| /* gauss.c - gaussian random numbers, using the Ziggurat method |
| * |
| * Copyright (C) 2005 Jochen Voss. |
| * |
| * This program is free software; you can redistribute it and/or modify |
| * it under the terms of the GNU General Public License as published by |
| * the Free Software Foundation; either version 2 of the License, or |
| * (at your option) any later version. |
| * |
| * This program is distributed in the hope that it will be useful, |
| * but WITHOUT ANY WARRANTY; without even the implied warranty of |
| * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the |
| * GNU General Public License for more details. |
| * |
| * You should have received a copy of the GNU General Public License |
| * along with this program; if not, write to the Free Software |
| * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA |
| */ |
| |
| /* |
| * This routine is based on the following article, with a couple of |
| * modifications which simplify the implementation. |
| * |
| * George Marsaglia, Wai Wan Tsang |
| * The Ziggurat Method for Generating Random Variables |
| * Journal of Statistical Software, vol. 5 (2000), no. 8 |
| * http://www.jstatsoft.org/v05/i08/ |
| * |
| * The modifications are: |
| * |
| * 1) use 128 steps instead of 256 to decrease the amount of static |
| * data necessary. |
| * |
| * 2) use an acceptance sampling from an exponential wedge |
| * exp(-R*(x-R/2)) for the tail of the base strip to simplify the |
| * implementation. The area of exponential wedge is used in |
| * calculating 'v' and the coefficients in ziggurat table, so the |
| * coefficients differ slightly from those in the Marsaglia and Tsang |
| * paper. |
| * |
| * See also Leong et al, "A Comment on the Implementation of the |
| * Ziggurat Method", Journal of Statistical Software, vol 5 (2005), no 7. |
| * |
| */ |
| |
| |
| #include <config.h> |
| #include <math.h> |
| #include <gsl/gsl_math.h> |
| #include <gsl/gsl_rng.h> |
| #include <gsl/gsl_randist.h> |
| |
| /* position of right-most step */ |
| #define PARAM_R 3.44428647676 |
| |
| /* tabulated values for the heigt of the Ziggurat levels */ |
| static const double ytab[128] = { |
| 1, 0.963598623011, 0.936280813353, 0.913041104253, |
| 0.892278506696, 0.873239356919, 0.855496407634, 0.838778928349, |
| 0.822902083699, 0.807732738234, 0.793171045519, 0.779139726505, |
| 0.765577436082, 0.752434456248, 0.739669787677, 0.727249120285, |
| 0.715143377413, 0.703327646455, 0.691780377035, 0.68048276891, |
| 0.669418297233, 0.65857233912, 0.647931876189, 0.637485254896, |
| 0.62722199145, 0.617132611532, 0.607208517467, 0.597441877296, |
| 0.587825531465, 0.578352913803, 0.569017984198, 0.559815170911, |
| 0.550739320877, 0.541785656682, 0.532949739145, 0.524227434628, |
| 0.515614886373, 0.507108489253, 0.498704867478, 0.490400854812, |
| 0.482193476986, 0.47407993601, 0.466057596125, 0.458123971214, |
| 0.450276713467, 0.442513603171, 0.434832539473, 0.427231532022, |
| 0.419708693379, 0.41226223212, 0.404890446548, 0.397591718955, |
| 0.390364510382, 0.383207355816, 0.376118859788, 0.369097692334, |
| 0.362142585282, 0.355252328834, 0.348425768415, 0.341661801776, |
