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- /* specfunc/bessel_Kn.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 3 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 "gsl__config.h"
- #include "gsl_math.h"
- #include "gsl_errno.h"
- #include "gsl_sf_gamma.h"
- #include "gsl_sf_psi.h"
- #include "gsl_sf_bessel.h"
- #include "gsl_specfunc__error.h"
- #include "gsl_specfunc__bessel.h"
- /*-*-*-*-*-*-*-*-*-*-*-* Private Section *-*-*-*-*-*-*-*-*-*-*-*/
- /* [Abramowitz+Stegun, 9.6.11]
- * assumes n >= 1
- */
- static
- int
- bessel_Kn_scaled_small_x(const int n, const double x, gsl_sf_result * result)
- {
- int k;
- double y = 0.25 * x * x;
- double ln_x_2 = log(0.5*x);
- double ex = exp(x);
- gsl_sf_result ln_nm1_fact;
- double k_term;
- double term1, sum1, ln_pre1;
- double term2, sum2, pre2;
- gsl_sf_lnfact_e((unsigned int)(n-1), &ln_nm1_fact);
- ln_pre1 = -n*ln_x_2 + ln_nm1_fact.val;
- if(ln_pre1 > GSL_LOG_DBL_MAX - 3.0) GSL_ERROR ("error", GSL_EOVRFLW);
- sum1 = 1.0;
- k_term = 1.0;
- for(k=1; k<=n-1; k++) {
- k_term *= -y/(k * (n-k));
- sum1 += k_term;
- }
- term1 = 0.5 * exp(ln_pre1) * sum1;
- pre2 = 0.5 * exp(n*ln_x_2);
- if(pre2 > 0.0) {
- const int KMAX = 20;
- gsl_sf_result psi_n;
- gsl_sf_result npk_fact;
- double yk = 1.0;
- double k_fact = 1.0;
- double psi_kp1 = -M_EULER;
- double psi_npkp1;
- gsl_sf_psi_int_e(n, &psi_n);
- gsl_sf_fact_e((unsigned int)n, &npk_fact);
- psi_npkp1 = psi_n.val + 1.0/n;
- sum2 = (psi_kp1 + psi_npkp1 - 2.0*ln_x_2)/npk_fact.val;
- for(k=1; k<KMAX; k++) {
- psi_kp1 += 1.0/k;
- psi_npkp1 += 1.0/(n+k);
- k_fact *= k;
- npk_fact.val *= n+k;
- yk *= y;
- k_term = yk*(psi_kp1 + psi_npkp1 - 2.0*ln_x_2)/(k_fact*npk_fact.val);
- sum2 += k_term;
- }
- term2 = ( GSL_IS_ODD(n) ? -1.0 : 1.0 ) * pre2 * sum2;
- }
- else {
- term2 = 0.0;
- }
- result->val = ex * (term1 + term2);
- result->err = ex * GSL_DBL_EPSILON * (fabs(ln_pre1)*fabs(term1) + fabs(term2));
- result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
- return GSL_SUCCESS;
- }
- /*-*-*-*-*-*-*-*-*-*-*-* Functions with Error Codes *-*-*-*-*-*-*-*-*-*-*-*/
- int gsl_sf_bessel_Kn_scaled_e(int n, const double x, gsl_sf_result * result)
- {
- n = abs(n); /* K(-n, z) = K(n, z) */
- /* CHECK_POINTER(result) */
- if(x <= 0.0) {
- DOMAIN_ERROR(result);
- }
- else if(n == 0) {
- return gsl_sf_bessel_K0_scaled_e(x, result);
- }
- else if(n == 1) {
- return gsl_sf_bessel_K1_scaled_e(x, result);
- }
- else if(x <= 5.0) {
- return bessel_Kn_scaled_small_x(n, x, result);
- }
- else if(GSL_ROOT3_DBL_EPSILON * x > 0.25 * (n*n + 1)) {
- return gsl_sf_bessel_Knu_scaled_asympx_e((double)n, x, result);
- }
- else if(GSL_MIN(0.29/(n*n), 0.5/(n*n + x*x)) < GSL_ROOT3_DBL_EPSILON) {
- return gsl_sf_bessel_Knu_scaled_asymp_unif_e((double)n, x, result);
- }
- else {
- /* Upward recurrence. [Gradshteyn + Ryzhik, 8.471.1] */
- double two_over_x = 2.0/x;
