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- /* specfunc/bessel_i.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_pow_int.h"
- #include "gsl_sf_bessel.h"
- #include "gsl_specfunc__error.h"
- #include "gsl_specfunc__bessel.h"
- /* i_{l+1}/i_l
- */
- static
- int
- bessel_il_CF1(const int l, const double x, const double threshold, double * ratio)
- {
- const int kmax = 2000;
- double tk = 1.0;
- double sum = 1.0;
- double rhok = 0.0;
- int k;
- for(k=1; k<=kmax; k++) {
- double ak = (x/(2.0*l+1.0+2.0*k)) * (x/(2.0*l+3.0+2.0*k));
- rhok = -ak*(1.0 + rhok)/(1.0 + ak*(1.0 + rhok));
- tk *= rhok;
- sum += tk;
- if(fabs(tk/sum) < threshold) break;
- }
- *ratio = x/(2.0*l+3.0) * sum;
- if(k == kmax)
- GSL_ERROR ("error", GSL_EMAXITER);
- else
- return GSL_SUCCESS;
- }
- /*-*-*-*-*-*-*-*-*-*-*-* Functions with Error Codes *-*-*-*-*-*-*-*-*-*-*-*/
- int gsl_sf_bessel_i0_scaled_e(const double x, gsl_sf_result * result)
- {
- double ax = fabs(x);
- /* CHECK_POINTER(result) */
- if(x == 0.0) {
- result->val = 1.0;
- result->err = 0.0;
- return GSL_SUCCESS;
- }
- else if(ax < 0.2) {
- const double eax = exp(-ax);
- const double y = ax*ax;
- const double c1 = 1.0/6.0;
- const double c2 = 1.0/120.0;
- const double c3 = 1.0/5040.0;
- const double c4 = 1.0/362880.0;
- const double c5 = 1.0/39916800.0;
- const double sum = 1.0 + y*(c1 + y*(c2 + y*(c3 + y*(c4 + y*c5))));
- result->val = eax * sum;
- result->err = 2.0 * GSL_DBL_EPSILON * result->val;
- }
- else if(ax < -0.5*GSL_LOG_DBL_EPSILON) {
- result->val = (1.0 - exp(-2.0*ax))/(2.0*ax);
- result->err = 2.0 * GSL_DBL_EPSILON * result->val;
- }
- else {
- result->val = 1.0/(2.0*ax);
- result->err = 2.0 * GSL_DBL_EPSILON * result->val;
- }
- return GSL_SUCCESS;
- }
- int gsl_sf_bessel_i1_scaled_e(const double x, gsl_sf_result * result)
- {
- double ax = fabs(x);
- /* CHECK_POINTER(result) */
- if(x == 0.0) {
- result->val = 0.0;
- result->err = 0.0;
- return GSL_SUCCESS;
- }
- else if(ax < 3.0*GSL_DBL_MIN) {
- UNDERFLOW_ERROR(result);
- }
- else if(ax < 0.25) {
- const double eax = exp(-ax);
- const double y = x*x;
- const double c1 = 1.0/10.0;
- const double c2 = 1.0/280.0;
- const double c3 = 1.0/15120.0;
- const double c4 = 1.0/1330560.0;
- const double c5 = 1.0/172972800.0;
- const double sum = 1.0 + y*(c1 + y*(c2 + y*(c3 + y*(c4 + y*c5))));
- result->val = eax * x/3.0 * sum;
- result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
- return GSL_SUCCESS;
- }
- else {
- double ex = exp(-2.0*ax);
- result->val = 0.5 * (ax*(1.0+ex) - (1.0-ex)) / (ax*ax);
- result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
- if(x < 0.0) result->val = -result->val;
- return GSL_SUCCESS;
- }
- }
- int gsl_sf_bessel_i2_scaled_e(const double x, gsl_sf_result * result)
- {
- double ax = fabs(x);
- /* CHECK_POINTER(result) */
- if(x == 0.0) {
- result->val = 0.0;
- result->err = 0.0;
- return GSL_SUCCESS;
- }
- else if(ax < 4.0*GSL_SQRT_DBL_MIN) {
- UNDERFLOW_ERROR(result);
- }
- else if(ax < 0.25) {
- const double y = x*x;
- const double c1 = 1.0/14.0;
- const double c2 = 1.0/504.0;
- const double c3 = 1.0/33264.0;
- const double c4 = 1.0/3459456.0;
- const double c5 = 1.0/518918400.0;
- const double sum = 1.0 + y*(c1 + y*(c2 + y*(c3 + y*(c4 + y*c5))));
- const double pre = exp(-ax) * x*x/15.0;
- result->val = pre * sum;
- result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
- return GSL_SUCCESS;
- }
- else {
- double ex = exp(-2.0*ax);
- double x2 = x*x;
- result->val = 0.5 * ((3.0+x2)*(1.0-ex) - 3.0*ax*(1.0+ex))/(ax*ax*ax);
- result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
- return GSL_SUCCESS;
- }
- }
- int gsl_sf_bessel_il_scaled_e(const int l, double x, gsl_sf_result * result)
- {
- double sgn = 1.0;
- double ax = fabs(x);
- if(x < 0.0) {
- /* i_l(-x) = (-1)^l i_l(x) */
- sgn = ( GSL_IS_ODD(l) ? -1.0 : 1.0 );
- x = -x;
- }
- if(l < 0) {
- DOMAIN_ERROR(result);
- }
- else if(x == 0.0) {
- result->val = ( l == 0 ? 1.0 : 0.0 );
- result->err = 0.0;
