/************************************************************************* Cephes Math Library Release 2.8: June, 2000 Copyright by Stephen L. Moshier Contributors: * Sergey Bochkanov (ALGLIB project). Translation from C to pseudocode. See subroutines comments for additional copyrights. Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met: - Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer. - Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer listed in this license in the documentation and/or other materials provided with the distribution. - Neither the name of the copyright holders nor the names of its contributors may be used to endorse or promote products derived from this software without specific prior written permission. THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. *************************************************************************/ #include "gammaf.h" #include double gammastirf(double x); double ap_pi(){ return 3.14159265358979323846; } int ap_round(double x) { int a=(int)(floor(x+0.5)); return a; } int ap_ifloor(double x) { int a=(int)(floor(x)); return a; } /************************************************************************* Вычисление гамма-функции. Входные параметры: X - аргумент Допустимые значения: 0 < X < 171.6 -170 < X < 0, X не целое. Относительная погрешность вычисления: ОБЛАСТЬ #ТЕСТОВ МАКС. СРЕДН. (-170, -33) 20000 2.3e-15 3.3e-16 (-33, 33) 20000 9.4e-16 2.2e-16 ( 33, 171.6) 20000 2.3e-15 3.2e-16 Cephes Math Library Release 2.8: June, 2000 Original copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier Translated to AlgoPascal by Bochkanov Sergey (2005, 2006). *************************************************************************/ double gamma(double x) { double result; double p; double pp; double q; double qq; double z; int i; double sgngam; sgngam = 1; q = fabs(x); if( q>33.0 ) { if( x<0.0 ) { p = ap_ifloor(q); i = ap_round(p); if( i%2==0 ) { sgngam = -1; } z = q-p; if( z>0.5 ) { p = p+1; z = q-p; } z = q*sin(ap_pi()*z); z = fabs(z); z = ap_pi()/(z*gammastirf(q)); } else { z = gammastirf(x); } result = sgngam*z; return result; } z = 1; while(x>=3) { x = x-1; z = z*x; } while(x<0) { if( x>-0.000000001 ) { result = z/((1+0.5772156649015329*x)*x); return result; } z = z/x; x = x+1; } while(x<2) { if( x<0.000000001 ) { result = z/((1+0.5772156649015329*x)*x); return result; } z = z/x; x = x+1.0; } if( x==2 ) { result = z; return result; } x = x-2.0; pp = 1.60119522476751861407E-4; pp = 1.19135147006586384913E-3+x*pp; pp = 1.04213797561761569935E-2+x*pp; pp = 4.76367800457137231464E-2+x*pp; pp = 2.07448227648435975150E-1+x*pp; pp = 4.94214826801497100753E-1+x*pp; pp = 9.99999999999999996796E-1+x*pp; qq = -2.31581873324120129819E-5; qq = 5.39605580493303397842E-4+x*qq; qq = -4.45641913851797240494E-3+x*qq; qq = 1.18139785222060435552E-2+x*qq; qq = 3.58236398605498653373E-2+x*qq; qq = -2.34591795718243348568E-1+x*qq; qq = 7.14304917030273074085E-2+x*qq; qq = 1.00000000000000000320+x*qq; result = z*pp/qq; return result; } /************************************************************************* Вычисление натурального логарифма модуля гамма-функции. Входные параметры: x - точка, в которой вычисляем логарифм: Выходные параметры: SgnGam - знак, который имеет гамма-функция. Либо +1, либо -1. Результат: натуральный логарифм модуля гамма-функции Допустимые значения: 0 < X < 2.55e305 -2.55e305 < X < 0, X не целое. Погрешность аппроксимации (относительная для значений функции больших 1, иначе - абсолютная): domain #trials peak rms 0, 3 28000 5.4e-16 1.1e-16 2.718, 2.556e305 40000 3.5e-16 8.3e-17 Следующий тест приводит относительную погрешность, хотя для некоторых точек ошибка может быть больше, чем указано в таблице: -200, -4 10000 4.8e-16 1.3e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier Translated to AlgoPascal by Bochkanov Sergey (2005, 2006). *************************************************************************/ /* double lngamma(double x, double &sgngam) { double result; double a; double b; double c; double p; double q; double u; double w; double z; int i; double logpi; double ls2pi; double tmp; sgngam = 1; logpi = 1.14472988584940017414; ls2pi = 0.91893853320467274178; if( x<-34.0 ) { q = -x; w = lngamma(q, tmp); p = ap_ifloor(q); i = ap_round(p); if( i%2==0 ) { sgngam = -1; } else { sgngam = 1; } z = q-p; if( z>0.5 ) { p = p+1; z = p-q; } z = q*sin(ap_pi()*z); result = logpi-log(z)-w; return result; } if( x<13 ) { z = 1; p = 0; u = x; while(u>=3) { p = p-1; u = x+p; z = z*u; } while(u<2) { z = z/u; p = p+1; u = x+p; } if( z<0 ) { sgngam = -1; z = -z; } else { sgngam = 1; } if( u==2 ) { result = log(z); return result; } p = p-2; x = x+p; b = -1378.25152569120859100; b = -38801.6315134637840924+x*b; b = -331612.992738871184744+x*b; b = -1162370.97492762307383+x*b; b = -1721737.00820839662146+x*b; b = -853555.664245765465627+x*b; c = 1; c = -351.815701436523470549+x*c; c = -17064.2106651881159223+x*c; c = -220528.590553854454839+x*c; c = -1139334.44367982507207+x*c; c = -2532523.07177582951285+x*c; c = -2018891.41433532773231+x*c; p = x*b/c; result = log(z)+p; return result; } q = (x-0.5)*log(x)-x+ls2pi; if( x>100000000 ) { result = q; return result; } p = 1/(x*x); if( x>=1000.0 ) { q = q+((7.9365079365079365079365*0.0001*p-2.7777777777777777777778*0.001)*p+0.0833333333333333333333)/x; } else { a = 8.11614167470508450300*0.0001; a = -5.95061904284301438324*0.0001+p*a; a = 7.93650340457716943945*0.0001+p*a; a = -2.77777777730099687205*0.001+p*a; a = 8.33333333333331927722*0.01+p*a; q = q+a/x; } result = q; return result; } */ /************************************************************************* Формула Стирлинга Для внутреннего использования Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier *************************************************************************/ double gammastirf(double x) { double result; double y; double w; double v; double stir; w = 1/x; stir = 7.87311395793093628397E-4; stir = -2.29549961613378126380E-4+w*stir; stir = -2.68132617805781232825E-3+w*stir; stir = 3.47222221605458667310E-3+w*stir; stir = 8.33333333333482257126E-2+w*stir; w = 1+w*stir; y = exp(x); if( x>143.01608 ) { v = pow(x, 0.5*x-0.25); y = v*(v/y); } else { y = pow(x, x-0.5)/y; } result = 2.50662827463100050242*y*w; return result; }