/* 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 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 */ /* Miscellaneous support functions for Bessel function evaluations. */ #include #include #include #include #include #include #include #include #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); } } /* Hankel's Asymptotic Expansion - A&S 9.2.5 * * 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). * * Sat Mar 15 05:16:18 GMT 2008 [BG] * Extended to use additional terms in the series to gain * higher accuracy. * */ int gsl_sf_bessel_Jnu_asympx_e(const double nu, const double x, gsl_sf_result * result) { double mu = 4.0*nu*nu; double chi = x - (0.5*nu + 0.25)*M_PI; double P = 0.0; double Q = 0.0; double k = 0, t = 1; int convP, convQ; do { t *= (k == 0) ? 1 : -(mu - (2*k-1)*(2*k-1)) / (k * (8 * x)); convP = fabs(t) < GSL_DBL_EPSILON * fabs(P); P += t; k++; t *= (mu - (2*k-1)*(2*k-1)) / (k * (8 * x)); convQ = fabs(t) < GSL_DBL_EPSILON * fabs(Q); Q += t; /* To preserve the consistency of the series we need to exit when P and Q have the same number of terms */ if (convP && convQ && k > (nu / 2)) break; k++; } while (k < 1000); { double pre = sqrt(2.0/(M_PI*x)); double c = cos(chi); double s = sin(chi); result->val = pre * (c*P - s*Q); result->err = pre * GSL_DBL_EPSILON * (fabs(c*P) + fabs(s*Q) + fabs(t)) * (1 + fabs(x)); /* NB: final term accounts for phase error with large x */ } 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, &l); 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 double RECUR_SMALL = GSL_SQRT_DBL_MIN; 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; } else if(fabs(An) < RECUR_SMALL || fabs(Bn) < RECUR_SMALL) { An /= RECUR_SMALL; Bn /= RECUR_SMALL; Anm1 /= RECUR_SMALL; Bnm1 /= RECUR_SMALL; Anm2 /= RECUR_SMALL; Bnm2 /= RECUR_SMALL; } 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; } /* FIXME: we should return an error term here as well, because the error from this recurrence affects the overall error estimate. */ *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 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