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+/* specfunc/bessel.c
+ *
+ * Copyright (C) 1996,1997,1998,1999,2000,2001,2002,2003 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 */
+/* Miscellaneous support functions for Bessel function evaluations.
+ */
+#include <config.h>
+#include <gsl/gsl_math.h>
+#include <gsl/gsl_errno.h>
+#include <gsl/gsl_sf_airy.h>
+#include <gsl/gsl_sf_elementary.h>
+#include <gsl/gsl_sf_exp.h>
+#include <gsl/gsl_sf_gamma.h>
+#include <gsl/gsl_sf_trig.h>
+
+#include "error.h"
+
+#include "bessel_amp_phase.h"
+#include "bessel_temme.h"
+#include "bessel.h"
+
+#define CubeRoot2_ 1.25992104989487316476721060728
+
+
+
+/* Debye functions [Abramowitz+Stegun, 9.3.9-10] */
+
+inline static double
+debye_u1(const double * tpow)
+{
+ return (3.0*tpow[1] - 5.0*tpow[3])/24.0;
+}
+
+inline static double
+debye_u2(const double * tpow)
+{
+ return (81.0*tpow[2] - 462.0*tpow[4] + 385.0*tpow[6])/1152.0;
+}
+
+inline
+static double debye_u3(const double * tpow)
+{
+ return (30375.0*tpow[3] - 369603.0*tpow[5] + 765765.0*tpow[7] - 425425.0*tpow[9])/414720.0;
+}
+
+inline
+static double debye_u4(const double * tpow)
+{
+ return (4465125.0*tpow[4] - 94121676.0*tpow[6] + 349922430.0*tpow[8] -
+ 446185740.0*tpow[10] + 185910725.0*tpow[12])/39813120.0;
+}
+
+inline
+static double debye_u5(const double * tpow)
+{
+ return (1519035525.0*tpow[5] - 49286948607.0*tpow[7] +
+ 284499769554.0*tpow[9] - 614135872350.0*tpow[11] +
+ 566098157625.0*tpow[13] - 188699385875.0*tpow[15])/6688604160.0;
+}
+
+#if 0
+inline
+static double debye_u6(const double * tpow)
+{
+ return (2757049477875.0*tpow[6] - 127577298354750.0*tpow[8] +
+ 1050760774457901.0*tpow[10] - 3369032068261860.0*tpow[12] +
+ 5104696716244125.0*tpow[14] - 3685299006138750.0*tpow[16] +
+ 1023694168371875.0*tpow[18])/4815794995200.0;
+}
+#endif
+
+
+/*-*-*-*-*-*-*-*-*-*-*-* Functions with Error Codes *-*-*-*-*-*-*-*-*-*-*-*/
+
+int
+gsl_sf_bessel_IJ_taylor_e(const double nu, const double x,
+ const int sign,
+ const int kmax,
+ const double threshold,
+ gsl_sf_result * result
+ )
+{
+ /* CHECK_POINTER(result) */
+
+ if(nu < 0.0 || x < 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 {
+ gsl_sf_result prefactor; /* (x/2)^nu / Gamma(nu+1) */
+ gsl_sf_result sum;
+
+ int stat_pre;
+ int stat_sum;
+ int stat_mul;
+
+ if(nu == 0.0) {
+ prefactor.val = 1.0;
+ prefactor.err = 0.0;
+ stat_pre = GSL_SUCCESS;
+ }
+ else if(nu < INT_MAX-1) {
+ /* Separate the integer part and use
+ * y^nu / Gamma(nu+1) = y^N /N! y^f / (N+1)_f,
+ * to control the error.
+ */
+ const int N = (int)floor(nu + 0.5);
+ const double f = nu - N;
+ gsl_sf_result poch_factor;
+ gsl_sf_result tc_factor;
+ const int stat_poch = gsl_sf_poch_e(N+1.0, f, &poch_factor);
+ const int stat_tc = gsl_sf_taylorcoeff_e(N, 0.5*x, &tc_factor);
+ const double p = pow(0.5*x,f);
+ prefactor.val = tc_factor.val * p / poch_factor.val;
+ prefactor.err = tc_factor.err * p / poch_factor.val;
+ prefactor.err += fabs(prefactor.val) / poch_factor.val * poch_factor.err;
+ prefactor.err += 2.0 * GSL_DBL_EPSILON * fabs(prefactor.val);
+ stat_pre = GSL_ERROR_SELECT_2(stat_tc, stat_poch);
+ }
+ else {
+ gsl_sf_result lg;
+ const int stat_lg = gsl_sf_lngamma_e(nu+1.0, &lg);
+ const double term1 = nu*log(0.5*x);
+ const double term2 = lg.val;
+ const double ln_pre = term1 - term2;
+ const double ln_pre_err = GSL_DBL_EPSILON * (fabs(term1)+fabs(term2)) + lg.err;
+ const int stat_ex = gsl_sf_exp_err_e(ln_pre, ln_pre_err, &prefactor);
+ stat_pre = GSL_ERROR_SELECT_2(stat_ex, stat_lg);
+ }
+
+ /* Evaluate the sum.
