diff options
Diffstat (limited to 'gsl-1.9/specfunc/bessel_temme.c')
-rw-r--r-- | gsl-1.9/specfunc/bessel_temme.c | 219 |
1 files changed, 219 insertions, 0 deletions
diff --git a/gsl-1.9/specfunc/bessel_temme.c b/gsl-1.9/specfunc/bessel_temme.c new file mode 100644 index 0000000..0f2adf3 --- /dev/null +++ b/gsl-1.9/specfunc/bessel_temme.c @@ -0,0 +1,219 @@ +/* specfunc/bessel_temme.c + * + * Copyright (C) 1996, 1997, 1998, 1999, 2000 Gerard Jungman + * + * 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., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA. + */ + +/* Author: G. Jungman */ + +/* Calculate series for Y_nu and K_nu for small x and nu. + * This is applicable for x < 2 and |nu|<=1/2. + * These functions assume x > 0. + */ +#include <config.h> +#include <gsl/gsl_math.h> +#include <gsl/gsl_errno.h> +#include <gsl/gsl_mode.h> +#include "bessel_temme.h" + +#include "chebyshev.h" +#include "cheb_eval.c" + +/* nu = (x+1)/4, -1<x<1, 1/(2nu)(1/Gamma[1-nu]-1/Gamma[1+nu]) */ +static double g1_dat[14] = { + -1.14516408366268311786898152867, + 0.00636085311347084238122955495, + 0.00186245193007206848934643657, + 0.000152833085873453507081227824, + 0.000017017464011802038795324732, + -6.4597502923347254354668326451e-07, + -5.1819848432519380894104312968e-08, + 4.5189092894858183051123180797e-10, + 3.2433227371020873043666259180e-11, + 6.8309434024947522875432400828e-13, + 2.8353502755172101513119628130e-14, + -7.9883905769323592875638087541e-16, + -3.3726677300771949833341213457e-17, + -3.6586334809210520744054437104e-20 +}; +static cheb_series g1_cs = { + g1_dat, + 13, + -1, 1, + 7 +}; + +/* nu = (x+1)/4, -1<x<1, 1/2 (1/Gamma[1-nu]+1/Gamma[1+nu]) */ +static double g2_dat[15] = +{ + 1.882645524949671835019616975350, + -0.077490658396167518329547945212, + -0.018256714847324929419579340950, + 0.0006338030209074895795923971731, + 0.0000762290543508729021194461175, + -9.5501647561720443519853993526e-07, + -8.8927268107886351912431512955e-08, + -1.9521334772319613740511880132e-09, + -9.4003052735885162111769579771e-11, + 4.6875133849532393179290879101e-12, + 2.2658535746925759582447545145e-13, + -1.1725509698488015111878735251e-15, + -7.0441338200245222530843155877e-17, + -2.4377878310107693650659740228e-18, + -7.5225243218253901727164675011e-20 +}; +static cheb_series g2_cs = { + g2_dat, + 14, + -1, 1, + 8 +}; + + +static +int +gsl_sf_temme_gamma(const double nu, double * g_1pnu, double * g_1mnu, double * g1, double * g2) +{ + const double anu = fabs(nu); /* functions are even */ + const double x = 4.0*anu - 1.0; + gsl_sf_result r_g1; + gsl_sf_result r_g2; + cheb_eval_e(&g1_cs, x, &r_g1); + cheb_eval_e(&g2_cs, x, &r_g2); + *g1 = r_g1.val; + *g2 = r_g2.val; + *g_1mnu = 1.0/(r_g2.val + nu * r_g1.val); + *g_1pnu = 1.0/(r_g2.val - nu * r_g1.val); + return GSL_SUCCESS; +} + + +int +gsl_sf_bessel_Y_temme(const double nu, const double x, + gsl_sf_result * Ynu, + gsl_sf_result * Ynup1) +{ + const int max_iter = 15000; + + const double half_x = 0.5 * x; + const double ln_half_x = log(half_x); + const double half_x_nu = exp(nu*ln_half_x); + const double pi_nu = M_PI * nu; + const double alpha = pi_nu / 2.0; + const double sigma = -nu * ln_half_x; + const double sinrat = (fabs(pi_nu) < GSL_DBL_EPSILON ? 