| /* specfunc/legendre_con.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 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 */ |
| |
| #include <config.h> |
| #include <gsl/gsl_math.h> |
| #include <gsl/gsl_errno.h> |
| #include <gsl/gsl_poly.h> |
| #include <gsl/gsl_sf_exp.h> |
| #include <gsl/gsl_sf_trig.h> |
| #include <gsl/gsl_sf_gamma.h> |
| #include <gsl/gsl_sf_ellint.h> |
| #include <gsl/gsl_sf_pow_int.h> |
| #include <gsl/gsl_sf_bessel.h> |
| #include <gsl/gsl_sf_hyperg.h> |
| #include <gsl/gsl_sf_legendre.h> |
| |
| #include "error.h" |
| #include "legendre.h" |
| |
| #define Root_2OverPi_ 0.797884560802865355879892 |
| #define locEPS (1000.0*GSL_DBL_EPSILON) |
| |
| |
| /*-*-*-*-*-*-*-*-*-*-*-* Private Section *-*-*-*-*-*-*-*-*-*-*-*/ |
| |
| |
| #define RECURSE_LARGE (1.0e-5*GSL_DBL_MAX) |
| #define RECURSE_SMALL (1.0e+5*GSL_DBL_MIN) |
| |
| |
| /* Continued fraction for f_{ell+1}/f_ell |
| * f_ell := P^{-mu-ell}_{-1/2 + I tau}(x), x < 1.0 |
| * |
| * Uses standard CF method from Temme's book. |
| */ |
| static |
| int |
| conicalP_negmu_xlt1_CF1(const double mu, const int ell, const double tau, |
| const double x, gsl_sf_result * result) |
| { |
| const double RECUR_BIG = GSL_SQRT_DBL_MAX; |
| const int maxiter = 5000; |
| int n = 1; |
| double xi = x/(sqrt(1.0-x)*sqrt(1.0+x)); |
| double Anm2 = 1.0; |
| double Bnm2 = 0.0; |
| double Anm1 = 0.0; |
| double Bnm1 = 1.0; |
| double a1 = 1.0; |
| double b1 = 2.0*(mu + ell + 1.0) * xi; |
| double An = b1*Anm1 + a1*Anm2; |
| double Bn = b1*Bnm1 + a1*Bnm2; |
| double an, bn; |
| double fn = An/Bn; |
| |
| while(n < maxiter) { |
| double old_fn; |
| double del; |
| n++; |
| Anm2 = Anm1; |
| Bnm2 = Bnm1; |
| Anm1 = An; |
| Bnm1 = Bn; |
| an = tau*tau + (mu - 0.5 + ell + n)*(mu - 0.5 + ell + n); |
| bn = 2.0*(ell + mu + n) * xi; |
| An = bn*Anm1 + an*Anm2; |
| Bn = 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; |
| |
| if(fabs(del - 1.0) < 2.0*GSL_DBL_EPSILON) break; |
| } |
| |
| result->val = fn; |
| result->err = 4.0 * GSL_DBL_EPSILON * (sqrt(n) + 1.0) * fabs(fn); |
| |
| if(n >= maxiter) |
| GSL_ERROR ("error", GSL_EMAXITER); |
| else |
| return GSL_SUCCESS; |
| } |
| |
| |
| /* Continued fraction for f_{ell+1}/f_ell |
| * f_ell := P^{-mu-ell}_{-1/2 + I tau}(x), x >= 1.0 |
| * |
| * Uses Gautschi (Euler) equivalent series. |
| */ |
| static |
| int |
| conicalP_negmu_xgt1_CF1(const double mu, const int ell, const double tau, |
| const double x, gsl_sf_result * result) |
| { |
| const int maxk = 20000; |
| const double gamma = 1.0-1.0/(x*x); |
| const double pre = sqrt(x-1.0)*sqrt(x+1.0) / (x*(2.0*(ell+mu+1.0))); |
| double tk = 1.0; |
| double sum = 1.0; |
| double rhok = 0.0; |
| int k; |
| |
| for(k=1; k<maxk; k++) { |
| double tlk = 2.0*(ell + mu + k); |
| double l1k = (ell + mu - 0.5 + 1.0 + k); |
| double ak = -(tau*tau + l1k*l1k)/(tlk*(tlk+2.0)) * gamma; |
| rhok = -ak*(1.0 + rhok)/(1.0 + ak*(1.0 + rhok)); |
| tk *= rhok; |
| sum += tk; |
| if(fabs(tk/sum) < GSL_DBL_EPSILON) break; |
| } |
| |
| result->val = pre * sum; |
| result->err = fabs(pre * tk); |
| result->err += 2.0 * GSL_DBL_EPSILON * (sqrt(k) + 1.0) * fabs(pre*sum); |
| |
| if(k >= maxk) |
| GSL_ERROR ("error", GSL_EMAXITER); |
| else |
| return GSL_SUCCESS; |
| } |
| |
| |
| /* Implementation of large negative mu asymptotic |
| * [Dunster, Proc. Roy. Soc. Edinburgh 119A, 311 (1991), p. 326] |
| */ |
| |
| inline |
| static double olver_U1(double beta2, double p) |
| { |
| return (p-1.0)/(24.0*(1.0+beta2)) * (3.0 + beta2*(2.0 + 5.0*p*(1.0+p))); |
| } |
| |
| inline |
| static double olver_U2(double beta2, double p) |
| { |
| double beta4 = beta2*beta2; |
| double p2 = p*p; |
| double poly1 = 4.0*beta4 + 84.0*beta2 - 63.0; |
| double poly2 = 16.0*beta4 + 90.0*beta2 - 81.0; |
| double poly3 = beta2*p2*(97.0*beta2 - 432.0 + 77.0*p*(beta2-6.0) - 385.0*beta2*p2*(1.0 + p)); |
| return (1.0-p)/(1152.0*(1.0+beta2)) * (poly1 + poly2 + poly3); |
| } |
| |
| static const double U3c1[] = { -1307.0, -1647.0, 3375.0, 3675.0 }; |
| static const double U3c2[] = { 29366.0, 35835.0, -252360.0, -272630.0, |
| 276810.0, 290499.0 }; |
| static const double U3c3[] = { -29748.0, -8840.0, 1725295.0, 1767025.0, |
| -7313470.0, -754778.0, 6309875.0, 6480045.0 }; |
| static const double U3c4[] = { 2696.0, -16740.0, -524250.0, -183975.0, |
| 14670540.0, 14172939.0, -48206730.0, -48461985.0, |
| 36756720.0, 37182145.0 }; |
| static const double U3c5[] = { 9136.0, 22480.0, 12760.0, |
| -252480.0, -4662165.0, -1705341.0, |
| 92370135.0, 86244015.0, -263678415.0, |
| -260275015.0, 185910725.0, 185910725.0 }; |
| |
| #if 0 |
| static double olver_U3(double beta2, double p) |
| { |
| double beta4 = beta2*beta2; |
| double beta6 = beta4*beta2; |
| double opb2s = (1.0+beta2)*(1.0+beta2); |
| double den = 39813120.0 * opb2s*opb2s; |
| double poly1 = gsl_poly_eval(U3c1, 4, p); |
| double poly2 = gsl_poly_eval(U3c2, 6, p); |
| double poly3 = gsl_poly_eval(U3c3, 8, p); |
| double poly4 = gsl_poly_eval(U3c4, 10, p); |
| double poly5 = gsl_poly_eval(U3c5, 12, p); |
| |
| return (p-1.0)*( 1215.0*poly1 + 324.0*beta2*poly2 |
| + 54.0*beta4*poly3 + 12.0*beta6*poly4 |
| + beta4*beta4*poly5 |
| ) / den; |