+ * [Abramowitz+Stegun, 9.1.10]
+ * [Abramowitz+Stegun, 9.6.7]
+ */
+ {
+ const double y = sign * 0.25 * x*x;
+ double sumk = 1.0;
+ double term = 1.0;
+ int k;
+
+ for(k=1; k<=kmax; k++) {
+ term *= y/((nu+k)*k);
+ sumk += term;
+ if(fabs(term/sumk) < threshold) break;
+ }
+
+ sum.val = sumk;
+ sum.err = threshold * fabs(sumk);
+
+ stat_sum = ( k >= kmax ? GSL_EMAXITER : GSL_SUCCESS );
+ }
+
+ stat_mul = gsl_sf_multiply_err_e(prefactor.val, prefactor.err,
+ sum.val, sum.err,
+ result);
+
+ return GSL_ERROR_SELECT_3(stat_mul, stat_pre, stat_sum);
+ }
+}
+
+
+/* x >> nu*nu+1
+ * error ~ O( ((nu*nu+1)/x)^4 )
+ *
+ * empirical error analysis:
+ * choose GSL_ROOT4_MACH_EPS * x > (nu*nu + 1)
+ *
+ * This is not especially useful. When the argument gets
+ * large enough for this to apply, the cos() and sin()
+ * start loosing digits. However, this seems inevitable
+ * for this particular method.
+ *
+ * Wed Jun 25 14:39:38 MDT 2003 [GJ]
+ * This function was inconsistent since the Q term did not
+ * go to relative order eps^2. That's why the error estimate
+ * originally given was screwy (it didn't make sense that the
+ * "empirical" error was coming out O(eps^3)).
+ * With Q to proper order, the error is O(eps^4).
+ */
+int
+gsl_sf_bessel_Jnu_asympx_e(const double nu, const double x, gsl_sf_result * result)
+{
+ double mu = 4.0*nu*nu;
+ double mum1 = mu-1.0;
+ double mum9 = mu-9.0;
+ double mum25 = mu-25.0;
+ double chi = x - (0.5*nu + 0.25)*M_PI;
+ double P = 1.0 - mum1*mum9/(128.0*x*x);
+ double Q = mum1/(8.0*x) * (1.0 - mum9*mum25/(384.0*x*x));
+ double pre = sqrt(2.0/(M_PI*x));
+ double c = cos(chi);
+ double s = sin(chi);
+ double r = mu/x;
+ result->val = pre * (c*P - s*Q);
+ result->err = pre * GSL_DBL_EPSILON * (1.0 + fabs(x)) * (fabs(c*P) + fabs(s*Q));
+ result->err += pre * fabs(0.1*r*r*r*r);
+ return GSL_SUCCESS;
+}
+
+
+/* x >> nu*nu+1
+ */
+int
+gsl_sf_bessel_Ynu_asympx_e(const double nu, const double x, gsl_sf_result * result)
+{
+ double ampl;
+ double theta;
+ double alpha = x;
+ double beta = -0.5*nu*M_PI;
+ int stat_a = gsl_sf_bessel_asymp_Mnu_e(nu, x, &ampl);
+ int stat_t = gsl_sf_bessel_asymp_thetanu_corr_e(nu, x, &theta);
+ double sin_alpha = sin(alpha);
+ double cos_alpha = cos(alpha);
+ double sin_chi = sin(beta + theta);
+ double cos_chi = cos(beta + theta);
+ double sin_term = sin_alpha * cos_chi + sin_chi * cos_alpha;
+ double sin_term_mag = fabs(sin_alpha * cos_chi) + fabs(sin_chi * cos_alpha);
+ result->val = ampl * sin_term;
+ result->err = fabs(ampl) * GSL_DBL_EPSILON * sin_term_mag;
+ result->err += fabs(result->val) * 2.0 * GSL_DBL_EPSILON;
+
+ if(fabs(alpha) > 1.0/GSL_DBL_EPSILON) {
+ result->err *= 0.5 * fabs(alpha);
+ }
+ else if(fabs(alpha) > 1.0/GSL_SQRT_DBL_EPSILON) {
+ result->err *= 256.0 * fabs(alpha) * GSL_SQRT_DBL_EPSILON;
