.016, 44540, 44557}, { 2.5, -64000.006399999998, 17808, 17824}, { 2.5, -25600.002560000001, 7115, 7131}, { 2.5, -10240.001024000001, 2838, 2854}, { 2.5, -4096.0004096000002, 1127, 1144}, { 2.5, -1638.40016384, 445, 461}, { 2.5, -655.36006553599998, 172, 188}, { 2.5, -262.14402621440001, 63, 79}, { 2.5, -104.85761048576001, 20, 35}, { 2.5, -41.943044194304001, 4, 19}, { 2.5, -16.777217677721602, 0, 13}, { 2.5, -6.7108870710886404, 0, 13}, { 2.5, -2.6843548284354561, 0, 15}, { 2.5, -1.0737419313741825, 2, 16}, { 6.25, -1000000.1, 278440, 278456}, { 6.25, -400000.03999999998, 111362, 111378}, { 6.25, -160000.016, 44531, 44548}, { 6.25, -64000.006399999998, 17800, 17816}, { 6.25, -25600.002560000001, 7108, 7124}, { 6.25, -10240.001024000001, 2831, 2848}, { 6.25, -4096.0004096000002, 1121, 1138}, { 6.25, -1638.40016384, 440, 456}, { 6.25, -655.36006553599998, 167, 183}, { 6.25, -262.14402621440001, 59, 75}, { 6.25, -104.85761048576001, 18, 32}, { 6.25, -41.943044194304001, 3, 16}, { 6.25, -16.777217677721602, 0, 10}, { 6.25, -6.7108870710886404, 0, 9}, { 6.25, -2.6843548284354561, 0, 9}, { 6.25, -1.0737419313741825, 4, 9}, { 15.625, -1000000.1, 278415, 278432}, { 15.625, -400000.03999999998, 111339, 111425}, { 15.625, -160000.016, 44510, 44527}, { 15.625, -64000.006399999998, 17781, 17797}, { 15.625, -25600.002560000001, 7091, 7107}, { 15.625, -10240.001024000001, 2816, 2833}, { 15.625, -4096.0004096000002, 1108, 1125}, { 15.625, -1638.40016384, 429, 445}, { 15.625, -655.36006553599998, 159, 174}, { 15.625, -262.14402621440001, 53, 67}, { 15.625, -104.85761048576001, 14, 27}, { 15.625, -41.943044194304001, 2, 11}, { 15.625, -16.777217677721602, 0, 7}, { 15.625, -6.7108870710886404, 0, 5}, { 15.625, -2.6843548284354561, 5, 5}, { 15.625, -1.0737419313741825, 5, 5}, { 39.0625, -1000000.1, 278359, 431061}, { 39.0625, -400000.03999999998, 111287, 111304}, { 39.0625, -160000.016, 44463, 44480}, { 39.0625, -64000.006399999998, 17738, 17755}, { 39.0625, -25600.002560000001, 7053, 7069}, { 39.0625, -10240.001024000001, 2784, 2800}, { 39.0625, -4096.0004096000002, 1081, 1097}, { 39.0625, -1638.40016384, 407, 422}, { 39.0625, -655.36006553599998, 143, 157}, { 39.0625, -262.14402621440001, 43, 56}, { 39.0625, -104.85761048576001, 10, 19}, { 39.0625, -41.943044194304001, 0, 7}, { 39.0625, -16.777217677721602, 0, 4}, { 39.0625, -6.7108870710886404, 0, 3}, { 39.0625, -2.6843548284354561, 3, 3}, { 39.0625, -1.0737419313741825, 3, 3}, { 97.65625, -1000000.1, 278230, 278277}, { 97.65625, -400000.03999999998, 111170, 111425}, { 97.65625, -160000.016, 44358, 44374}, { 97.65625, -64000.006399999998, 17645, 17661}, { 97.65625, -25600.002560000001, 6972, 6988}, { 97.65625, -10240.001024000001, 2715, 2731}, { 97.65625, -4096.0004096000002, 1026, 1041}, { 97.65625, -1638.40016384, 366, 380}, { 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3814.697265625, -262.14402621440001, 0, 4}, { 3814.697265625, -104.85761048576001, 0, 1}, { 3814.697265625, -41.943044194304001, 0, 1}, { 3814.697265625, -16.777217677721602, 0, 1}, { 3814.697265625, -6.7108870710886404, 0, 1}, { 3814.697265625, -2.6843548284354561, 0, 1}, { 3814.697265625, -1.0737419313741825, 0, 1}, { 9536.7431640625, -1000000.1, 265953, 266155}, { 9536.7431640625, -400000.03999999998, 100987, 101002}, { 9536.7431640625, -160000.016, 36374, 36388}, { 9536.7431640625, -64000.006399999998, 11921, 11934}, { 9536.7431640625, -25600.002560000001, 3374, 3384}, { 9536.7431640625, -10240.001024000001, 795, 802}, { 9536.7431640625, -4096.0004096000002, 157, 162}, { 9536.7431640625, -1638.40016384, 27, 30}, { 9536.7431640625, -655.36006553599998, 3, 6}, { 9536.7431640625, -262.14402621440001, 0, 1}, { 9536.7431640625, -104.85761048576001, 0, 1}, { 9536.7431640625, -41.943044194304001, 0, 1}, { 9536.7431640625, -16.777217677721602, 0, 1}, { 9536.7431640625, -6.7108870710886404, 0, 1}, { 9536.7431640625, -2.6843548284354561, 0, 1}, { 9536.7431640625, -1.0737419313741825, 0, 1}, }; static const unsigned total_elements = sizeof(domain) / sizeof(domain[0]); static const