| } |
| #endif /* 0 */ |
| |
| |
| /* Large negative mu asymptotic |
| * P^{-mu}_{-1/2 + I tau}, mu -> Inf |
| * |x| < 1 |
| * |
| * [Dunster, Proc. Roy. Soc. Edinburgh 119A, 311 (1991), p. 326] |
| */ |
| int |
| gsl_sf_conicalP_xlt1_large_neg_mu_e(double mu, double tau, double x, |
| gsl_sf_result * result, double * ln_multiplier) |
| { |
| double beta = tau/mu; |
| double beta2 = beta*beta; |
| double S = beta * acos((1.0-beta2)/(1.0+beta2)); |
| double p = x/sqrt(beta2*(1.0-x*x) + 1.0); |
| gsl_sf_result lg_mup1; |
| int lg_stat = gsl_sf_lngamma_e(mu+1.0, &lg_mup1); |
| double ln_pre_1 = 0.5*mu*(S - log(1.0+beta2) + log((1.0-p)/(1.0+p))) - lg_mup1.val; |
| double ln_pre_2 = -0.25 * log(1.0 + beta2*(1.0-x)); |
| double ln_pre_3 = -tau * atan(p*beta); |
| double ln_pre = ln_pre_1 + ln_pre_2 + ln_pre_3; |
| double sum = 1.0 - olver_U1(beta2, p)/mu + olver_U2(beta2, p)/(mu*mu); |
| |
| if(sum == 0.0) { |
| result->val = 0.0; |
| result->err = 0.0; |
| *ln_multiplier = 0.0; |
| return GSL_SUCCESS; |
| } |
| else { |
| int stat_e = gsl_sf_exp_mult_e(ln_pre, sum, result); |
| if(stat_e != GSL_SUCCESS) { |
| result->val = sum; |
| result->err = 2.0 * GSL_DBL_EPSILON * fabs(sum); |
| *ln_multiplier = ln_pre; |
| } |
| else { |
| *ln_multiplier = 0.0; |
| } |
| return lg_stat; |
| } |
| } |
| |
| |
| /* Implementation of large tau asymptotic |
| * |
| * A_n^{-mu}, B_n^{-mu} [Olver, p.465, 469] |
| */ |
| |
| inline |
| static double olver_B0_xi(double mu, double xi) |
| { |
| return (1.0 - 4.0*mu*mu)/(8.0*xi) * (1.0/tanh(xi) - 1.0/xi); |
| } |
| |
| static double olver_A1_xi(double mu, double xi, double x) |
| { |
| double B = olver_B0_xi(mu, xi); |
| double psi; |
| if(fabs(x - 1.0) < GSL_ROOT4_DBL_EPSILON) { |
| double y = x - 1.0; |
| double s = -1.0/3.0 + y*(2.0/15.0 - y *(61.0/945.0 - 452.0/14175.0*y)); |
| psi = (4.0*mu*mu - 1.0)/16.0 * s; |
| } |
| else { |
| psi = (4.0*mu*mu - 1.0)/16.0 * (1.0/(x*x-1.0) - 1.0/(xi*xi)); |
| } |
| return 0.5*xi*xi*B*B + (mu+0.5)*B - psi + mu/6.0*(0.25 - mu*mu); |
| } |
| |
| inline |
| static double olver_B0_th(double mu, double theta) |
| { |
| return -(1.0 - 4.0*mu*mu)/(8.0*theta) * (1.0/tan(theta) - 1.0/theta); |
| } |
| |
| static double olver_A1_th(double mu, double theta, double x) |
| { |
| double B = olver_B0_th(mu, theta); |
| double psi; |
| if(fabs(x - 1.0) < GSL_ROOT4_DBL_EPSILON) { |
| double y = 1.0 - x; |
| double s = -1.0/3.0 + y*(2.0/15.0 - y *(61.0/945.0 - 452.0/14175.0*y)); |
| psi = (4.0*mu*mu - 1.0)/16.0 * s; |
| } |
| else { |
| psi = (4.0*mu*mu - 1.0)/16.0 * (1.0/(x*x-1.0) + 1.0/(theta*theta)); |
| } |
| return -0.5*theta*theta*B*B + (mu+0.5)*B - psi + mu/6.0*(0.25 - mu*mu); |
| } |
| |
| |
| /* Large tau uniform asymptotics |
| * P^{-mu}_{-1/2 + I tau} |
| * 1 < x |
| * tau -> Inf |
| * [Olver, p. 469] |
| */ |
| int |
| gsl_sf_conicalP_xgt1_neg_mu_largetau_e(const double mu, const double tau, |
| const double x, double acosh_x, |
| gsl_sf_result * result, double * ln_multiplier) |
| { |
| double xi = acosh_x; |
| double ln_xi_pre; |
| double ln_pre; |
| double sumA, sumB, sum; |
| double arg; |
| gsl_sf_result J_mup1; |
| gsl_sf_result J_mu; |
| double J_mum1; |
| |
| if(xi < GSL_ROOT4_DBL_EPSILON) { |
| ln_xi_pre = -xi*xi/6.0; /* log(1.0 - xi*xi/6.0) */ |
| } |
| else { |
| gsl_sf_result lnshxi; |
| gsl_sf_lnsinh_e(xi, &lnshxi); |
| ln_xi_pre = log(xi) - lnshxi.val; /* log(xi/sinh(xi) */ |
| } |
| |
| ln_pre = 0.5*ln_xi_pre - mu*log(tau); |
| |
| arg = tau*xi; |
| |
| gsl_sf_bessel_Jnu_e(mu + 1.0, arg, &J_mup1); |
| gsl_sf_bessel_Jnu_e(mu, arg, &J_mu); |
| J_mum1 = -J_mup1.val + 2.0*mu/arg*J_mu.val; /* careful of mu < 1 */ |
| |
| sumA = 1.0 - olver_A1_xi(-mu, xi, x)/(tau*tau); |
| sumB = olver_B0_xi(-mu, xi); |
| sum = J_mu.val * sumA - xi/tau * J_mum1 * sumB; |
| |
| if(sum == 0.0) { |
| result->val = 0.0; |
| result->err = 0.0; |
| *ln_multiplier = 0.0; |
| return GSL_SUCCESS; |
| } |
| else { |
| int stat_e = gsl_sf_exp_mult_e(ln_pre, sum, result); |
| if(stat_e != GSL_SUCCESS) { |
| result->val = sum; |
| result->err = 2.0 * GSL_DBL_EPSILON * fabs(sum); |
| *ln_multiplier = ln_pre; |
| } |
| else { |
| *ln_multiplier = 0.0; |
| } |
| return GSL_SUCCESS; |
| } |
| } |
| |
| |
| /* Large tau uniform asymptotics |
| * P^{-mu}_{-1/2 + I tau} |
| * -1 < x < 1 |
| * tau -> Inf |
| * [Olver, p. 473] |
| */ |
| int |
| gsl_sf_conicalP_xlt1_neg_mu_largetau_e(const double mu, const double tau, |
| const double x, const double acos_x, |
| gsl_sf_result * result, double * ln_multiplier) |
| { |
| double theta = acos_x; |
| double ln_th_pre; |
| double ln_pre; |
| double sumA, sumB, sum, sumerr; |
| double arg; |
| gsl_sf_result I_mup1, I_mu; |
| double I_mum1; |
| |
| if(theta < GSL_ROOT4_DBL_EPSILON) { |
| ln_th_pre = theta*theta/6.0; /* log(1.0 + theta*theta/6.0) */ |
| } |
| else { |
| ln_th_pre = log(theta/sin(theta)); |
| } |
| |
| ln_pre = 0.5 * ln_th_pre - mu * log(tau); |
| |
| arg = tau*theta; |
| gsl_sf_bessel_Inu_e(mu + 1.0, arg, &I_mup1); |
| gsl_sf_bessel_Inu_e(mu, arg, &I_mu); |
| I_mum1 = I_mup1.val + 2.0*mu/arg * I_mu.val; /* careful of mu < 1 */ |
| |
| sumA = 1.0 - olver_A1_th(-mu, theta, x)/(tau*tau); |
| sumB = olver_B0_th(-mu, theta); |