+ }
+
+ return GSL_ERROR_SELECT_2(stat_t, stat_a);
+}
+
+
+/* x >> nu*nu+1
+ */
+int
+gsl_sf_bessel_Inu_scaled_asympx_e(const double nu, const double x, gsl_sf_result * result)
+{
+ double mu = 4.0*nu*nu;
+ double mum1 = mu-1.0;
+ double mum9 = mu-9.0;
+ double pre = 1.0/sqrt(2.0*M_PI*x);
+ double r = mu/x;
+ result->val = pre * (1.0 - mum1/(8.0*x) + mum1*mum9/(128.0*x*x));
+ result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val) + pre * fabs(0.1*r*r*r);
+ return GSL_SUCCESS;
+}
+
+/* x >> nu*nu+1
+ */
+int
+gsl_sf_bessel_Knu_scaled_asympx_e(const double nu, const double x, gsl_sf_result * result)
+{
+ double mu = 4.0*nu*nu;
+ double mum1 = mu-1.0;
+ double mum9 = mu-9.0;
+ double pre = sqrt(M_PI/(2.0*x));
+ double r = nu/x;
+ result->val = pre * (1.0 + mum1/(8.0*x) + mum1*mum9/(128.0*x*x));
+ result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val) + pre * fabs(0.1*r*r*r);
+ return GSL_SUCCESS;
+}
+
+
+/* nu -> Inf; uniform in x > 0 [Abramowitz+Stegun, 9.7.7]
+ *
+ * error:
+ * The error has the form u_N(t)/nu^N where 0 <= t <= 1.
+ * It is not hard to show that |u_N(t)| is small for such t.
+ * We have N=6 here, and |u_6(t)| < 0.025, so the error is clearly
+ * bounded by 0.025/nu^6. This gives the asymptotic bound on nu
+ * seen below as nu ~ 100. For general MACH_EPS it will be
+ * nu > 0.5 / MACH_EPS^(1/6)
+ * When t is small, the bound is even better because |u_N(t)| vanishes
+ * as t->0. In fact u_N(t) ~ C t^N as t->0, with C ~= 0.1.
+ * We write
+ * err_N <= min(0.025, C(1/(1+(x/nu)^2))^3) / nu^6
+ * therefore
+ * min(0.29/nu^2, 0.5/(nu^2+x^2)) < MACH_EPS^{1/3}
+ * and this is the general form.
+ *
+ * empirical error analysis, assuming 14 digit requirement:
+ * choose x > 50.000 nu ==> nu > 3
+ * choose x > 10.000 nu ==> nu > 15
+ * choose x > 2.000 nu ==> nu > 50
+ * choose x > 1.000 nu ==> nu > 75
+ * choose x > 0.500 nu ==> nu > 80
+ * choose x > 0.100 nu ==> nu > 83
+ *
+ * This makes sense. For x << nu, the error will be of the form u_N(1)/nu^N,
+ * since the polynomial term will be evaluated near t=1, so the bound
+ * on nu will become constant for small x. Furthermore, increasing x with
+ * nu fixed will decrease the error.
+ */
+int
+gsl_sf_bessel_Inu_scaled_asymp_unif_e(const double nu, const double x, gsl_sf_result * result)
+{
+ int i;
+ double z = x/nu;
+ double root_term = hypot(1.0,z);
+ double pre = 1.0/sqrt(2.0*M_PI*nu * root_term);
+ double eta = root_term + log(z/(1.0+root_term));
+ double ex_arg = ( z < 1.0/GSL_ROOT3_DBL_EPSILON ? nu*(-z + eta) : -0.5*nu/z*(1.0 - 1.0/(12.0*z*z)) );
+ gsl_sf_result ex_result;
+ int stat_ex = gsl_sf_exp_e(ex_arg, &ex_result);
+ if(stat_ex == GSL_SUCCESS) {
+ double t = 1.0/root_term;
+ double sum;
+ double tpow[16];
+ tpow[0] = 1.0;