- return GSL_SUCCESS;
- }
- else if(l == 0) {
- gsl_sf_result il;
- int stat_il = gsl_sf_bessel_i0_scaled_e(x, &il);
- result->val = sgn * il.val;
- result->err = il.err;
- return stat_il;
- }
- else if(l == 1) {
- gsl_sf_result il;
- int stat_il = gsl_sf_bessel_i1_scaled_e(x, &il);
- result->val = sgn * il.val;
- result->err = il.err;
- return stat_il;
- }
- else if(l == 2) {
- gsl_sf_result il;
- int stat_il = gsl_sf_bessel_i2_scaled_e(x, &il);
- result->val = sgn * il.val;
- result->err = il.err;
- return stat_il;
- }
- else if(x*x < 10.0*(l+1.5)/M_E) {
- gsl_sf_result b;
- int stat = gsl_sf_bessel_IJ_taylor_e(l+0.5, x, 1, 50, GSL_DBL_EPSILON, &b);
- double pre = exp(-ax) * sqrt((0.5*M_PI)/x);
- result->val = sgn * pre * b.val;
- result->err = pre * b.err;
- result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
- return stat;
- }
- else if(l < 150) {
- gsl_sf_result i0_scaled;
- int stat_i0 = gsl_sf_bessel_i0_scaled_e(ax, &i0_scaled);
- double rat;
- int stat_CF1 = bessel_il_CF1(l, ax, GSL_DBL_EPSILON, &rat);
- double iellp1 = rat * GSL_SQRT_DBL_MIN;
- double iell = GSL_SQRT_DBL_MIN;
- double iellm1;
- int ell;
- for(ell = l; ell >= 1; ell--) {
- iellm1 = iellp1 + (2*ell + 1)/x * iell;
- iellp1 = iell;
- iell = iellm1;
- }
- result->val = sgn * i0_scaled.val * (GSL_SQRT_DBL_MIN / iell);
- result->err = i0_scaled.err * (GSL_SQRT_DBL_MIN / iell);
- result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
- return GSL_ERROR_SELECT_2(stat_i0, stat_CF1);
- }
- else if(GSL_MIN(0.29/(l*l+1.0), 0.5/(l*l+1.0+x*x)) < 0.5*GSL_ROOT3_DBL_EPSILON) {
- int status = gsl_sf_bessel_Inu_scaled_asymp_unif_e(l + 0.5, x, result);
- double pre = sqrt((0.5*M_PI)/x);
- result->val *= sgn * pre;
- result->err *= pre;
- return status;
- }
- else {
- /* recurse down from safe values */
- double rt_term = sqrt((0.5*M_PI)/x);
- const int LMAX = 2 + (int) (1.2 / GSL_ROOT6_DBL_EPSILON);
- gsl_sf_result r_iellp1;
- gsl_sf_result r_iell;
- int stat_a1 = gsl_sf_bessel_Inu_scaled_asymp_unif_e(LMAX + 1 + 0.5, x, &r_iellp1);
- int stat_a2 = gsl_sf_bessel_Inu_scaled_asymp_unif_e(LMAX + 0.5, x, &r_iell);
- double iellp1 = r_iellp1.val;
- double iell = r_iell.val;
- double iellm1 = 0.0;
- int ell;
- iellp1 *= rt_term;
- iell *= rt_term;
- for(ell = LMAX; ell >= l+1; ell--) {
- iellm1 = iellp1 + (2*ell + 1)/x * iell;
- iellp1 = iell;
- iell = iellm1;
- }
- result->val = sgn * iellm1;
- result->err = fabs(result->val)*(GSL_DBL_EPSILON + fabs(r_iellp1.err/r_iellp1.val) + fabs(r_iell.err/r_iell.val));
- result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
- return GSL_ERROR_SELECT_2(stat_a1, stat_a2);
- }
- }
- int gsl_sf_bessel_il_scaled_array(const int lmax, const double x, double * result_array)
- {
- if(x == 0.0) {
- int ell;
- result_array[0] = 1.0;
- for (ell = lmax; ell >= 1; ell--) {
- result_array[ell] = 0.0;
- };
- return GSL_SUCCESS;
- } else {
- int ell;
- gsl_sf_result r_iellp1;
- gsl_sf_result r_iell;
- int stat_0 = gsl_sf_bessel_il_scaled_e(lmax+1, x, &r_iellp1);
- int stat_1 = gsl_sf_bessel_il_scaled_e(lmax, x, &r_iell);
- double iellp1 = r_iellp1.val;
- double iell = r_iell.val;
- double iellm1;
- result_array[lmax] = iell;
- for(ell = lmax; ell >= 1; ell--) {
- iellm1 = iellp1 + (2*ell + 1)/x * iell;
- iellp1 = iell;
- iell = iellm1;
- result_array[ell-1] = iellm1;
- }
- return GSL_ERROR_SELECT_2(stat_0, stat_1);
- }
- }
- /*-*-*-*-*-*-*-*-*-* Functions w/ Natural Prototypes *-*-*-*-*-*-*-*-*-*-*/
- #include "gsl_specfunc__eval.h"
- double gsl_sf_bessel_i0_scaled(const double x)
- {
- EVAL_RESULT(gsl_sf_bessel_i0_scaled_e(x, &result));
- }
- double gsl_sf_bessel_i1_scaled(const double x)
- {
- EVAL_RESULT(gsl_sf_bessel_i1_scaled_e(x, &result));
- }
- double gsl_sf_bessel_i2_scaled(const double x)
- {
- EVAL_RESULT(gsl_sf_bessel_i2_scaled_e(x, &result));
- }
- double gsl_sf_bessel_il_scaled(const int l, const double x)
- {
- EVAL_RESULT(gsl_sf_bessel_il_scaled_e(l, x, &result));
- }
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