| sum = I_mu.val * sumA - theta/tau * I_mum1 * sumB; |
| sumerr = fabs(I_mu.err * sumA); |
| sumerr += fabs(I_mup1.err * theta/tau * sumB); |
| sumerr += fabs(I_mu.err * theta/tau * sumB * 2.0 * mu/arg); |
| |
| if(sum == 0.0) { |
| result->val = 0.0; |
| result->err = 0.0; |
| *ln_multiplier = 0.0; |
| return GSL_SUCCESS; |
| } |
| else { |
| int stat_e = gsl_sf_exp_mult_e(ln_pre, sum, result); |
| if(stat_e != GSL_SUCCESS) { |
| result->val = sum; |
| result->err = sumerr; |
| result->err += GSL_DBL_EPSILON * fabs(sum); |
| *ln_multiplier = ln_pre; |
| } |
| else { |
| *ln_multiplier = 0.0; |
| } |
| return GSL_SUCCESS; |
| } |
| } |
| |
| |
| /* Hypergeometric function which appears in the |
| * large x expansion below: |
| * |
| * 2F1(1/4 - mu/2 - I tau/2, 3/4 - mu/2 - I tau/2, 1 - I tau, y) |
| * |
| * Note that for the usage below y = 1/x^2; |
| */ |
| static |
| int |
| conicalP_hyperg_large_x(const double mu, const double tau, const double y, |
| double * reF, double * imF) |
| { |
| const int kmax = 1000; |
| const double re_a = 0.25 - 0.5*mu; |
| const double re_b = 0.75 - 0.5*mu; |
| const double re_c = 1.0; |
| const double im_a = -0.5*tau; |
| const double im_b = -0.5*tau; |
| const double im_c = -tau; |
| |
| double re_sum = 1.0; |
| double im_sum = 0.0; |
| double re_term = 1.0; |
| double im_term = 0.0; |
| int k; |
| |
| for(k=1; k<=kmax; k++) { |
| double re_ak = re_a + k - 1.0; |
| double re_bk = re_b + k - 1.0; |
| double re_ck = re_c + k - 1.0; |
| double im_ak = im_a; |
| double im_bk = im_b; |
| double im_ck = im_c; |
| double den = re_ck*re_ck + im_ck*im_ck; |
| double re_multiplier = ((re_ak*re_bk - im_ak*im_bk)*re_ck + im_ck*(im_ak*re_bk + re_ak*im_bk)) / den; |
| double im_multiplier = ((im_ak*re_bk + re_ak*im_bk)*re_ck - im_ck*(re_ak*re_bk - im_ak*im_bk)) / den; |
| double re_tmp = re_multiplier*re_term - im_multiplier*im_term; |
| double im_tmp = im_multiplier*re_term + re_multiplier*im_term; |
| double asum = fabs(re_sum) + fabs(im_sum); |
| re_term = y/k * re_tmp; |
| im_term = y/k * im_tmp; |
| if(fabs(re_term/asum) < GSL_DBL_EPSILON && fabs(im_term/asum) < GSL_DBL_EPSILON) break; |
| re_sum += re_term; |
| im_sum += im_term; |
| } |
| |
| *reF = re_sum; |
| *imF = im_sum; |
| |
| if(k == kmax) |
| GSL_ERROR ("error", GSL_EMAXITER); |
| else |
| return GSL_SUCCESS; |
| } |
| |
| |
| /* P^{mu}_{-1/2 + I tau} |
| * x->Inf |
| */ |
| int |
| gsl_sf_conicalP_large_x_e(const double mu, const double tau, const double x, |
| gsl_sf_result * result, double * ln_multiplier) |
| { |
| /* 2F1 term |
| */ |
| double y = ( x < 0.5*GSL_SQRT_DBL_MAX ? 1.0/(x*x) : 0.0 ); |
| double reF, imF; |
| int stat_F = conicalP_hyperg_large_x(mu, tau, y, &reF, &imF); |
| |
| /* f = Gamma(+i tau)/Gamma(1/2 - mu + i tau) |
| * FIXME: shift so it's better for tau-> 0 |
| */ |
| gsl_sf_result lgr_num, lgth_num; |
| gsl_sf_result lgr_den, lgth_den; |
| int stat_gn = gsl_sf_lngamma_complex_e(0.0,tau,&lgr_num,&lgth_num); |
| int stat_gd = gsl_sf_lngamma_complex_e(0.5-mu,tau,&lgr_den,&lgth_den); |
| |
| double angle = lgth_num.val - lgth_den.val + atan2(imF,reF); |
| |
| double lnx = log(x); |
| double lnxp1 = log(x+1.0); |
| double lnxm1 = log(x-1.0); |
| double lnpre_const = 0.5*M_LN2 - 0.5*M_LNPI; |
| double lnpre_comm = (mu-0.5)*lnx - 0.5*mu*(lnxp1 + lnxm1); |
| double lnpre_err = GSL_DBL_EPSILON * (0.5*M_LN2 + 0.5*M_LNPI) |
| + GSL_DBL_EPSILON * fabs((mu-0.5)*lnx) |
| + GSL_DBL_EPSILON * fabs(0.5*mu)*(fabs(lnxp1)+fabs(lnxm1)); |
| |
| /* result = pre*|F|*|f| * cos(angle - tau * (log(x)+M_LN2)) |
| */ |
| gsl_sf_result cos_result; |
| int stat_cos = gsl_sf_cos_e(angle + tau*(log(x) + M_LN2), &cos_result); |
| int status = GSL_ERROR_SELECT_4(stat_cos, stat_gd, stat_gn, stat_F); |
| if(cos_result.val == 0.0) { |
| result->val = 0.0; |
| result->err = 0.0; |
| return status; |
| } |
| else { |
| double lnFf_val = 0.5*log(reF*reF+imF*imF) + lgr_num.val - lgr_den.val; |
| double lnFf_err = lgr_num.err + lgr_den.err + GSL_DBL_EPSILON * fabs(lnFf_val); |
| double lnnoc_val = lnpre_const + lnpre_comm + lnFf_val; |
| double lnnoc_err = lnpre_err + lnFf_err + GSL_DBL_EPSILON * fabs(lnnoc_val); |
| int stat_e = gsl_sf_exp_mult_err_e(lnnoc_val, lnnoc_err, |
| cos_result.val, cos_result.err, |
| result); |
| if(stat_e == GSL_SUCCESS) { |
| *ln_multiplier = 0.0; |
| } |
| else { |
| result->val = cos_result.val; |
| result->err = cos_result.err; |
| result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val); |
| *ln_multiplier = lnnoc_val; |
| } |
| return status; |
| } |
| } |
| |
| |
| /* P^{mu}_{-1/2 + I tau} first hypergeometric representation |
| * -1 < x < 1 |
| * This is more effective for |x| small, however it will work w/o |
| * reservation for any x < 0 because everything is positive |
| * definite in that case. |
| * |
| * [Kolbig, (3)] (note typo in args of gamma functions) |
| * [Bateman, (22)] (correct form) |
| */ |
| static |
| int |
| conicalP_xlt1_hyperg_A(double mu, double tau, double x, gsl_sf_result * result) |
| { |
| double x2 = x*x; |
| double err_amp = 1.0 + 1.0/(GSL_DBL_EPSILON + fabs(1.0-fabs(x))); |
| double pre_val = M_SQRTPI / pow(0.5*sqrt(1-x2), mu); |