| 0.334959376311, 0.328317486588, 0.321735172063, 0.31521151497, |
| 0.308745638367, 0.302336704338, 0.29598391232, 0.289686497571, |
| 0.283443729739, 0.27725491156, 0.271119377649, 0.265036493387, |
| 0.259005653912, 0.253026283183, 0.247097833139, 0.241219782932, |
| 0.235391638239, 0.229612930649, 0.223883217122, 0.218202079518, |
| 0.212569124201, 0.206983981709, 0.201446306496, 0.195955776745, |
| 0.190512094256, 0.185114984406, 0.179764196185, 0.174459502324, |
| 0.169200699492, 0.1639876086, 0.158820075195, 0.153697969964, |
| 0.148621189348, 0.143589656295, 0.138603321143, 0.133662162669, |
| 0.128766189309, 0.123915440582, 0.119109988745, 0.114349940703, |
| 0.10963544023, 0.104966670533, 0.100343857232, 0.0957672718266, |
| 0.0912372357329, 0.0867541250127, 0.082318375932, 0.0779304915295, |
| 0.0735910494266, 0.0693007111742, 0.065060233529, 0.0608704821745, |
| 0.056732448584, 0.05264727098, 0.0486162607163, 0.0446409359769, |
| 0.0407230655415, 0.0368647267386, 0.0330683839378, 0.0293369977411, |
| 0.0256741818288, 0.0220844372634, 0.0185735200577, 0.0151490552854, |
| 0.0118216532614, 0.00860719483079, 0.00553245272614, 0.00265435214565 |
| }; |
| |
| /* tabulated values for 2^24 times x[i]/x[i+1], |
| * used to accept for U*x[i+1]<=x[i] without any floating point operations */ |
| static const unsigned long ktab[128] = { |
| 0, 12590644, 14272653, 14988939, |
| 15384584, 15635009, 15807561, 15933577, |
| 16029594, 16105155, 16166147, 16216399, |
| 16258508, 16294295, 16325078, 16351831, |
| 16375291, 16396026, 16414479, 16431002, |
| 16445880, 16459343, 16471578, 16482744, |
| 16492970, 16502368, 16511031, 16519039, |
| 16526459, 16533352, 16539769, 16545755, |
| 16551348, 16556584, 16561493, 16566101, |
| 16570433, 16574511, 16578353, 16581977, |
| 16585398, 16588629, 16591685, 16594575, |
| 16597311, 16599901, 16602354, 16604679, |
| 16606881, 16608968, 16610945, 16612818, |
| 16614592, 16616272, 16617861, 16619363, |
| 16620782, 16622121, 16623383, 16624570, |
| 16625685, 16626730, 16627708, 16628619, |
| 16629465, 16630248, 16630969, 16631628, |
| 16632228, 16632768, 16633248, 16633671, |
| 16634034, 16634340, 16634586, 16634774, |
| 16634903, 16634972, 16634980, 16634926, |
| 16634810, 16634628, 16634381, 16634066, |
| 16633680, 16633222, 16632688, 16632075, |
| 16631380, 16630598, 16629726, 16628757, |
| 16627686, 16626507, 16625212, 16623794, |
| 16622243, 16620548, 16618698, 16616679, |
| 16614476, 16612071, 16609444, 16606571, |
| 16603425, 16599973, 16596178, 16591995, |
| 16587369, 16582237, 16576520, 16570120, |
| 16562917, 16554758, 16545450, 16534739, |
| 16522287, 16507638, 16490152, 16468907, |
| 16442518, 16408804, 16364095, 16301683, |
| 16207738, 16047994, 15704248, 15472926 |
| }; |
| |
| /* tabulated values of 2^{-24}*x[i] */ |
| static const double wtab[128] = { |
| 1.62318314817e-08, 2.16291505214e-08, 2.54246305087e-08, 2.84579525938e-08, |
| 3.10340022482e-08, 3.33011726243e-08, 3.53439060345e-08, 3.72152672658e-08, |
| 3.8950989572e-08, 4.05763964764e-08, 4.21101548915e-08, 4.35664624904e-08, |