1.0 : pi_nu/sin(pi_nu)); + const double sinhrat = (fabs(sigma) < GSL_DBL_EPSILON ? 1.0 : sinh(sigma)/sigma); + const double sinhalf = (fabs(alpha) < GSL_DBL_EPSILON ? 1.0 : sin(alpha)/alpha); + const double sin_sqr = nu*M_PI*M_PI*0.5 * sinhalf*sinhalf; + + double sum0, sum1; + double fk, pk, qk, hk, ck; + int k = 0; + int stat_iter; + + double g_1pnu, g_1mnu, g1, g2; + int stat_g = gsl_sf_temme_gamma(nu, &g_1pnu, &g_1mnu, &g1, &g2); + + fk = 2.0/M_PI * sinrat * (cosh(sigma)*g1 - sinhrat*ln_half_x*g2); + pk = 1.0/M_PI /half_x_nu * g_1pnu; + qk = 1.0/M_PI *half_x_nu * g_1mnu; + hk = pk; + ck = 1.0; + + sum0 = fk + sin_sqr * qk; + sum1 = pk; + + while(k < max_iter) { + double del0; + double del1; + double gk; + k++; + fk = (k*fk + pk + qk)/(k*k-nu*nu); + ck *= -half_x*half_x/k; + pk /= (k - nu); + qk /= (k + nu); + gk = fk + sin_sqr * qk; + hk = -k*gk + pk; + del0 = ck * gk; + del1 = ck * hk; + sum0 += del0; + sum1 += del1; + if(fabs(del0) < 0.5*(1.0 + fabs(sum0))*GSL_DBL_EPSILON) break; + } + + Ynu->val = -sum0; + Ynu->err = (2.0 + 0.5*k) * GSL_DBL_EPSILON * fabs(Ynu->val); + Ynup1->val = -sum1 * 2.0/x; + Ynup1->err = (2.0 + 0.5*k) * GSL_DBL_EPSILON * fabs(Ynup1->val); + + stat_iter = ( k >= max_iter ? GSL_EMAXITER : GSL_SUCCESS ); + return GSL_ERROR_SELECT_2(stat_iter, stat_g); +} + + +int +gsl_sf_bessel_K_scaled_temme(const double nu, const double x, + double * K_nu, double * K_nup1, double * Kp_nu) +{ + const int max_iter = 15000; + + const double half_x = 0.5 * x; + const double ln_half_x = log(half_x); + const double half_x_nu = exp(nu*ln_half_x); + const double pi_nu = M_PI * nu; + const double sigma = -nu * ln_half_x; + const double sinrat = (fabs(pi_nu) < GSL_DBL_EPSILON ? 1.0 : pi_nu/sin(pi_nu)); + const double sinhrat = (fabs(sigma) < GSL_DBL_EPSILON ? 1.0 : sinh(sigma)/sigma); + const double ex = exp(x); + + double sum0, sum1; + double fk, pk, qk, hk, ck; + int k = 0; + int stat_iter; + + double g_1pnu, g_1mnu, g1, g2; + int stat_g = gsl_sf_temme_gamma(nu, &g_1pnu, &g_1mnu, &g1, &g2); + + fk = sinrat * (cosh(sigma)*g1 - sinhrat*ln_half_x*g2); + pk = 0.5/half_x_nu * g_1pnu; + qk = 0.5*half_x_nu * g_1mnu; + hk = pk; + ck = 1.0; + sum0 = fk; + sum1 = hk; + while(k < max_iter) { + double del0; + double del1; + k++; + fk = (k*fk + pk + qk)/(k*k-nu*nu); + ck *= half_x*half_x/k; + pk /= (k - nu); + qk /= (k + nu); + hk = -k*fk + pk; + del0 = ck * fk; + del1 = ck * hk; + sum0 += del0; + sum1 += del1; + if(fabs(del0) < 0.5*fabs(sum0)*GSL_DBL_EPSILON) break; + } + + *K_nu = sum0 * ex; + *K_nup1 = sum1 * 2.0/x * ex; + *Kp_nu = - *K_nup1 + nu/x * *K_nu; + + stat_iter = ( k == max_iter ? GSL_EMAXITER : GSL_SUCCESS ); + return GSL_ERROR_SELECT_2(stat_iter, stat_g); +} |