+ for(i=1; i<16; i++) tpow[i] = t * tpow[i-1];
+ sum = 1.0 + debye_u1(tpow)/nu + debye_u2(tpow)/(nu*nu) + debye_u3(tpow)/(nu*nu*nu)
+ + debye_u4(tpow)/(nu*nu*nu*nu) + debye_u5(tpow)/(nu*nu*nu*nu*nu);
+ result->val = pre * ex_result.val * sum;
+ result->err = pre * ex_result.val / (nu*nu*nu*nu*nu*nu);
+ result->err += pre * ex_result.err * fabs(sum);
+ result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
+ return GSL_SUCCESS;
+ }
+ else {
+ result->val = 0.0;
+ result->err = 0.0;
+ return stat_ex;
+ }
+}
+
+
+/* nu -> Inf; uniform in x > 0 [Abramowitz+Stegun, 9.7.8]
+ *
+ * error:
+ * identical to that above for Inu_scaled
+ */
+int
+gsl_sf_bessel_Knu_scaled_asymp_unif_e(const double nu, const double x, gsl_sf_result * result)
+{
+ int i;
+ double z = x/nu;
+ double root_term = hypot(1.0,z);
+ double pre = sqrt(M_PI/(2.0*nu*root_term));
+ double eta = root_term + log(z/(1.0+root_term));
+ double ex_arg = ( z < 1.0/GSL_ROOT3_DBL_EPSILON ? nu*(z - eta) : 0.5*nu/z*(1.0 + 1.0/(12.0*z*z)) );
+ gsl_sf_result ex_result;
+ int stat_ex = gsl_sf_exp_e(ex_arg, &ex_result);
+ if(stat_ex == GSL_SUCCESS) {
+ double t = 1.0/root_term;
+ double sum;
+ double tpow[16];
+ tpow[0] = 1.0;
+ for(i=1; i<16; i++) tpow[i] = t * tpow[i-1];
+ sum = 1.0 - debye_u1(tpow)/nu + debye_u2(tpow)/(nu*nu) - debye_u3(tpow)/(nu*nu*nu)
+ + debye_u4(tpow)/(nu*nu*nu*nu) - debye_u5(tpow)/(nu*nu*nu*nu*nu);
+ result->val = pre * ex_result.val * sum;
+ result->err = pre * ex_result.err * fabs(sum);
+ result->err += pre * ex_result.val / (nu*nu*nu*nu*nu*nu);
+ result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
+ return GSL_SUCCESS;
+ }
+ else {
+ result->val = 0.0;
+ result->err = 0.0;
+ return stat_ex;
+ }
+}
+
+
+/* Evaluate J_mu(x),J_{mu+1}(x) and Y_mu(x),Y_{mu+1}(x) for |mu| < 1/2
+ */
+int
+gsl_sf_bessel_JY_mu_restricted(const double mu, const double x,
+ gsl_sf_result * Jmu, gsl_sf_result * Jmup1,
+ gsl_sf_result * Ymu, gsl_sf_result * Ymup1)
+{
+ /* CHECK_POINTER(Jmu) */
+ /* CHECK_POINTER(Jmup1) */
+ /* CHECK_POINTER(Ymu) */
+ /* CHECK_POINTER(Ymup1) */
+
+ if(x < 0.0 || fabs(mu) > 0.5) {
+ Jmu->val = 0.0;
+ Jmu->err = 0.0;
+ Jmup1->val = 0.0;
+ Jmup1->err = 0.0;
+ Ymu->val = 0.0;
+ Ymu->err = 0.0;
+ Ymup1->val = 0.0;
+ Ymup1->err = 0.0;
+ GSL_ERROR ("error", GSL_EDOM);
+ }
+ else if(x == 0.0) {
+ if(mu == 0.0) {
+ Jmu->val = 1.0;
+ Jmu->err = 0.0;
+ }
+ else {
+ Jmu->val = 0.0;
+ Jmu->err = 0.0;
+ }
+ Jmup1->val = 0.0;
+ Jmup1->err = 0.0;
+ Ymu->val = 0.0;
+ Ymu->err = 0.0;
+ Ymup1->val = 0.0;
+ Ymup1->err = 0.0;
+ GSL_ERROR ("error", GSL_EDOM);
+ }
+ else {
+ int stat_Y;
+ int stat_J;
+
+ if(x < 2.0) {
+ /* Use Taylor series for J and the Temme series for Y.
+ * The Taylor series for J requires nu > 0, so we shift
+ * up one and use the recursion relation to get Jmu, in
+ * case mu < 0.
