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- /* specfunc/bessel_Jnu.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_bessel.h"
- #include "gsl_specfunc__error.h"
- #include "gsl_specfunc__bessel.h"
- #include "gsl_specfunc__bessel_olver.h"
- #include "gsl_specfunc__bessel_temme.h"
- /* Evaluate at large enough nu to apply asymptotic
- * results and apply backward recurrence.
- */
- #if 0
- static
- int
- bessel_J_recur_asymp(const double nu, const double x,
- gsl_sf_result * Jnu, gsl_sf_result * Jnup1)
- {
- const double nu_cut = 25.0;
- int n;
- int steps = ceil(nu_cut - nu) + 1;
- gsl_sf_result r_Jnp1;
- gsl_sf_result r_Jn;
- int stat_O1 = gsl_sf_bessel_Jnu_asymp_Olver_e(nu + steps + 1.0, x, &r_Jnp1);
- int stat_O2 = gsl_sf_bessel_Jnu_asymp_Olver_e(nu + steps, x, &r_Jn);
- double r_fe = fabs(r_Jnp1.err/r_Jnp1.val) + fabs(r_Jn.err/r_Jn.val);
- double Jnp1 = r_Jnp1.val;
- double Jn = r_Jn.val;
- double Jnm1;
- double Jnp1_save;
- for(n=steps; n>0; n--) {
- Jnm1 = 2.0*(nu+n)/x * Jn - Jnp1;
- Jnp1 = Jn;
- Jnp1_save = Jn;
- Jn = Jnm1;
- }
- Jnu->val = Jn;
- Jnu->err = (r_fe + GSL_DBL_EPSILON * (steps + 1.0)) * fabs(Jn);
- Jnup1->val = Jnp1_save;
- Jnup1->err = (r_fe + GSL_DBL_EPSILON * (steps + 1.0)) * fabs(Jnp1_save);
- return GSL_ERROR_SELECT_2(stat_O1, stat_O2);
- }
- #endif
- /*-*-*-*-*-*-*-*-*-*-*-* Functions with Error Codes *-*-*-*-*-*-*-*-*-*-*-*/
- int
- gsl_sf_bessel_Jnu_e(const double nu, const double x, gsl_sf_result * result)
- {
- /* CHECK_POINTER(result) */
- if(x < 0.0 || nu < 0.0) {
- DOMAIN_ERROR(result);
- }
- else if(x == 0.0) {
- if(nu == 0.0) {
- result->val = 1.0;
- result->err = 0.0;
- }
- else {
- result->val = 0.0;
- result->err = 0.0;
- }
- return GSL_SUCCESS;
- }
- else if(x*x < 10.0*(nu+1.0)) {
- return gsl_sf_bessel_IJ_taylor_e(nu, x, -1, 100, GSL_DBL_EPSILON, result);
- }
- else if(nu > 50.0) {
- return gsl_sf_bessel_Jnu_asymp_Olver_e(nu, x, result);
- }
- else if(x > 1000.0)
- {
- /* We need this to avoid feeding large x to CF1; note that
- * due to the above check, we know that n <= 50. See similar
- * block in bessel_Jn.c.
- */
- return gsl_sf_bessel_Jnu_asympx_e(nu, x, result);
- }
- else {
- /* -1/2 <= mu <= 1/2 */
- int N = (int)(nu + 0.5);
- double mu = nu - N;
- /* Determine the J ratio at nu.
- */
- double Jnup1_Jnu;
- double sgn_Jnu;
- const int stat_CF1 = gsl_sf_bessel_J_CF1(nu, x, &Jnup1_Jnu, &sgn_Jnu);
- if(x < 2.0) {
- /* Determine Y_mu, Y_mup1 directly and recurse forward to nu.
- * Then use the CF1 information to solve for J_nu and J_nup1.
- */
- gsl_sf_result Y_mu, Y_mup1;
- const int stat_mu = gsl_sf_bessel_Y_temme(mu, x, &Y_mu, &Y_mup1);
-
- double Ynm1 = Y_mu.val;
- double Yn = Y_mup1.val;
- double Ynp1 = 0.0;
- int n;
- for(n=1; n<N; n++) {
- Ynp1 = 2.0*(mu+n)/x * Yn - Ynm1;
- Ynm1 = Yn;
- Yn = Ynp1;
- }
- result->val = 2.0/(M_PI*x) / (Jnup1_Jnu*Yn - Ynp1);
- result->err = GSL_DBL_EPSILON * (N + 2.0) * fabs(result->val);
- return GSL_ERROR_SELECT_2(stat_mu, stat_CF1);
- }
- else {
- /* Recurse backward from nu to mu, determining the J ratio
- * at mu. Use this together with a Steed method CF2 to
- * determine the actual J_mu, and thus obtain the normalization.
- */
- double Jmu;
- double Jmup1_Jmu;
- double sgn_Jmu;
- double Jmuprime_Jmu;
- double P, Q;
- const int stat_CF2 = gsl_sf_bessel_JY_steed_CF2(mu, x, &P, &Q);
- double gamma;
-
- double Jnp1 = sgn_Jnu * GSL_SQRT_DBL_MIN * Jnup1_Jnu;
- double Jn = sgn_Jnu * GSL_SQRT_DBL_MIN;
- double Jnm1;
- int n;
- for(n=N; n>0; n--) {
- Jnm1 = 2.0*(mu+n)/x * Jn - Jnp1;
- Jnp1 = Jn;
- Jn = Jnm1;
- }
- Jmup1_Jmu = Jnp1/Jn;
- sgn_Jmu = GSL_SIGN(Jn);
- Jmuprime_Jmu = mu/x - Jmup1_Jmu;
- gamma = (P - Jmuprime_Jmu)/Q;
- Jmu = sgn_Jmu * sqrt(2.0/(M_PI*x) / (Q + gamma*(P-Jmuprime_Jmu)));
- result->val = Jmu * (sgn_Jnu * GSL_SQRT_DBL_MIN) / Jn;
- result->err = 2.0 * GSL_DBL_EPSILON * (N + 2.0) * fabs(result->val);
- return GSL_ERROR_SELECT_2(stat_CF2, stat_CF1);
- }
- }
- }
- /*-*-*-*-*-*-*-*-*-* Functions w/ Natural Prototypes *-*-*-*-*-*-*-*-*-*-*/
- #include "gsl_specfunc__eval.h"
- double gsl_sf_bessel_Jnu(const double nu, const double x)
- {
- EVAL_RESULT(gsl_sf_bessel_Jnu_e(nu, x, &result));
- }
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