- gsl_sf_result r_b_jm1;
- gsl_sf_result r_b_j;
- int stat_0 = gsl_sf_bessel_K0_scaled_e(x, &r_b_jm1);
- int stat_1 = gsl_sf_bessel_K1_scaled_e(x, &r_b_j);
- double b_jm1 = r_b_jm1.val;
- double b_j = r_b_j.val;
- double b_jp1;
- int j;
- for(j=1; j<n; j++) {
- b_jp1 = b_jm1 + j * two_over_x * b_j;
- b_jm1 = b_j;
- b_j = b_jp1;
- }
-
- result->val = b_j;
- result->err = n * (fabs(b_j) * (fabs(r_b_jm1.err/r_b_jm1.val) + fabs(r_b_j.err/r_b_j.val)));
- result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
- return GSL_ERROR_SELECT_2(stat_0, stat_1);
- }
- }
- int gsl_sf_bessel_Kn_e(const int n, const double x, gsl_sf_result * result)
- {
- const int status = gsl_sf_bessel_Kn_scaled_e(n, x, result);
- const double ex = exp(-x);
- result->val *= ex;
- result->err *= ex;
- result->err += x * GSL_DBL_EPSILON * fabs(result->val);
- return status;
- }
- int gsl_sf_bessel_Kn_scaled_array(const int nmin, const int nmax, const double x, double * result_array)
- {
- /* CHECK_POINTER(result_array) */
- if(nmin < 0 || nmax < nmin || x <= 0.0) {
- int j;
- for(j=0; j<=nmax-nmin; j++) result_array[j] = 0.0;
- GSL_ERROR ("domain error", GSL_EDOM);
- }
- else if(nmax == 0) {
- gsl_sf_result b;
- int stat = gsl_sf_bessel_K0_scaled_e(x, &b);
- result_array[0] = b.val;
- return stat;
- }
- else {
- double two_over_x = 2.0/x;
- gsl_sf_result r_Knm1;
- gsl_sf_result r_Kn;
- int stat_0 = gsl_sf_bessel_Kn_scaled_e(nmin, x, &r_Knm1);
- int stat_1 = gsl_sf_bessel_Kn_scaled_e(nmin+1, x, &r_Kn);
- int stat = GSL_ERROR_SELECT_2(stat_0, stat_1);
- double Knp1;
- double Kn = r_Kn.val;
- double Knm1 = r_Knm1.val;
- int n;
- for(n=nmin+1; n<=nmax+1; n++) {
- if(Knm1 < GSL_DBL_MAX) {
- result_array[n-1-nmin] = Knm1;
- Knp1 = Knm1 + n * two_over_x * Kn;
- Knm1 = Kn;
- Kn = Knp1;
- }
- else {
- /* Overflow. Set the rest of the elements to
- * zero and bug out.
- * FIXME: Note: this relies on the convention
- * that the test x < DBL_MIN fails for x not
- * a number. This may be only an IEEE convention,
- * so the portability is unclear.
- */
- int j;
- for(j=n; j<=nmax+1; j++) result_array[j-1-nmin] = 0.0;
- GSL_ERROR ("overflow", GSL_EOVRFLW);
- }
- }
- return stat;
- }
- }
- int
- gsl_sf_bessel_Kn_array(const int nmin, const int nmax, const double x, double * result_array)
- {
- int status = gsl_sf_bessel_Kn_scaled_array(nmin, nmax, x, result_array);
- double ex = exp(-x);
- int i;
- for(i=0; i<=nmax-nmin; i++) result_array[i] *= ex;
- return status;
- }
- /*-*-*-*-*-*-*-*-*-* Functions w/ Natural Prototypes *-*-*-*-*-*-*-*-*-*-*/
- #include "gsl_specfunc__eval.h"
- double gsl_sf_bessel_Kn_scaled(const int n, const double x)
- {
- EVAL_RESULT(gsl_sf_bessel_Kn_scaled_e(n, x, &result));
- }
- double gsl_sf_bessel_Kn(const int n, const double x)
- {
- EVAL_RESULT(gsl_sf_bessel_Kn_e(n, x, &result));
- }
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