+ */
+ gsl_sf_result Jmup2;
+ int stat_J1 = gsl_sf_bessel_IJ_taylor_e(mu+1.0, x, -1, 100, GSL_DBL_EPSILON, Jmup1);
+ int stat_J2 = gsl_sf_bessel_IJ_taylor_e(mu+2.0, x, -1, 100, GSL_DBL_EPSILON, &Jmup2);
+ double c = 2.0*(mu+1.0)/x;
+ Jmu->val = c * Jmup1->val - Jmup2.val;
+ Jmu->err = c * Jmup1->err + Jmup2.err;
+ Jmu->err += 2.0 * GSL_DBL_EPSILON * fabs(Jmu->val);
+ stat_J = GSL_ERROR_SELECT_2(stat_J1, stat_J2);
+ stat_Y = gsl_sf_bessel_Y_temme(mu, x, Ymu, Ymup1);
+ return GSL_ERROR_SELECT_2(stat_J, stat_Y);
+ }
+ else if(x < 1000.0) {
+ double P, Q;
+ double J_ratio;
+ double J_sgn;
+ const int stat_CF1 = gsl_sf_bessel_J_CF1(mu, x, &J_ratio, &J_sgn);
+ const int stat_CF2 = gsl_sf_bessel_JY_steed_CF2(mu, x, &P, &Q);
+ double Jprime_J_ratio = mu/x - J_ratio;
+ double gamma = (P - Jprime_J_ratio)/Q;
+ Jmu->val = J_sgn * sqrt(2.0/(M_PI*x) / (Q + gamma*(P-Jprime_J_ratio)));
+ Jmu->err = 4.0 * GSL_DBL_EPSILON * fabs(Jmu->val);
+ Jmup1->val = J_ratio * Jmu->val;
+ Jmup1->err = fabs(J_ratio) * Jmu->err;
+ Ymu->val = gamma * Jmu->val;
+ Ymu->err = fabs(gamma) * Jmu->err;
+ Ymup1->val = Ymu->val * (mu/x - P - Q/gamma);
+ Ymup1->err = Ymu->err * fabs(mu/x - P - Q/gamma) + 4.0*GSL_DBL_EPSILON*fabs(Ymup1->val);
+ return GSL_ERROR_SELECT_2(stat_CF1, stat_CF2);
+ }
+ else {
+ /* Use asymptotics for large argument.
+ */
+ const int stat_J0 = gsl_sf_bessel_Jnu_asympx_e(mu, x, Jmu);
+ const int stat_J1 = gsl_sf_bessel_Jnu_asympx_e(mu+1.0, x, Jmup1);
+ const int stat_Y0 = gsl_sf_bessel_Ynu_asympx_e(mu, x, Ymu);
+ const int stat_Y1 = gsl_sf_bessel_Ynu_asympx_e(mu+1.0, x, Ymup1);
+ stat_J = GSL_ERROR_SELECT_2(stat_J0, stat_J1);
+ stat_Y = GSL_ERROR_SELECT_2(stat_Y0, stat_Y1);
+ return GSL_ERROR_SELECT_2(stat_J, stat_Y);
+ }
+ }
+}
+
+
+int
+gsl_sf_bessel_J_CF1(const double nu, const double x,
+ double * ratio, double * sgn)
+{
+ const double RECUR_BIG = GSL_SQRT_DBL_MAX;
+ const int maxiter = 10000;
+ int n = 1;
+ double Anm2 = 1.0;
+ double Bnm2 = 0.0;
+ double Anm1 = 0.0;
+ double Bnm1 = 1.0;
+ double a1 = x/(2.0*(nu+1.0));
+ double An = Anm1 + a1*Anm2;
+ double Bn = Bnm1 + a1*Bnm2;
+ double an;
+ double fn = An/Bn;
+ double dn = a1;
+ double s = 1.0;
+
+ while(n < maxiter) {
+ double old_fn;
+ double del;
+ n++;
+ Anm2 = Anm1;
+ Bnm2 = Bnm1;
+ Anm1 = An;
+ Bnm1 = Bn;
+ an = -x*x/(4.0*(nu+n-1.0)*(nu+n));
+ An = Anm1 + an*Anm2;
+ 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;
+
+ dn = 1.0 / (2.0*(nu+n)/x - dn);
+ if(dn < 0.0) s = -s;
+
+ if(fabs(del - 1.0) < 2.0*GSL_DBL_EPSILON) break;
+ }
+
+ *ratio = fn;
+ *sgn = s;
+
+ if(n >= maxiter)
+ GSL_ERROR ("error", GSL_EMAXITER);
+ else
+ return GSL_SUCCESS;
+}
+
+
+
+/* Evaluate the continued fraction CF1 for J_{nu+1}/J_nu
+ * using Gautschi (Euler) equivalent series.
+ * This exhibits an annoying problem because the
+ * a_k are not positive definite (in fact they are all negative).
+ * There are cases when rho_k blows up. Example: nu=1,x=4.