| double pre_err = err_amp * GSL_DBL_EPSILON * (fabs(mu)+1.0) * fabs(pre_val) ; |
| gsl_sf_result ln_g1, ln_g2, arg_g1, arg_g2; |
| gsl_sf_result F1, F2; |
| gsl_sf_result pre1, pre2; |
| double t1_val, t1_err; |
| double t2_val, t2_err; |
| |
| int stat_F1 = gsl_sf_hyperg_2F1_conj_e(0.25 - 0.5*mu, 0.5*tau, 0.5, x2, &F1); |
| int stat_F2 = gsl_sf_hyperg_2F1_conj_e(0.75 - 0.5*mu, 0.5*tau, 1.5, x2, &F2); |
| int status = GSL_ERROR_SELECT_2(stat_F1, stat_F2); |
| |
| gsl_sf_lngamma_complex_e(0.75 - 0.5*mu, -0.5*tau, &ln_g1, &arg_g1); |
| gsl_sf_lngamma_complex_e(0.25 - 0.5*mu, -0.5*tau, &ln_g2, &arg_g2); |
| |
| gsl_sf_exp_err_e(-2.0*ln_g1.val, 2.0*ln_g1.err, &pre1); |
| gsl_sf_exp_err_e(-2.0*ln_g2.val, 2.0*ln_g2.err, &pre2); |
| pre2.val *= -2.0*x; |
| pre2.err *= 2.0*fabs(x); |
| pre2.err += GSL_DBL_EPSILON * fabs(pre2.val); |
| |
| t1_val = pre1.val * F1.val; |
| t1_err = fabs(pre1.val) * F1.err + pre1.err * fabs(F1.val); |
| t2_val = pre2.val * F2.val; |
| t2_err = fabs(pre2.val) * F2.err + pre2.err * fabs(F2.val); |
| |
| result->val = pre_val * (t1_val + t2_val); |
| result->err = pre_val * (t1_err + t2_err); |
| result->err += pre_err * fabs(t1_val + t2_val); |
| result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val); |
| |
| return status; |
| } |
| |
| |
| /* P^{mu}_{-1/2 + I tau} |
| * defining hypergeometric representation |
| * [Abramowitz+Stegun, 8.1.2] |
| * 1 < x < 3 |
| * effective for x near 1 |
| * |
| */ |
| #if 0 |
| static |
| int |
| conicalP_def_hyperg(double mu, double tau, double x, double * result) |
| { |
| double F; |
| int stat_F = gsl_sf_hyperg_2F1_conj_renorm_e(0.5, tau, 1.0-mu, 0.5*(1.0-x), &F); |
| *result = pow((x+1.0)/(x-1.0), 0.5*mu) * F; |
| return stat_F; |
| } |
| #endif /* 0 */ |
| |
| |
| /* P^{mu}_{-1/2 + I tau} second hypergeometric representation |
| * [Zhurina+Karmazina, (3.1)] |
| * -1 < x < 3 |
| * effective for x near 1 |
| * |
| */ |
| #if 0 |
| static |
| int |
| conicalP_xnear1_hyperg_C(double mu, double tau, double x, double * result) |
| { |
| double ln_pre, arg_pre; |
| double ln_g1, arg_g1; |
| double ln_g2, arg_g2; |
| double F; |
| |
| int stat_F = gsl_sf_hyperg_2F1_conj_renorm_e(0.5+mu, tau, 1.0+mu, 0.5*(1.0-x), &F); |
| |
| gsl_sf_lngamma_complex_e(0.5+mu, tau, &ln_g1, &arg_g1); |
| gsl_sf_lngamma_complex_e(0.5-mu, tau, &ln_g2, &arg_g2); |
| |
| ln_pre = mu*M_LN2 - 0.5*mu*log(fabs(x*x-1.0)) + ln_g1 - ln_g2; |
| arg_pre = arg_g1 - arg_g2; |
| |
| *result = exp(ln_pre) * F; |
| return stat_F; |
| } |
| #endif /* 0 */ |
| |
| |
| /* V0, V1 from Kolbig, m = 0 |
| */ |
| static |
| int |
| conicalP_0_V(const double t, const double f, const double tau, const double sgn, |
| double * V0, double * V1) |
| { |
| double C[8]; |
| double T[8]; |
| double H[8]; |
| double V[12]; |
| int i; |
| T[0] = 1.0; |
| H[0] = 1.0; |
| V[0] = 1.0; |
| for(i=1; i<=7; i++) { |
| T[i] = T[i-1] * t; |
| H[i] = H[i-1] * (t*f); |
| } |
| for(i=1; i<=11; i++) { |
| V[i] = V[i-1] * tau; |
| } |
| |
| C[0] = 1.0; |
| C[1] = (H[1]-1.0)/(8.0*T[1]); |
| C[2] = (9.0*H[2] + 6.0*H[1] - 15.0 - sgn*8.0*T[2])/(128.0*T[2]); |
| C[3] = 5.0*(15.0*H[3] + 27.0*H[2] + 21.0*H[1] - 63.0 - sgn*T[2]*(16.0*H[1]+24.0))/(1024.0*T[3]); |
| C[4] = 7.0*(525.0*H[4] + 1500.0*H[3] + 2430.0*H[2] + 1980.0*H[1] - 6435.0 |
| + 192.0*T[4] - sgn*T[2]*(720.0*H[2]+1600.0*H[1]+2160.0) |
| ) / (32768.0*T[4]); |
| C[5] = 21.0*(2835.0*H[5] + 11025.0*H[4] + 24750.0*H[3] + 38610.0*H[2] |
| + 32175.0*H[1] - 109395.0 + T[4]*(1984.0*H[1]+4032.0) |
| - sgn*T[2]*(4800.0*H[3]+15120.0*H[2]+26400.0*H[1]+34320.0) |
| ) / (262144.0*T[5]); |
| C[6] = 11.0*(218295.0*H[6] + 1071630.0*H[5] + 3009825.0*H[4] + 6142500.0*H[3] |
| + 9398025.0*H[2] + 7936110.0*H[1] - 27776385.0 |
| + T[4]*(254016.0*H[2]+749952.0*H[1]+1100736.0) |
| - sgn*T[2]*(441000.0*H[4] + 1814400.0*H[3] + 4127760.0*H[2] |
| + 6552000.0*H[1] + 8353800.0 + 31232.0*T[4] |
| ) |
| ) / (4194304.0*T[6]); |
| |
| *V0 = C[0] + (-4.0*C[3]/T[1]+C[4])/V[4] |
| + (-192.0*C[5]/T[3]+144.0*C[6]/T[2])/V[8] |
| + sgn * (-C[2]/V[2] |
| + (-24.0*C[4]/T[2]+12.0*C[5]/T[1]-C[6])/V[6] |
| + (-1920.0*C[6]/T[4])/V[10] |
| ); |
| *V1 = C[1]/V[1] + (8.0*(C[3]/T[2]-C[4]/T[1])+C[5])/V[5] |
| + (384.0*C[5]/T[4] - 768.0*C[6]/T[3])/V[9] |
| + sgn * ((2.0*C[2]/T[1]-C[3])/V[3] |
| + (48.0*C[4]/T[3]-72.0*C[5]/T[2] + 18.0*C[6]/T[1])/V[7] |
| + (3840.0*C[6]/T[5])/V[11] |
| ); |
| |
| return GSL_SUCCESS; |
| } |
| |
| |
| /* V0, V1 from Kolbig, m = 1 |
| */ |
| static |
| int |
| conicalP_1_V(const double t, const double f, const double tau, const double sgn, |
| double * V0, double * V1) |
| { |
| double Cm1; |
| double C[8]; |
| double T[8]; |
| double H[8]; |
| double V[12]; |
| int i; |
| T[0] = 1.0; |
| H[0] = 1.0; |
| V[0] = 1.0; |
| for(i=1; i<=7; i++) { |
| T[i] = T[i-1] * t; |
| H[i] = H[i-1] * (t*f); |
| } |
| for(i=1; i<=11; i++) { |
| V[i] = V[i-1] * tau; |
| } |
| |
| Cm1 = -1.0; |
| C[0] = 3.0*(1.0-H[1])/(8.0*T[1]); |
| C[1] = (-15.0*H[2]+6.0*H[1]+9.0+sgn*8.0*T[2])/(128.0*T[2]); |
| C[2] = 3.0*(-35.0*H[3] - 15.0*H[2] + 15.0*H[1] + 35.0 + sgn*T[2]*(32.0*H[1]+8.0))/(1024.0*T[3]); |
| C[3] = (-4725.0*H[4] - 6300.0*H[3] - 3150.0*H[2] + 3780.0*H[1] + 10395.0 |
| -1216.0*T[4] + sgn*T[2]*(6000.0*H[2]+5760.0*H[1]+1680.0)) / (32768.0*T[4]); |