unsigned stride = 16; //static const unsigned a_elements = total_elements / stride; BOOST_MATH_ASSERT(total_elements % stride == 0); static const double a_max = domain[total_elements - 1][0]; static const double a_min = domain[0][0]; static const double b_max = domain[stride - 1][1]; static const double b_min = domain[0][1]; if (a < a_min) return 0; // TODO: Use series? if (b < b_min) { // // This is a general heuristic that's more or less correct, // but a bit too safe in most cases. Checked OK with // random testing. // if (z > -b) return 1; if (a < 100) return z < -b / (4 - 5 * (log(a)) * a / b) ? -1 : 0; else return z < -b / (4 - 5 * sqrt(log(a)) * a / b) ? -1 : 0; } if (a > a_max) { // // Crossover point is decreasing with increasing a // upper limit is fine, lower limit is not: // if (b > b_max) return 0; // TODO: don't know what else to do??? unsigned index = total_elements - stride; BOOST_MATH_ASSERT(domain[index][0] == a_max); while (domain[index][1] < b) ++index; double b0 = domain[index - 1][1]; double b1 = domain[index][1]; double z0 = domain[index - 1][3]; double z1 = domain[index][3]; T upper_z_limit = static_cast<T>((z0 * (b1 - b) + z1 * (b - b0)) / (b1 - b0)); if (z > upper_z_limit) return 1; return z < -b / (4 - 5 * sqrt(log(a)) * a / b) ? -1 : 0; } if (b > b_max) { return 0; // TODO: is there a better way??? } // // Bi-linear interpolation case: // unsigned index = 0; while (domain[index][0] < a) index += stride; while (domain[index][1] < b) ++index; // // We now have 4 corners: // double x1 = domain[index - stride - 1][0]; double x2 = domain[index][0]; double y1 = domain[index - 1][1]; double y2 = domain[index][1]; double f11 = domain[index - stride - 1][2]; BOOST_MATH_ASSERT(domain[index - stride - 1][0] == x1); BOOST_MATH_ASSERT(domain[index - stride - 1][1] == y1); double f12 = domain[index - stride][2]; BOOST_MATH_ASSERT(domain[index - stride][0] == x1); BOOST_MATH_ASSERT(domain[index - stride][1] == y2); double f21 = domain[index - 1][2]; BOOST_MATH_ASSERT(domain[index - 1][0] == x2); BOOST_MATH_ASSERT(domain[index - 1][1] == y1); double f22 = domain[index][2]; BOOST_MATH_ASSERT(domain[index][0] == x2); BOOST_MATH_ASSERT(domain[index][1] == y2); // // Interpolation is a crude tool and our corners have quite tight bounds, // and the "impossible region" is bounded by convex planes. For the upper // bound this is fine - bilinear interpolation will be over-conservative // but safe. For the lower limit though we may calculate that the "safe" // zone is in an unsafe place. To work around this we move our coordinates // closer to the lowest corner of our bounding box by an amount // proportional to the distance from the nearest corner. This has the effect // of making our lower bound smaller than it would otherwise be, and more so nearer // the centre of the bounding box. // T effective_x = static_cast<T>(a + (std::min)(T(a - x1), T(x2 - a)) * 0.25); T effective_y = static_cast<T>(b + (std::min)(T(b - y1), T(y2 - b)) * 0.25); T lower_limit = static_cast<T>(1 / ((x2 - x1) * (y2 - y1)) * (f11 * (x2 - effective_x) * (y2 - effective_y) + f21 * (effective_x - x1) * (y2 - effective_y) + f12 * (x2 - effective_x) * (effective_y - y1) + f22 * (effective_x - x1) * (effective_y - y1))); // // If one f value was zero, take the whole box as zero: // double min_f = (std::min)((std::min)(f11, f12), (std::min)(f21, f22)); if (min_f == 0) lower_limit = 0; // // Now we can test! // if (z < lower_limit) return -1; BOOST_MATH_ASSERT(f11 <= domain[index - stride - 1][3]); f11 = domain[index - stride - 1][3]; BOOST_MATH_ASSERT(f12 <= domain[index - stride][3]); f12 = domain[index - stride][3]; BOOST_MATH_ASSERT(f21 <= domain[index - 1][3]); f21 = domain[index - 1][3]; BOOST_MATH_ASSERT(f22 <= domain[index][3]); f22 = domain[index][3]; T upper_limit = static_cast<T>(1 / ((x2 - x1) * (y2 - y1)) * (f11 * (x2 - a) * (y2 - b) + f21 * (a - x1) * (y2 - b) + f12 * (x2 - a) * (b - y1) + f22 * (a - x1) * (b - y1))); if (z > upper_limit) return 1; return 0; } } } } #endif // BOOST_MATH_DETIAL_1F1_MAP_NEG_B_HPP