+ */
+#if 0
+int
+gsl_sf_bessel_J_CF1_ser(const double nu, const double x,
+ double * ratio, double * sgn)
+{
+ const int maxk = 20000;
+ double tk = 1.0;
+ double sum = 1.0;
+ double rhok = 0.0;
+ double dk = 0.0;
+ double s = 1.0;
+ int k;
+
+ for(k=1; k<maxk; k++) {
+ double ak = -0.25 * (x/(nu+k)) * x/(nu+k+1.0);
+ rhok = -ak*(1.0 + rhok)/(1.0 + ak*(1.0 + rhok));
+ tk *= rhok;
+ sum += tk;
+
+ dk = 1.0 / (2.0/x - (nu+k-1.0)/(nu+k) * dk);
+ if(dk < 0.0) s = -s;
+
+ if(fabs(tk/sum) < GSL_DBL_EPSILON) break;
+ }
+
+ *ratio = x/(2.0*(nu+1.0)) * sum;
+ *sgn = s;
+
+ if(k == maxk)
+ GSL_ERROR ("error", GSL_EMAXITER);
+ else
+ return GSL_SUCCESS;
+}
+#endif
+
+
+/* Evaluate the continued fraction CF1 for I_{nu+1}/I_nu
+ * using Gautschi (Euler) equivalent series.
+ */
+int
+gsl_sf_bessel_I_CF1_ser(const double nu, const double x, double * ratio)
+{
+ const int maxk = 20000;
+ double tk = 1.0;
+ double sum = 1.0;
+ double rhok = 0.0;
+ int k;
+
+ for(k=1; k<maxk; k++) {
+ double ak = 0.25 * (x/(nu+k)) * x/(nu+k+1.0);
+ rhok = -ak*(1.0 + rhok)/(1.0 + ak*(1.0 + rhok));
+ tk *= rhok;
+ sum += tk;
+ if(fabs(tk/sum) < GSL_DBL_EPSILON) break;
+ }
+
+ *ratio = x/(2.0*(nu+1.0)) * sum;
+
+ if(k == maxk)
+ GSL_ERROR ("error", GSL_EMAXITER);
+ else
+ return GSL_SUCCESS;
+}
+
+
+int
+gsl_sf_bessel_JY_steed_CF2(const double nu, const double x,
+ double * P, double * Q)
+{
+ const int max_iter = 10000;
+ const double SMALL = 1.0e-100;
+
+ int i = 1;
+
+ double x_inv = 1.0/x;
+ double a = 0.25 - nu*nu;
+ double p = -0.5*x_inv;
+ double q = 1.0;
+ double br = 2.0*x;
+ double bi = 2.0;
+ double fact = a*x_inv/(p*p + q*q);
+ double cr = br + q*fact;
+ double ci = bi + p*fact;
+ double den = br*br + bi*bi;
+ double dr = br/den;
+ double di = -bi/den;
+ double dlr = cr*dr - ci*di;
+ double dli = cr*di + ci*dr;
+ double temp = p*dlr - q*dli;
+ q = p*dli + q*dlr;
+ p = temp;
+ for (i=2; i<=max_iter; i++) {
+ a += 2*(i-1);
+ bi += 2.0;
+ dr = a*dr + br;
+ di = a*di + bi;
+ if(fabs(dr)+fabs(di) < SMALL) dr = SMALL;
+ fact = a/(cr*cr+ci*ci);
+ cr = br + cr*fact;
+ ci = bi - ci*fact;
+ if(fabs(cr)+fabs(ci) < SMALL) cr = SMALL;
+ den = dr*dr + di*di;
+ dr /= den;
+ di /= -den;
+ dlr = cr*dr - ci*di;
+ dli = cr*di + ci*dr;
+ temp = p*dlr - q*dli;
+ q = p*dli + q*dlr;
+ p = temp;
+ if(fabs(dlr-1.0)+fabs(dli) < GSL_DBL_EPSILON) break;
+ }
+
+ *P = p;
+ *Q = q;
+
+ if(i == max_iter)
+ GSL_ERROR ("error", GSL_EMAXITER);
+ else
+ return GSL_SUCCESS;
+}
+
+
+/* Evaluate continued fraction CF2, using Thompson-Barnett-Temme method,
+ * to obtain values of exp(x)*K_nu and exp(x)*K_{nu+1}.
+ *
+ * This is unstable for small x; x > 2 is a good cutoff.
+ * Also requires |nu| < 1/2.