| C[4] = 7.0*(-10395.0*H[5] - 23625.0*H[4] - 28350.0*H[3] - 14850.0*H[2] |
| +19305.0*H[1] + 57915.0 - T[4]*(6336.0*H[1]+6080.0) |
| + sgn*T[2]*(16800.0*H[3] + 30000.0*H[2] + 25920.0*H[1] + 7920.0) |
| ) / (262144.0*T[5]); |
| C[5] = (-2837835.0*H[6] - 9168390.0*H[5] - 16372125.0*H[4] - 18918900*H[3] |
| -10135125.0*H[2] + 13783770.0*H[1] + 43648605.0 |
| -T[4]*(3044160.0*H[2] + 5588352.0*H[1] + 4213440.0) |
| +sgn*T[2]*(5556600.0*H[4] + 14817600.0*H[3] + 20790000.0*H[2] |
| + 17297280.0*H[1] + 5405400.0 + 323072.0*T[4] |
| ) |
| ) / (4194304.0*T[6]); |
| C[6] = 0.0; |
| |
| *V0 = C[0] + (-4.0*C[3]/T[1]+C[4])/V[4] |
| + (-192.0*C[5]/T[3]+144.0*C[6]/T[2])/V[8] |
| + sgn * (-C[2]/V[2] |
| + (-24.0*C[4]/T[2]+12.0*C[5]/T[1]-C[6])/V[6] |
| ); |
| *V1 = C[1]/V[1] + (8.0*(C[3]/T[2]-C[4]/T[1])+C[5])/V[5] |
| + (384.0*C[5]/T[4] - 768.0*C[6]/T[3])/V[9] |
| + sgn * (Cm1*V[1] + (2.0*C[2]/T[1]-C[3])/V[3] |
| + (48.0*C[4]/T[3]-72.0*C[5]/T[2] + 18.0*C[6]/T[1])/V[7] |
| ); |
| |
| return GSL_SUCCESS; |
| } |
| |
| |
| |
| /*-*-*-*-*-*-*-*-*-*-*-* Functions with Error Codes *-*-*-*-*-*-*-*-*-*-*-*/ |
| |
| /* P^0_{-1/2 + I lambda} |
| */ |
| int |
| gsl_sf_conicalP_0_e(const double lambda, const double x, gsl_sf_result * result) |
| { |
| /* CHECK_POINTER(result) */ |
| |
| if(x <= -1.0) { |
| DOMAIN_ERROR(result); |
| } |
| else if(x == 1.0) { |
| result->val = 1.0; |
| result->err = 0.0; |
| return GSL_SUCCESS; |
| } |
| else if(lambda == 0.0) { |
| gsl_sf_result K; |
| int stat_K; |
| if(x < 1.0) { |
| const double th = acos(x); |
| const double s = sin(0.5*th); |
| stat_K = gsl_sf_ellint_Kcomp_e(s, GSL_MODE_DEFAULT, &K); |
| result->val = 2.0/M_PI * K.val; |
| result->err = 2.0/M_PI * K.err; |
| result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val); |
| return stat_K; |
| } |
| else { |
| const double xi = acosh(x); |
| const double c = cosh(0.5*xi); |
| const double t = tanh(0.5*xi); |
| stat_K = gsl_sf_ellint_Kcomp_e(t, GSL_MODE_DEFAULT, &K); |
| result->val = 2.0/M_PI / c * K.val; |
| result->err = 2.0/M_PI / c * K.err; |
| result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val); |
| return stat_K; |
| } |
| } |
| else if( (x <= 0.0 && lambda < 1000.0) |
| || (x < 0.1 && lambda < 17.0) |
| || (x < 0.2 && lambda < 5.0 ) |
| ) { |
| return conicalP_xlt1_hyperg_A(0.0, lambda, x, result); |
| } |
| else if( (x <= 0.2 && lambda < 17.0) |
| || (x <= 1.5 && lambda < 20.0) |
| ) { |
| return gsl_sf_hyperg_2F1_conj_e(0.5, lambda, 1.0, (1.0-x)/2, result); |
| } |
| else if(1.5 < x && lambda < GSL_MAX(x,20.0)) { |
| gsl_sf_result P; |
| double lm; |
| int stat_P = gsl_sf_conicalP_large_x_e(0.0, lambda, x, |
| &P, &lm |
| ); |
| int stat_e = gsl_sf_exp_mult_err_e(lm, 2.0*GSL_DBL_EPSILON * fabs(lm), |
| P.val, P.err, |
| result); |
| return GSL_ERROR_SELECT_2(stat_e, stat_P); |
| } |
| else { |
| double V0, V1; |
| if(x < 1.0) { |
| double th = acos(x); |
| double sth = sqrt(1.0-x*x); /* sin(th) */ |
| gsl_sf_result I0, I1; |
| int stat_I0 = gsl_sf_bessel_I0_scaled_e(th * lambda, &I0); |
| int stat_I1 = gsl_sf_bessel_I1_scaled_e(th * lambda, &I1); |
| int stat_I = GSL_ERROR_SELECT_2(stat_I0, stat_I1); |
| int stat_V = conicalP_0_V(th, x/sth, lambda, -1.0, &V0, &V1); |
| double bessterm = V0 * I0.val + V1 * I1.val; |
| double besserr = fabs(V0) * I0.err + fabs(V1) * I1.err; |
| double arg1 = th*lambda; |
| double sqts = sqrt(th/sth); |
| int stat_e = gsl_sf_exp_mult_err_e(arg1, 4.0 * GSL_DBL_EPSILON * fabs(arg1), |
| sqts * bessterm, sqts * besserr, |
| result); |
| return GSL_ERROR_SELECT_3(stat_e, stat_V, stat_I); |
| } |
| else { |
| double sh = sqrt(x-1.0)*sqrt(x+1.0); /* sinh(xi) */ |
| double xi = log(x + sh); /* xi = acosh(x) */ |
| gsl_sf_result J0, J1; |
| int stat_J0 = gsl_sf_bessel_J0_e(xi * lambda, &J0); |
| int stat_J1 = gsl_sf_bessel_J1_e(xi * lambda, &J1); |
| int stat_J = GSL_ERROR_SELECT_2(stat_J0, stat_J1); |
| int stat_V = conicalP_0_V(xi, x/sh, lambda, 1.0, &V0, &V1); |
| double bessterm = V0 * J0.val + V1 * J1.val; |
| double besserr = fabs(V0) * J0.err + fabs(V1) * J1.err; |
| double pre_val = sqrt(xi/sh); |
| double pre_err = 2.0 * fabs(pre_val); |
| result->val = pre_val * bessterm; |
| result->err = pre_val * besserr; |
| result->err += pre_err * fabs(bessterm); |
| result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val); |
| return GSL_ERROR_SELECT_2(stat_V, stat_J); |
| } |
| } |
| } |
| |
| |
| /* P^1_{-1/2 + I lambda} |
| */ |
| int |
| gsl_sf_conicalP_1_e(const double lambda, const double x, gsl_sf_result * result) |
| { |
| /* CHECK_POINTER(result) */ |
| |
| if(x <= -1.0) { |
| DOMAIN_ERROR(result); |
| } |
| else if(lambda == 0.0) { |
| gsl_sf_result K, E; |
| int stat_K, stat_E; |
| if(x == 1.0) { |
| result->val = 0.0; |
| result->err = 0.0; |
| return GSL_SUCCESS; |
| } |
| else if(x < 1.0) { |
| if(1.0-x < GSL_SQRT_DBL_EPSILON) { |
| double err_amp = GSL_MAX_DBL(1.0, 1.0/(GSL_DBL_EPSILON + fabs(1.0-x))); |
| result->val = 0.25/M_SQRT2 * sqrt(1.0-x) * (1.0 + 5.0/16.0 * (1.0-x)); |
| result->err = err_amp * 3.0 * GSL_DBL_EPSILON * fabs(result->val); |
| return GSL_SUCCESS; |
| } |
| else { |
| const double th = acos(x); |