| 4.49563968336e-08, 4.62887864029e-08, 4.75707945735e-08, 4.88083237257e-08, |
| 5.00063025384e-08, 5.11688950428e-08, 5.22996558616e-08, 5.34016475624e-08, |
| 5.44775307871e-08, 5.55296344581e-08, 5.65600111659e-08, 5.75704813695e-08, |
| 5.85626690412e-08, 5.95380306862e-08, 6.04978791776e-08, 6.14434034901e-08, |
| 6.23756851626e-08, 6.32957121259e-08, 6.42043903937e-08, 6.51025540077e-08, |
| 6.59909735447e-08, 6.68703634341e-08, 6.77413882848e-08, 6.8604668381e-08, |
| 6.94607844804e-08, 7.03102820203e-08, 7.11536748229e-08, 7.1991448372e-08, |
| 7.2824062723e-08, 7.36519550992e-08, 7.44755422158e-08, 7.52952223703e-08, |
| 7.61113773308e-08, 7.69243740467e-08, 7.77345662086e-08, 7.85422956743e-08, |
| 7.93478937793e-08, 8.01516825471e-08, 8.09539758128e-08, 8.17550802699e-08, |
| 8.25552964535e-08, 8.33549196661e-08, 8.41542408569e-08, 8.49535474601e-08, |
| 8.57531242006e-08, 8.65532538723e-08, 8.73542180955e-08, 8.8156298059e-08, |
| 8.89597752521e-08, 8.97649321908e-08, 9.05720531451e-08, 9.138142487e-08, |
| 9.21933373471e-08, 9.30080845407e-08, 9.38259651738e-08, 9.46472835298e-08, |
| 9.54723502847e-08, 9.63014833769e-08, 9.71350089201e-08, 9.79732621669e-08, |
| 9.88165885297e-08, 9.96653446693e-08, 1.00519899658e-07, 1.0138063623e-07, |
| 1.02247952126e-07, 1.03122261554e-07, 1.04003996769e-07, 1.04893609795e-07, |
| 1.05791574313e-07, 1.06698387725e-07, 1.07614573423e-07, 1.08540683296e-07, |
| 1.09477300508e-07, 1.1042504257e-07, 1.11384564771e-07, 1.12356564007e-07, |
| 1.13341783071e-07, 1.14341015475e-07, 1.15355110887e-07, 1.16384981291e-07, |
| 1.17431607977e-07, 1.18496049514e-07, 1.19579450872e-07, 1.20683053909e-07, |
| 1.21808209468e-07, 1.2295639141e-07, 1.24129212952e-07, 1.25328445797e-07, |
| 1.26556042658e-07, 1.27814163916e-07, 1.29105209375e-07, 1.30431856341e-07, |
| 1.31797105598e-07, 1.3320433736e-07, 1.34657379914e-07, 1.36160594606e-07, |
| 1.37718982103e-07, 1.39338316679e-07, 1.41025317971e-07, 1.42787873535e-07, |
| 1.44635331499e-07, 1.4657889173e-07, 1.48632138436e-07, 1.50811780719e-07, |
| 1.53138707402e-07, 1.55639532047e-07, 1.58348931426e-07, 1.61313325908e-07, |
| 1.64596952856e-07, 1.68292495203e-07, 1.72541128694e-07, 1.77574279496e-07, |
| 1.83813550477e-07, 1.92166040885e-07, 2.05295471952e-07, 2.22600839893e-07 |
| }; |
| |
| |
| double |
| gsl_ran_gaussian_ziggurat (const gsl_rng * r, const double sigma) |
| { |
| unsigned long int i, j; |
| int sign; |
| double x, y; |
| |
| while (1) |
| { |
| i = gsl_rng_uniform_int (r, 256); /* choose the step */ |
| j = gsl_rng_uniform_int (r, 16777216); /* sample from 2^24 */ |
| sign = (i & 0x80) ? +1 : -1; |
| i &= 0x7f; |
| |
| x = j * wtab[i]; |
| |
| if (j < ktab[i]) |
| break; |
| |
| if (i < 127) |
| { |
| double y0, y1, U1; |
| y0 = ytab[i]; |
| y1 = ytab[i + 1]; |
| U1 = gsl_rng_uniform (r); |
| y = y1 + (y0 - y1) * U1; |
| } |
| else |
| { |
| double U1, U2; |
| U1 = 1.0 - gsl_rng_uniform (r); |
| U2 = gsl_rng_uniform (r); |
| x = PARAM_R - log (U1) / PARAM_R; |
| y = exp (-PARAM_R * (x - 0.5 * PARAM_R)) * U2; |
| } |
| |
| if (y < exp (-0.5 * x * x)) |
| break; |
| } |
| |
| return sign * sigma * x; |
| } |