+ */
+int
+gsl_sf_bessel_K_scaled_steed_temme_CF2(const double nu, const double x,
+ double * K_nu, double * K_nup1,
+ double * Kp_nu)
+{
+ const int maxiter = 10000;
+
+ int i = 1;
+ double bi = 2.0*(1.0 + x);
+ double di = 1.0/bi;
+ double delhi = di;
+ double hi = di;
+
+ double qi = 0.0;
+ double qip1 = 1.0;
+
+ double ai = -(0.25 - nu*nu);
+ double a1 = ai;
+ double ci = -ai;
+ double Qi = -ai;
+
+ double s = 1.0 + Qi*delhi;
+
+ for(i=2; i<=maxiter; i++) {
+ double dels;
+ double tmp;
+ ai -= 2.0*(i-1);
+ ci = -ai*ci/i;
+ tmp = (qi - bi*qip1)/ai;
+ qi = qip1;
+ qip1 = tmp;
+ Qi += ci*qip1;
+ bi += 2.0;
+ di = 1.0/(bi + ai*di);
+ delhi = (bi*di - 1.0) * delhi;
+ hi += delhi;
+ dels = Qi*delhi;
+ s += dels;
+ if(fabs(dels/s) < GSL_DBL_EPSILON) break;
+ }
+
+ hi *= -a1;
+
+ *K_nu = sqrt(M_PI/(2.0*x)) / s;
+ *K_nup1 = *K_nu * (nu + x + 0.5 - hi)/x;
+ *Kp_nu = - *K_nup1 + nu/x * *K_nu;
+ if(i == maxiter)
+ GSL_ERROR ("error", GSL_EMAXITER);
+ else
+ return GSL_SUCCESS;
+}
+
+
+int gsl_sf_bessel_cos_pi4_e(double y, double eps, gsl_sf_result * result)
+{
+ const double sy = sin(y);
+ const double cy = cos(y);
+ const double s = sy + cy;
+ const double d = sy - cy;
+ const double abs_sum = fabs(cy) + fabs(sy);
+ double seps;
+ double ceps;
+ if(fabs(eps) < GSL_ROOT5_DBL_EPSILON) {
+ const double e2 = eps*eps;
+ seps = eps * (1.0 - e2/6.0 * (1.0 - e2/20.0));
+ ceps = 1.0 - e2/2.0 * (1.0 - e2/12.0);
+ }
+ else {
+ seps = sin(eps);
+ ceps = cos(eps);
+ }
+ result->val = (ceps * s - seps * d)/ M_SQRT2;
+ result->err = 2.0 * GSL_DBL_EPSILON * (fabs(ceps) + fabs(seps)) * abs_sum / M_SQRT2;
+
+ /* Try to account for error in evaluation of sin(y), cos(y).
+ * This is a little sticky because we don't really know
+ * how the library routines are doing their argument reduction.
+ * However, we will make a reasonable guess.
+ * FIXME ?
+ */
+ if(y > 1.0/GSL_DBL_EPSILON) {
+ result->err *= 0.5 * y;
+ }
+ else if(y > 1.0/GSL_SQRT_DBL_EPSILON) {
+ result->err *= 256.0 * y * GSL_SQRT_DBL_EPSILON;
+ }
+
+ return GSL_SUCCESS;
+}
+
+
+int gsl_sf_bessel_sin_pi4_e(double y, double eps, gsl_sf_result * result)
+{
+ const double sy = sin(y);
+ const double cy = cos(y);
+ const double s = sy + cy;
+ const double d = sy - cy;
+ const double abs_sum = fabs(cy) + fabs(sy);
+ double seps;
+ double ceps;
+ if(fabs(eps) < GSL_ROOT5_DBL_EPSILON) {
+ const double e2 = eps*eps;
+ seps = eps * (1.0 - e2/6.0 * (1.0 - e2/20.0));
+ ceps = 1.0 - e2/2.0 * (1.0 - e2/12.0);
+ }
+ else {
+ seps = sin(eps);
+ ceps = cos(eps);
+ }
+ result->val = (ceps * d + seps * s)/ M_SQRT2;
+ result->err = 2.0 * GSL_DBL_EPSILON * (fabs(ceps) + fabs(seps)) * abs_sum / M_SQRT2;
+
+ /* Try to account for error in evaluation of sin(y), cos(y).
+ * See above.
+ * FIXME ?
+ */
+ if(y > 1.0/GSL_DBL_EPSILON) {
+ result->err *= 0.5 * y;
+ }
+ else if(y > 1.0/GSL_SQRT_DBL_EPSILON) {
+ result->err *= 256.0 * y * GSL_SQRT_DBL_EPSILON;
+ }
+
+ return GSL_SUCCESS;
+}
+
+
+/************************************************************************
+ * *
+ Asymptotic approximations 8.11.5, 8.12.5, and 8.42.7 from
+ G.N.Watson, A Treatise on the Theory of Bessel Functions,
+ 2nd Edition (Cambridge University Press, 1944).
+ Higher terms in expansion for x near l given by
+ Airey in Phil. Mag. 31, 520 (1916).
+
+ This approximation is accurate to near 0.1% at the boundaries
+ between the asymptotic regions; well away from the boundaries
+ the accuracy is better than 10^{-5}.
+ * *
+ ************************************************************************/
+#if 0
+double besselJ_meissel(double nu, double x)
+{
+ double beta = pow(nu, 0.325);
+ double result;
+
+ /* Fitted matching points. */
+ double llimit = 1.1 * beta;
+ double ulimit = 1.3 * beta;
+
+ double nu2 = nu * nu;
+
+ if (nu < 5. && x < 1.)