| const double s = sin(0.5*th); |
| const double c2 = 1.0 - s*s; |
| const double sth = sin(th); |
| const double pre = 2.0/(M_PI*sth); |
| stat_K = gsl_sf_ellint_Kcomp_e(s, GSL_MODE_DEFAULT, &K); |
| stat_E = gsl_sf_ellint_Ecomp_e(s, GSL_MODE_DEFAULT, &E); |
| result->val = pre * (E.val - c2 * K.val); |
| result->err = pre * (E.err + fabs(c2) * K.err); |
| result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val); |
| return stat_K; |
| } |
| } |
| else { |
| if(x-1.0 < GSL_SQRT_DBL_EPSILON) { |
| double err_amp = GSL_MAX_DBL(1.0, 1.0/(GSL_DBL_EPSILON + fabs(1.0-x))); |
| result->val = -0.25/M_SQRT2 * sqrt(x-1.0) * (1.0 - 5.0/16.0 * (x-1.0)); |
| result->err = err_amp * 3.0 * GSL_DBL_EPSILON * fabs(result->val); |
| return GSL_SUCCESS; |
| } |
| else { |
| const double xi = acosh(x); |
| const double c = cosh(0.5*xi); |
| const double t = tanh(0.5*xi); |
| const double sxi = sinh(xi); |
| const double pre = 2.0/(M_PI*sxi) * c; |
| stat_K = gsl_sf_ellint_Kcomp_e(t, GSL_MODE_DEFAULT, &K); |
| stat_E = gsl_sf_ellint_Ecomp_e(t, GSL_MODE_DEFAULT, &E); |
| result->val = pre * (E.val - K.val); |
| result->err = pre * (E.err + K.err); |
| result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val); |
| return stat_K; |
| } |
| } |
| } |
| else if( (x <= 0.0 && lambda < 1000.0) |
| || (x < 0.1 && lambda < 17.0) |
| || (x < 0.2 && lambda < 5.0 ) |
| ) { |
| return conicalP_xlt1_hyperg_A(1.0, lambda, x, result); |
| } |
| else if( (x <= 0.2 && lambda < 17.0) |
| || (x < 1.5 && lambda < 20.0) |
| ) { |
| const double arg = fabs(x*x - 1.0); |
| const double sgn = GSL_SIGN(1.0 - x); |
| const double pre = 0.5*(lambda*lambda + 0.25) * sgn * sqrt(arg); |
| gsl_sf_result F; |
| int stat_F = gsl_sf_hyperg_2F1_conj_e(1.5, lambda, 2.0, (1.0-x)/2, &F); |
| result->val = pre * F.val; |
| result->err = fabs(pre) * F.err; |
| result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val); |
| return stat_F; |
| } |
| else if(1.5 <= x && lambda < GSL_MAX(x,20.0)) { |
| gsl_sf_result P; |
| double lm; |
| int stat_P = gsl_sf_conicalP_large_x_e(1.0, lambda, x, |
| &P, &lm |
| ); |
| int stat_e = gsl_sf_exp_mult_err_e(lm, 2.0 * GSL_DBL_EPSILON * fabs(lm), |
| P.val, P.err, |
| result); |
| return GSL_ERROR_SELECT_2(stat_e, stat_P); |
| } |
| else { |
| double V0, V1; |
| if(x < 1.0) { |
| const double sqrt_1mx = sqrt(1.0 - x); |
| const double sqrt_1px = sqrt(1.0 + x); |
| const double th = acos(x); |
| const double sth = sqrt_1mx * sqrt_1px; /* sin(th) */ |
| gsl_sf_result I0, I1; |
| int stat_I0 = gsl_sf_bessel_I0_scaled_e(th * lambda, &I0); |
| int stat_I1 = gsl_sf_bessel_I1_scaled_e(th * lambda, &I1); |
| int stat_I = GSL_ERROR_SELECT_2(stat_I0, stat_I1); |
| int stat_V = conicalP_1_V(th, x/sth, lambda, -1.0, &V0, &V1); |
| double bessterm = V0 * I0.val + V1 * I1.val; |
| double besserr = fabs(V0) * I0.err + fabs(V1) * I1.err |
| + 2.0 * GSL_DBL_EPSILON * fabs(V0 * I0.val) |
| + 2.0 * GSL_DBL_EPSILON * fabs(V1 * I1.val); |
| double arg1 = th * lambda; |
| double sqts = sqrt(th/sth); |
| int stat_e = gsl_sf_exp_mult_err_e(arg1, 2.0 * GSL_DBL_EPSILON * fabs(arg1), |
| sqts * bessterm, sqts * besserr, |
| result); |
| result->err *= 1.0/sqrt_1mx; |
| return GSL_ERROR_SELECT_3(stat_e, stat_V, stat_I); |
| } |
| else { |
| const double sqrt_xm1 = sqrt(x - 1.0); |
| const double sqrt_xp1 = sqrt(x + 1.0); |
| const double sh = sqrt_xm1 * sqrt_xp1; /* sinh(xi) */ |
| const double xi = log(x + sh); /* xi = acosh(x) */ |
| const double xi_lam = xi * lambda; |
| gsl_sf_result J0, J1; |
| const int stat_J0 = gsl_sf_bessel_J0_e(xi_lam, &J0); |
| const int stat_J1 = gsl_sf_bessel_J1_e(xi_lam, &J1); |
| const int stat_J = GSL_ERROR_SELECT_2(stat_J0, stat_J1); |
| const int stat_V = conicalP_1_V(xi, x/sh, lambda, 1.0, &V0, &V1); |
| const double bessterm = V0 * J0.val + V1 * J1.val; |
| const double besserr = fabs(V0) * J0.err + fabs(V1) * J1.err |
| + 512.0 * 2.0 * GSL_DBL_EPSILON * fabs(V0 * J0.val) |
| + 512.0 * 2.0 * GSL_DBL_EPSILON * fabs(V1 * J1.val) |
| + GSL_DBL_EPSILON * fabs(xi_lam * V0 * J1.val) |
| + GSL_DBL_EPSILON * fabs(xi_lam * V1 * J0.val); |
| const double pre = sqrt(xi/sh); |
| result->val = pre * bessterm; |
| result->err = pre * besserr * sqrt_xp1 / sqrt_xm1; |
| result->err += 4.0 * GSL_DBL_EPSILON * fabs(result->val); |
| return GSL_ERROR_SELECT_2(stat_V, stat_J); |
| } |
| } |
| } |
| |
| |
| /* P^{1/2}_{-1/2 + I lambda} (x) |
| * [Abramowitz+Stegun 8.6.8, 8.6.12] |
| * checked OK [GJ] Fri May 8 12:24:36 MDT 1998 |
| */ |
| int gsl_sf_conicalP_half_e(const double lambda, const double x, |
| gsl_sf_result * result |
| ) |
| { |
| /* CHECK_POINTER(result) */ |
| |
| if(x <= -1.0) { |
| DOMAIN_ERROR(result); |
| } |
| else if(x < 1.0) { |
| double err_amp = 1.0 + 1.0/(GSL_DBL_EPSILON + fabs(1.0-fabs(x))); |
| double ac = acos(x); |
| double den = sqrt(sqrt(1.0-x)*sqrt(1.0+x)); |
| result->val = Root_2OverPi_ / den * cosh(ac * lambda); |
| result->err = err_amp * 3.0 * GSL_DBL_EPSILON * fabs(result->val); |
| result->err *= fabs(ac * lambda) + 1.0; |
| return GSL_SUCCESS; |
| } |
| else if(x == 1.0) { |
| result->val = 0.0; |
| result->err = 0.0; |
| return GSL_SUCCESS; |
| } |
| else { |
| /* x > 1 */ |
