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Diffstat (limited to 'gsl-1.9/specfunc/legendre_Qn.c')
| -rw-r--r-- | gsl-1.9/specfunc/legendre_Qn.c | 366 |
1 files changed, 366 insertions, 0 deletions
diff --git a/gsl-1.9/specfunc/legendre_Qn.c b/gsl-1.9/specfunc/legendre_Qn.c new file mode 100644 index 0000000..2147f3c --- /dev/null +++ b/gsl-1.9/specfunc/legendre_Qn.c @@ -0,0 +1,366 @@ +/* specfunc/legendre_Qn.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 */ + +#include <config.h> +#include <gsl/gsl_math.h> +#include <gsl/gsl_errno.h> +#include <gsl/gsl_sf_bessel.h> +#include <gsl/gsl_sf_elementary.h> +#include <gsl/gsl_sf_exp.h> +#include <gsl/gsl_sf_pow_int.h> +#include <gsl/gsl_sf_legendre.h> + +#include "error.h" + +/* Evaluate f_{ell+1}/f_ell + * f_ell := Q^{b}_{a+ell}(x) + * x > 1 + */ +static +int +legendreQ_CF1_xgt1(int ell, double a, double b, double x, double * result) +{ + const double RECUR_BIG = GSL_SQRT_DBL_MAX; + const int maxiter = 5000; + int n = 1; + double Anm2 = 1.0; + double Bnm2 = 0.0; + double Anm1 = 0.0; + double Bnm1 = 1.0; + double a1 = ell + 1.0 + a + b; + double b1 = (2.0*(ell+1.0+a) + 1.0) * x; + double An = b1*Anm1 + a1*Anm2; + double Bn = b1*Bnm1 + a1*Bnm2; + double an, bn; + double fn = An/Bn; + + while(n < maxiter) { + double old_fn; + double del; + double lna; + n++; + Anm2 = Anm1; + Bnm2 = Bnm1; + Anm1 = An; + Bnm1 = Bn; + lna = ell + n + a; + an = b*b - lna*lna; + bn = (2.0*lna + 1.0) * x; + An = bn*Anm1 + an*Anm2; + Bn = bn*Bnm1 + an*Bnm2; + + if(fabs(An) > RECUR_BIG || fabs(Bn) > RECUR_BIG) { + An /= RECUR_BIG; + Bn /= RECUR_BIG; + Anm1 /= RECUR_BIG; + Bnm1 /= RECUR_BIG; + Anm2 /= RECUR_BIG; + Bnm2 /= RECUR_BIG; + } + + old_fn = fn; + fn = An/Bn; + del = old_fn/fn; + + if(fabs(del - 1.0) < 4.0*GSL_DBL_EPSILON) break; + } + + *result = fn; + + if(n == maxiter) + GSL_ERROR ("error", GSL_EMAXITER); + else + return GSL_SUCCESS; +} + + +/* Uniform asymptotic for Q_l(x). + * Assumes x > -1.0 and x != 1.0. + * Discards second order and higher terms. + */ +static +int +legendre_Ql_asymp_unif(const double ell, const double x, gsl_sf_result * result) +{ + if(x < 1.0) { + double u = ell + 0.5; + double th = acos(x); + gsl_sf_result Y0, Y1; + int stat_Y0, stat_Y1; + int stat_m; + double pre; + double B00; + double sum; + + /* B00 = 1/8 (1 - th cot(th) / th^2 + * pre = sqrt(th/sin(th)) + */ + if(th < GSL_ROOT4_DBL_EPSILON) { + B00 = (1.0 + th*th/15.0)/24.0; + pre = 1.0 + th*th/12.0; + } + else { + double sin_th = sqrt(1.0 - x*x); + double cot_th = x / sin_th; + B00 = 1.0/8.0 * (1.0 - th * cot_th) / (th*th); + pre = sqrt(th/sin_th); + } + + stat_Y0 = gsl_sf_bessel_Y0_e(u*th, &Y0); + stat_Y1 = gsl_sf_bessel_Y1_e(u*th, &Y1); + + sum = -0.5*M_PI * (Y0.val + th/u * Y1.val * B00); + + stat_m = gsl_sf_multiply_e(pre, sum, result); + result->err += 0.5*M_PI * fabs(pre) * (Y0.err + fabs(th/u*B00)*Y1.err); + result->err += GSL_DBL_EPSILON * fabs(result->val); + + return GSL_ERROR_SELECT_3(stat_m, stat_Y0, stat_Y1); + } + else { + double u = ell + 0.5; + double xi = acosh(x); + gsl_sf_result K0_scaled, K1_scaled; + int stat_K0, stat_K1; + int stat_e; + double pre; + double B00; + double sum; + + /* B00 = -1/8 (1 - xi coth(xi) / xi^2 + * pre = sqrt(xi/sinh(xi)) + */ + if(xi < GSL_ROOT4_DBL_EPSILON) { + B00 = (1.0-xi*xi/15.0)/24.0; + pre = 1.0 - xi*xi/12.0; + } + else { + double sinh_xi = sqrt(x*x - 1.0); + double coth_xi = x / sinh_xi; + B00 = -1.0/8.0 * (1.0 - xi * coth_xi) / (xi*xi); + pre = sqrt(xi/sinh_xi); + } + + stat_K0 = gsl_sf_bessel_K0_scaled_e(u*xi, &K0_scaled); + stat_K1 = gsl_sf_bessel_K1_scaled_e(u*xi, &K1_scaled); + + sum = K0_scaled.val - xi/u * K1_scaled.val * B00; + + stat_e = gsl_sf_exp_mult_e(-u*xi, pre * sum, result); + result->err = GSL_DBL_EPSILON * fabs(result->val) * fabs(u*xi); + result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val); + + return GSL_ERROR_SELECT_3(stat_e, stat_K0, stat_K1); + } +} + + + +/*-*-*-*-*-*-*-*-*-*-*-* Functions with Error Codes *-*-*-*-*-*-*-*-*-*-*-*/ + +int +gsl_sf_legendre_Q0_e(const double x, gsl_sf_result * result) +{ + /* CHECK_POINTER(result) */ + + if(x <= -1.0 || x == 1.0) { + DOMAIN_ERROR(result); + } + else if(x*x < GSL_ROOT6_DBL_EPSILON) { /* |x| <~ 0.05 */ + const double c3 = 1.0/3.0; + const double c5 = 1.0/5.0; + const double c7 = 1.0/7.0; + const double c9 = 1.0/9.0; + const double c11 = 1.0/11.0; + const double y = x * x; + const double series = 1.0 + y*(c3 + y*(c5 + y*(c7 + y*(c9 + y*c11)))); + result->val = x * series; + result->err = 2.0 * GSL_DBL_EPSILON * fabs(x); + return GSL_SUCCESS; + } + else if(x < 1.0) { + result->val = 0.5 * log((1.0+x)/(1.0-x)); + result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val); + return GSL_SUCCESS; + } + else if(x < 10.0) { + result->val = 0.5 * log((x+1.0)/(x-1.0)); + result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val); + return GSL_SUCCESS; + } + else if(x*GSL_DBL_MIN < 2.0) { + const double y = 1.0/(x*x); + const double c1 = 1.0/3.0; + const double c2 = 1.0/5.0; + const double c3 = 1.0/7.0; + const double c4 = 1.0/9.0; + const double c5 = 1.0/11.0; + const double c6 = 1.0/13.0; + const double c7 = 1.0/15.0; + result->val = (1.0/x) * (1.0 + y*(c1 + y*(c2 + y*(c3 + y*(c4 + y*(c5 + y*(c6 + y*c7))))))); + result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val); + return GSL_SUCCESS; + } + else { + UNDERFLOW_ERROR(result); + } +} + + +int +gsl_sf_legendre_Q1_e(const double x, gsl_sf_result * result) +{ + /* CHECK_POINTER(result) */ + + if(x <= -1.0 || x == 1.0) { + DOMAIN_ERROR(result); + } + else if(x*x < GSL_ROOT6_DBL_EPSILON) { /* |x| <~ 0.05 */ + const double c3 = 1.0/3.0; + const double c5 = 1.0/5.0; + const double c7 = 1.0/7.0; + const double c9 = 1.0/9.0; + const double c11 = 1.0/11.0; + const double y = x * x; + const double series = 1.0 + y*(c3 + y*(c5 + y*(c7 + y*(c9 + y*c11)))); + result->val = x * x * series - 1.0; + result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val); + return GSL_SUCCESS; + } + else if(x < 1.0){ + result->val = 0.5 * x * (log((1.0+x)/(1.0-x))) - 1.0; + result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val); + return GSL_SUCCESS; + } + else if(x < 6.0) { + result->val = 0.5 * x * log((x+1.0)/(x-1.0)) - 1.0; + result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val); + return GSL_SUCCESS; + } + else if(x*GSL_SQRT_DBL_MIN < 0.99/M_SQRT3) { + const double y = 1/(x*x); + const double c1 = 3.0/5.0; + const double c2 = 3.0/7.0; + const double c3 = 3.0/9.0; + const double c4 = 3.0/11.0; + const double c5 = 3.0/13.0; + const double c6 = 3.0/15.0; + const double c7 = 3.0/17.0; + const double c8 = 3.0/19.0; + const double sum = 1.0 + y*(c1 + y*(c2 + y*(c3 + y*(c4 + y*(c5 + y*(c6 + y*(c7 + y*c8))))))); + result->val = sum / (3.0*x*x); + result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val); + return GSL_SUCCESS; + } + else { + UNDERFLOW_ERROR(result); + } +} + + +int +gsl_sf_legendre_Ql_e(const int l, const double x, gsl_sf_result * result) +{ + /* CHECK_POINTER(result) */ + + if(x <= -1.0 || x == 1.0 || l < 0) { + DOMAIN_ERROR(result); + } + else if(l == 0) { + return gsl_sf_legendre_Q0_e(x, result); + } + else if(l == 1) { + return gsl_sf_legendre_Q1_e(x, result); + } + else if(l > 100000) { + return legendre_Ql_asymp_unif(l, x, result); + } + else if(x < 1.0){ + /* Forward recurrence. + */ + gsl_sf_result Q0, Q1; + int stat_Q0 = gsl_sf_legendre_Q0_e(x, &Q0); + int stat_Q1 = gsl_sf_legendre_Q1_e(x, &Q1); + double Qellm1 = Q0.val; + double Qell = Q1.val; + double Qellp1; + int ell; + for(ell=1; ell<l; ell++) { + Qellp1 = (x*(2.0*ell + 1.0) * Qell - ell * Qellm1) / (ell + 1.0); + Qellm1 = Qell; + Qell = Qellp1; + } + result->val = Qell; + result->err = GSL_DBL_EPSILON * l * fabs(result->val); + return GSL_ERROR_SELECT_2(stat_Q0, stat_Q1); + } + else { + /* x > 1.0 */ + + double rat; + int stat_CF1 = legendreQ_CF1_xgt1(l, 0.0, 0.0, x, &rat); + int stat_Q; + double Qellp1 = rat * GSL_SQRT_DBL_MIN; + double Qell = GSL_SQRT_DBL_MIN; + double Qellm1; + int ell; + for(ell=l; ell>0; ell--) { + Qellm1 = (x * (2.0*ell + 1.0) * Qell - (ell+1.0) * Qellp1) / ell; + Qellp1 = Qell; + Qell = Qellm1; + } + + if(fabs(Qell) > fabs(Qellp1)) { + gsl_sf_result Q0; + stat_Q = gsl_sf_legendre_Q0_e(x, &Q0); + result->val = GSL_SQRT_DBL_MIN * Q0.val / Qell; + result->err = l * GSL_DBL_EPSILON * fabs(result->val); + } + else { + gsl_sf_result Q1; + stat_Q = gsl_sf_legendre_Q1_e(x, &Q1); + result->val = GSL_SQRT_DBL_MIN * Q1.val / Qellp1; + result->err = l * GSL_DBL_EPSILON * fabs(result->val); + } + + return GSL_ERROR_SELECT_2(stat_Q, stat_CF1); + } +} + + +/*-*-*-*-*-*-*-*-*-* Functions w/ Natural Prototypes *-*-*-*-*-*-*-*-*-*-*/ + +#include "eval.h" + +double gsl_sf_legendre_Q0(const double x) +{ + EVAL_RESULT(gsl_sf_legendre_Q0_e(x, &result)); +} + +double gsl_sf_legendre_Q1(const double x) +{ + EVAL_RESULT(gsl_sf_legendre_Q1_e(x, &result)); +} + +double gsl_sf_legendre_Ql(const int l, const double x) +{ + EVAL_RESULT(gsl_sf_legendre_Ql_e(l, x, &result)); +} |