+ {
+ /* Small argument and order. Use a Taylor expansion. */
+ int k;
+ double xo2 = 0.5 * x;
+ double gamfactor = pow(nu,nu) * exp(-nu) * sqrt(nu * 2. * M_PI)
+ * (1. + 1./(12.*nu) + 1./(288.*nu*nu));
+ double prefactor = pow(xo2, nu) / gamfactor;
+ double C[5];
+
+ C[0] = 1.;
+ C[1] = -C[0] / (nu+1.);
+ C[2] = -C[1] / (2.*(nu+2.));
+ C[3] = -C[2] / (3.*(nu+3.));
+ C[4] = -C[3] / (4.*(nu+4.));
+
+ result = 0.;
+ for(k=0; k<5; k++)
+ result += C[k] * pow(xo2, 2.*k);
+
+ result *= prefactor;
+ }
+ else if(x < nu - llimit)
+ {
+ /* Small x region: x << l. */
+ double z = x / nu;
+ double z2 = z*z;
+ double rtomz2 = sqrt(1.-z2);
+ double omz2_2 = (1.-z2)*(1.-z2);
+
+ /* Calculate Meissel exponent. */
+ double term1 = 1./(24.*nu) * ((2.+3.*z2)/((1.-z2)*rtomz2) -2.);
+ double term2 = - z2*(4. + z2)/(16.*nu2*(1.-z2)*omz2_2);
+ double V_nu = term1 + term2;
+
+ /* Calculate the harmless prefactor. */
+ double sterlingsum = 1. + 1./(12.*nu) + 1./(288*nu2);
+ double harmless = 1. / (sqrt(rtomz2*2.*M_PI*nu) * sterlingsum);
+
+ /* Calculate the logarithm of the nu dependent prefactor. */
+ double ln_nupre = rtomz2 + log(z) - log(1. + rtomz2);
+
+ result = harmless * exp(nu*ln_nupre - V_nu);
+ }
+ else if(x < nu + ulimit)
+ {
+ /* Intermediate region 1: x near nu. */
+ double eps = 1.-nu/x;
+ double eps_x = eps * x;
+ double eps_x_2 = eps_x * eps_x;
+ double xo6 = x/6.;
+ double B[6];
+ static double gam[6] = {2.67894, 1.35412, 1., 0.89298, 0.902745, 1.};
+ static double sf[6] = {0.866025, 0.866025, 0., -0.866025, -0.866025, 0.};
+
+ /* Some terms are identically zero, because sf[] can be zero.
+ * Some terms do not appear in the result.
+ */
+ B[0] = 1.;
+ B[1] = eps_x;
+ /* B[2] = 0.5 * eps_x_2 - 1./20.; */
+ B[3] = eps_x * (eps_x_2/6. - 1./15.);
+ B[4] = eps_x_2 * (eps_x_2 - 1.)/24. + 1./280.;
+ /* B[5] = eps_x * (eps_x_2*(0.5*eps_x_2 - 1.)/60. + 43./8400.); */
+
+ result = B[0] * gam[0] * sf[0] / pow(xo6, 1./3.);
+ result += B[1] * gam[1] * sf[1] / pow(xo6, 2./3.);
+ result += B[3] * gam[3] * sf[3] / pow(xo6, 4./3.);
+ result += B[4] * gam[4] * sf[4] / pow(xo6, 5./3.);
+
+ result /= (3.*M_PI);
+ }
+ else
+ {
+ /* Region of very large argument. Use expansion
+ * for x>>l, and we need not be very exacting.
+ */
+ double secb = x/nu;
+ double sec2b= secb*secb;
+
+ double cotb = 1./sqrt(sec2b-1.); /* cotb=cot(beta) */
+
+ double beta = acos(nu/x);
+ double trigarg = nu/cotb - nu*beta - 0.25 * M_PI;
+
+ double cot3b = cotb * cotb * cotb;
+ double cot6b = cot3b * cot3b;
+
+ double sum1, sum2, expterm, prefactor, trigcos;
+
+ sum1 = 2.0 + 3.0 * sec2b;
+ trigarg -= sum1 * cot3b / (24.0 * nu);
+
+ trigcos = cos(trigarg);
+
+ sum2 = 4.0 + sec2b;
+ expterm = sum2 * sec2b * cot6b / (16.0 * nu2);
+
+ expterm = exp(-expterm);
+ prefactor = sqrt(2. * cotb / (nu * M_PI));
+
+ result = prefactor * expterm * trigcos;
+ }
+
+ return result;
+}
+#endif
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