| double err_amp = 1.0 + 1.0/(GSL_DBL_EPSILON + fabs(1.0-fabs(x))); |
| double sq_term = sqrt(x-1.0)*sqrt(x+1.0); |
| double ln_term = log(x + sq_term); |
| double den = sqrt(sq_term); |
| double carg_val = lambda * ln_term; |
| double carg_err = 2.0 * GSL_DBL_EPSILON * fabs(carg_val); |
| gsl_sf_result cos_result; |
| int stat_cos = gsl_sf_cos_err_e(carg_val, carg_err, &cos_result); |
| result->val = Root_2OverPi_ / den * cos_result.val; |
| result->err = err_amp * Root_2OverPi_ / den * cos_result.err; |
| result->err += 4.0 * GSL_DBL_EPSILON * fabs(result->val); |
| return stat_cos; |
| } |
| } |
| |
| |
| /* P^{-1/2}_{-1/2 + I lambda} (x) |
| * [Abramowitz+Stegun 8.6.9, 8.6.14] |
| * checked OK [GJ] Fri May 8 12:24:43 MDT 1998 |
| */ |
| int gsl_sf_conicalP_mhalf_e(const double lambda, const double x, gsl_sf_result * result) |
| { |
| /* CHECK_POINTER(result) */ |
| |
| if(x <= -1.0) { |
| DOMAIN_ERROR(result); |
| } |
| else if(x < 1.0) { |
| double ac = acos(x); |
| double den = sqrt(sqrt(1.0-x)*sqrt(1.0+x)); |
| double arg = ac * lambda; |
| double err_amp = 1.0 + 1.0/(GSL_DBL_EPSILON + fabs(1.0-fabs(x))); |
| if(fabs(arg) < GSL_SQRT_DBL_EPSILON) { |
| result->val = Root_2OverPi_ / den * ac; |
| result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val); |
| result->err *= err_amp; |
| } |
| else { |
| result->val = Root_2OverPi_ / (den*lambda) * sinh(arg); |
| result->err = GSL_DBL_EPSILON * (fabs(arg)+1.0) * fabs(result->val); |
| result->err *= err_amp; |
| result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val); |
| } |
| return GSL_SUCCESS; |
| } |
| else if(x == 1.0) { |
| result->val = 0.0; |
| result->err = 0.0; |
| return GSL_SUCCESS; |
| } |
| else { |
| /* x > 1 */ |
| double sq_term = sqrt(x-1.0)*sqrt(x+1.0); |
| double ln_term = log(x + sq_term); |
| double den = sqrt(sq_term); |
| double arg_val = lambda * ln_term; |
| double arg_err = 2.0 * GSL_DBL_EPSILON * fabs(arg_val); |
| if(arg_val < GSL_SQRT_DBL_EPSILON) { |
| result->val = Root_2OverPi_ / den * ln_term; |
| result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val); |
| return GSL_SUCCESS; |
| } |
| else { |
| gsl_sf_result sin_result; |
| int stat_sin = gsl_sf_sin_err_e(arg_val, arg_err, &sin_result); |
| result->val = Root_2OverPi_ / (den*lambda) * sin_result.val; |
| result->err = Root_2OverPi_ / fabs(den*lambda) * sin_result.err; |
| result->err += 3.0 * GSL_DBL_EPSILON * fabs(result->val); |
| return stat_sin; |
| } |
| } |
| } |
| |
| |
| int gsl_sf_conicalP_sph_reg_e(const int l, const double lambda, |
| const double x, |
| gsl_sf_result * result |
| ) |
| { |
| /* CHECK_POINTER(result) */ |
| |
| if(x <= -1.0 || l < -1) { |
| DOMAIN_ERROR(result); |
| } |
| else if(l == -1) { |
| return gsl_sf_conicalP_half_e(lambda, x, result); |
| } |
| else if(l == 0) { |
| return gsl_sf_conicalP_mhalf_e(lambda, x, result); |
| } |
| else if(x == 1.0) { |
| result->val = 0.0; |
| result->err = 0.0; |
| return GSL_SUCCESS; |
| } |
| else if(x < 0.0) { |
| double c = 1.0/sqrt(1.0-x*x); |
| gsl_sf_result r_Pellm1; |
| gsl_sf_result r_Pell; |
| int stat_0 = gsl_sf_conicalP_half_e(lambda, x, &r_Pellm1); /* P^( 1/2) */ |
| int stat_1 = gsl_sf_conicalP_mhalf_e(lambda, x, &r_Pell); /* P^(-1/2) */ |
| int stat_P = GSL_ERROR_SELECT_2(stat_0, stat_1); |
| double Pellm1 = r_Pellm1.val; |
| double Pell = r_Pell.val; |
| double Pellp1; |
| int ell; |
| |
| for(ell=0; ell<l; ell++) { |
| double d = (ell+1.0)*(ell+1.0) + lambda*lambda; |
| Pellp1 = (Pellm1 - (2.0*ell+1.0)*c*x * Pell) / d; |
| Pellm1 = Pell; |
| Pell = Pellp1; |
| } |
| |
| result->val = Pell; |
| result->err = (0.5*l + 1.0) * GSL_DBL_EPSILON * fabs(Pell); |
| result->err += GSL_DBL_EPSILON * l * fabs(result->val); |
| return stat_P; |
| } |
| else if(x < 1.0) { |
| const double xi = x/(sqrt(1.0-x)*sqrt(1.0+x)); |
| gsl_sf_result rat; |
| gsl_sf_result Phf; |
| int stat_CF1 = conicalP_negmu_xlt1_CF1(0.5, l, lambda, x, &rat); |
| int stat_Phf = gsl_sf_conicalP_half_e(lambda, x, &Phf); |
| double Pellp1 = rat.val * GSL_SQRT_DBL_MIN; |
| double Pell = GSL_SQRT_DBL_MIN; |
| double Pellm1; |
| int ell; |
| |
| for(ell=l; ell>=0; ell--) { |
| double d = (ell+1.0)*(ell+1.0) + lambda*lambda; |
| Pellm1 = (2.0*ell+1.0)*xi * Pell + d * Pellp1; |
| Pellp1 = Pell; |
| Pell = Pellm1; |
| } |
| |
| result->val = GSL_SQRT_DBL_MIN * Phf.val / Pell; |
| result->err = GSL_SQRT_DBL_MIN * Phf.err / fabs(Pell); |
| result->err += fabs(rat.err/rat.val) * (l + 1.0) * fabs(result->val); |
| result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val); |
| |
| return GSL_ERROR_SELECT_2(stat_Phf, stat_CF1); |
| } |
| else if(x == 1.0) { |
| result->val = 0.0; |
| result->err = 0.0; |
| return GSL_SUCCESS; |
| } |
| else { |
| /* x > 1.0 */ |
| |
| const double xi = x/sqrt((x-1.0)*(x+1.0)); |
| gsl_sf_result rat; |
| int stat_CF1 = conicalP_negmu_xgt1_CF1(0.5, l, lambda, x, &rat); |
| int stat_P; |
| double Pellp1 = rat.val * GSL_SQRT_DBL_MIN; |
| double Pell = GSL_SQRT_DBL_MIN; |
| double Pellm1; |
| int ell; |
| |
| for(ell=l; ell>=0; ell--) { |
| double d = (ell+1.0)*(ell+1.0) + lambda*lambda; |
| Pellm1 = (2.0*ell+1.0)*xi * Pell - d * Pellp1; |
| Pellp1 = Pell; |
| Pell = Pellm1; |
| } |
| |
| if(fabs(Pell) > fabs(Pellp1)){ |
| gsl_sf_result Phf; |
| stat_P = gsl_sf_conicalP_half_e(lambda, x, &Phf); |
| result->val = GSL_SQRT_DBL_MIN * Phf.val / Pell; |
| result->err = 2.0 * GSL_SQRT_DBL_MIN * Phf.err / fabs(Pell); |
| result->err += 2.0 * fabs(rat.err/rat.val) * (l + 1.0) * fabs(result->val); |
| result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val); |
| } |
| else { |
| gsl_sf_result Pmhf; |
| stat_P = gsl_sf_conicalP_mhalf_e(lambda, x, &Pmhf); |
| result->val = GSL_SQRT_DBL_MIN * Pmhf.val / Pellp1; |
| result->err = 2.0 * GSL_SQRT_DBL_MIN * Pmhf.err / fabs(Pellp1); |
| result->err += 2.0 * fabs(rat.err/rat.val) * (l + 1.0) * fabs(result->val); |
| result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val); |
| } |
| |
| return GSL_ERROR_SELECT_2(stat_P, stat_CF1); |
| } |
| } |
| |
| |
| int gsl_sf_conicalP_cyl_reg_e(const int m, const double lambda, |
| const double x, |
| gsl_sf_result * result |
| ) |
| { |
| /* CHECK_POINTER(result) */ |
| |
| if(x <= -1.0 || m < -1) { |
| DOMAIN_ERROR(result); |
| } |
| else if(m == -1) { |
| return gsl_sf_conicalP_1_e(lambda, x, result); |
| } |
| else if(m == 0) { |
| return gsl_sf_conicalP_0_e(lambda, x, result); |
| } |
| else if(x == 1.0) { |
| result->val = 0.0; |
| result->err = 0.0; |
| return GSL_SUCCESS; |
| } |
| else if(x < 0.0) { |
| double c = 1.0/sqrt(1.0-x*x); |
| gsl_sf_result r_Pkm1; |
| gsl_sf_result r_Pk; |
| int stat_0 = gsl_sf_conicalP_1_e(lambda, x, &r_Pkm1); /* P^1 */ |
| int stat_1 = gsl_sf_conicalP_0_e(lambda, x, &r_Pk); /* P^0 */ |
| int stat_P = GSL_ERROR_SELECT_2(stat_0, stat_1); |
| double Pkm1 = r_Pkm1.val; |
| double Pk = r_Pk.val; |
| double Pkp1; |
| int k; |
| |
| for(k=0; k<m; k++) { |
| double d = (k+0.5)*(k+0.5) + lambda*lambda; |
| Pkp1 = (Pkm1 - 2.0*k*c*x * Pk) / d; |
| Pkm1 = Pk; |
| Pk = Pkp1; |
| } |
| |
| result->val = Pk; |
| result->err = (m + 2.0) * GSL_DBL_EPSILON * fabs(Pk); |
| result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val); |
| |
| return stat_P; |
| } |
| else if(x < 1.0) { |
| const double xi = x/(sqrt(1.0-x)*sqrt(1.0+x)); |
| gsl_sf_result rat; |
| gsl_sf_result P0; |
| int stat_CF1 = conicalP_negmu_xlt1_CF1(0.0, m, lambda, x, &rat); |
| int stat_P0 = gsl_sf_conicalP_0_e(lambda, x, &P0); |
| double Pkp1 = rat.val * GSL_SQRT_DBL_MIN; |
| double Pk = GSL_SQRT_DBL_MIN; |
| double Pkm1; |
| int k; |
| |
| for(k=m; k>0; k--) { |
| double d = (k+0.5)*(k+0.5) + lambda*lambda; |
| Pkm1 = 2.0*k*xi * Pk + d * Pkp1; |
| Pkp1 = Pk; |
| Pk = Pkm1; |
| } |
| |
| result->val = GSL_SQRT_DBL_MIN * P0.val / Pk; |
| result->err = 2.0 * GSL_SQRT_DBL_MIN * P0.err / fabs(Pk); |
| result->err += 2.0 * fabs(rat.err/rat.val) * (m + 1.0) * fabs(result->val); |
| result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val); |
| |
| return GSL_ERROR_SELECT_2(stat_P0, stat_CF1); |
| } |
| else if(x == 1.0) { |
| result->val = 0.0; |
| result->err = 0.0; |
| return GSL_SUCCESS; |
| } |
| else { |
| /* x > 1.0 */ |
| |
| const double xi = x/sqrt((x-1.0)*(x+1.0)); |
| gsl_sf_result rat; |
| int stat_CF1 = conicalP_negmu_xgt1_CF1(0.0, m, lambda, x, &rat); |
| int stat_P; |
| double Pkp1 = rat.val * GSL_SQRT_DBL_MIN; |
| double Pk = GSL_SQRT_DBL_MIN; |
| double Pkm1; |
| int k; |
| |
| for(k=m; k>-1; k--) { |
| double d = (k+0.5)*(k+0.5) + lambda*lambda; |
| Pkm1 = 2.0*k*xi * Pk - d * Pkp1; |
| Pkp1 = Pk; |
| Pk = Pkm1; |
| } |
| |
| if(fabs(Pk) > fabs(Pkp1)){ |
| gsl_sf_result P1; |
| stat_P = gsl_sf_conicalP_1_e(lambda, x, &P1); |
| result->val = GSL_SQRT_DBL_MIN * P1.val / Pk; |
| result->err = 2.0 * GSL_SQRT_DBL_MIN * P1.err / fabs(Pk); |
| result->err += 2.0 * fabs(rat.err/rat.val) * (m+2.0) * fabs(result->val); |
| result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val); |
| } |
| else { |
| gsl_sf_result P0; |
| stat_P = gsl_sf_conicalP_0_e(lambda, x, &P0); |
| result->val = GSL_SQRT_DBL_MIN * P0.val / Pkp1; |
| result->err = 2.0 * GSL_SQRT_DBL_MIN * P0.err / fabs(Pkp1); |
| result->err += 2.0 * fabs(rat.err/rat.val) * (m+2.0) * fabs(result->val); |
| result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val); |
| } |
| |
| return GSL_ERROR_SELECT_2(stat_P, stat_CF1); |
| } |
| } |
| |
| |
| /*-*-*-*-*-*-*-*-*-* Functions w/ Natural Prototypes *-*-*-*-*-*-*-*-*-*-*/ |
| |
| #include "eval.h" |
| |
| double gsl_sf_conicalP_0(const double lambda, const double x) |
| { |
| EVAL_RESULT(gsl_sf_conicalP_0_e(lambda, x, &result)); |
| } |
| |
| double gsl_sf_conicalP_1(const double lambda, const double x) |
| { |
| EVAL_RESULT(gsl_sf_conicalP_1_e(lambda, x, &result)); |
| } |
| |
| double gsl_sf_conicalP_half(const double lambda, const double x) |
| { |
| EVAL_RESULT(gsl_sf_conicalP_half_e(lambda, x, &result)); |
| } |
| |
| double gsl_sf_conicalP_mhalf(const double lambda, const double x) |
| { |
| EVAL_RESULT(gsl_sf_conicalP_mhalf_e(lambda, x, &result)); |
| } |
| |
| double gsl_sf_conicalP_sph_reg(const int l, const double lambda, const double x) |
| { |
| EVAL_RESULT(gsl_sf_conicalP_sph_reg_e(l, lambda, x, &result)); |
| } |
| |
| double gsl_sf_conicalP_cyl_reg(const int m, const double lambda, const double x) |
| { |
| EVAL_RESULT(gsl_sf_conicalP_cyl_reg_e(m, lambda, x, &result)); |
| } |
