/* specfunc/coulomb.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 */ /* Evaluation of Coulomb wave functions F_L(eta, x), G_L(eta, x), * and their derivatives. A combination of Steed's method, asymptotic * results, and power series. * * Steed's method: * [Barnett, CPC 21, 297 (1981)] * Power series and other methods: * [Biedenharn et al., PR 97, 542 (1954)] * [Bardin et al., CPC 3, 73 (1972)] * [Abad+Sesma, CPC 71, 110 (1992)] */ #include #include #include #include #include #include #include #include #include #include "error.h" /* the L=0 normalization constant * [Abramowitz+Stegun 14.1.8] */ static double C0sq(double eta) { double twopieta = 2.0*M_PI*eta; if(fabs(eta) < GSL_DBL_EPSILON) { return 1.0; } else if(twopieta > GSL_LOG_DBL_MAX) { return 0.0; } else { gsl_sf_result scale; gsl_sf_expm1_e(twopieta, &scale); return twopieta/scale.val; } } /* the full definition of C_L(eta) for any valid L and eta * [Abramowitz and Stegun 14.1.7] * This depends on the complex gamma function. For large * arguments the phase of the complex gamma function is not * very accurately determined. However the modulus is, and that * is all that we need to calculate C_L. * * This is not valid for L <= -3/2 or L = -1. */ static int CLeta(double L, double eta, gsl_sf_result * result) { gsl_sf_result ln1; /* log of numerator Gamma function */ gsl_sf_result ln2; /* log of denominator Gamma function */ double sgn = 1.0; double arg_val, arg_err; if(fabs(eta/(L+1.0)) < GSL_DBL_EPSILON) { gsl_sf_lngamma_e(L+1.0, &ln1); } else { gsl_sf_result p1; /* phase of numerator Gamma -- not used */ gsl_sf_lngamma_complex_e(L+1.0, eta, &ln1, &p1); /* should be ok */ } gsl_sf_lngamma_e(2.0*(L+1.0), &ln2); if(L < -1.0) sgn = -sgn; arg_val = L*M_LN2 - 0.5*eta*M_PI + ln1.val - ln2.val; arg_err = ln1.err + ln2.err; arg_err += GSL_DBL_EPSILON * (fabs(L*M_LN2) + fabs(0.5*eta*M_PI)); return gsl_sf_exp_err_e(arg_val, arg_err, result); } int gsl_sf_coulomb_CL_e(double lam, double eta, gsl_sf_result * result) { /* CHECK_POINTER(result) */ if(lam <= -1.0) { DOMAIN_ERROR(result); } else if(fabs(lam) < GSL_DBL_EPSILON) { /* saves a calculation of complex_lngamma(), otherwise not necessary */ result->val = sqrt(C0sq(eta)); result->err = 2.0 * GSL_DBL_EPSILON * result->val; return GSL_SUCCESS; } else { return CLeta(lam, eta, result); } } /* cl[0] .. cl[kmax] = C_{lam_min}(eta) .. C_{lam_min+kmax}(eta) */ int gsl_sf_coulomb_CL_array(double lam_min, int kmax, double eta, double * cl) { int k; gsl_sf_result cl_0; gsl_sf_coulomb_CL_e(lam_min, eta, &cl_0); cl[0] = cl_0.val; for(k=1; k<=kmax; k++) { double L = lam_min + k; cl[k] = cl[k-1] * hypot(L, eta)/(L*(2.0*L+1.0)); } return GSL_SUCCESS; } /* Evaluate the series for Phi_L(eta,x) and Phi_L*(eta,x) * [Abramowitz+Stegun 14.1.5] * [Abramowitz+Stegun 14.1.13] * * The sequence of coefficients A_k^L is * manifestly well-controlled for L >= -1/2 * and eta < 10. * * This makes sense since this is the region * away from threshold, and you expect * the evaluation to become easier as you * get farther from threshold. * * Empirically, this is quite well-behaved for * L >= -1/2 * eta < 10 * x < 10 */ #if 0 static int coulomb_Phi_series(const double lam, const double eta, const double x, double * result, double * result_star) { int kmin = 5; int kmax = 200; int k; double Akm2 = 1.0; double Akm1 = eta/(lam+1.0); double Ak; double xpow = x; double sum = Akm2 + Akm1*x; double sump = (lam+1.0)*Akm2 + (lam+2.0)*Akm1*x; double prev_abs_del = fabs(Akm1*x); double prev_abs_del_p = (lam+2.0) * prev_abs_del; for(k=2; k kmin ) break; /* We need to keep track of the previous delta because when * eta is near zero the odd terms of the sum are very small * and this could lead to premature termination. */ prev_abs_del = abs_del; prev_abs_del_p = abs_del_p; Akm2 = Akm1; Akm1 = Ak; } *result = sum; *result_star = sump; if(k==kmax) { GSL_ERROR ("error", GSL_EMAXITER); } else { return GSL_SUCCESS; } } #endif /* 0 */ /* Determine the connection phase, phi_lambda. * See coulomb_FG_series() below. We have * to be careful about sin(phi)->0. Note that * there is an underflow condition for large * positive eta in any case. */ static int coulomb_connection(const double lam, const double eta, double * cos_phi, double * sin_phi) { if(eta > -GSL_LOG_DBL_MIN/2.0*M_PI-1.0) { *cos_phi = 1.0; *sin_phi = 0.0; GSL_ERROR ("error", GSL_EUNDRFLW); } else if(eta > -GSL_LOG_DBL_EPSILON/(4.0*M_PI)) { const double eps = 2.0 * exp(-2.0*M_PI*eta); const double tpl = tan(M_PI * lam); const double dth = eps * tpl / (tpl*tpl + 1.0); *cos_phi = -1.0 + 0.5 * dth*dth; *sin_phi = -dth; return GSL_SUCCESS; } else { double X = tanh(M_PI * eta) / tan(M_PI * lam); double phi = -atan(X) - (lam + 0.5) * M_PI; *cos_phi = cos(phi); *sin_phi = sin(phi); return GSL_SUCCESS; } } /* Evaluate the Frobenius series for F_lam(eta,x) and G_lam(eta,x). * Homegrown algebra. Evaluates the series for F_{lam} and * F_{-lam-1}, then uses * G_{lam} = (F_{lam} cos(phi) - F_{-lam-1}) / sin(phi) * where * phi = Arg[Gamma[1+lam+I eta]] - Arg[Gamma[-lam + I eta]] - (lam+1/2)Pi * = Arg[Sin[Pi(-lam+I eta)] - (lam+1/2)Pi * = atan2(-cos(lam Pi)sinh(eta Pi), -sin(lam Pi)cosh(eta Pi)) - (lam+1/2)Pi * * = -atan(X) - (lam+1/2) Pi, X = tanh(eta Pi)/tan(lam Pi) * * Not appropriate for lam <= -1/2, lam = 0, or lam >= 1/2. */ static int coulomb_FG_series(const double lam, const double eta, const double x, gsl_sf_result * F, gsl_sf_result * G) { const int max_iter = 800; gsl_sf_result ClamA; gsl_sf_result ClamB; int stat_A = CLeta(lam, eta, &ClamA); int stat_B = CLeta(-lam-1.0, eta, &ClamB); const double tlp1 = 2.0*lam + 1.0; const double pow_x = pow(x, lam); double cos_phi_lam; double sin_phi_lam; double uA_mm2 = 1.0; /* uA sum is for F_{lam} */ double uA_mm1 = x*eta/(lam+1.0); double uA_m; double uB_mm2 = 1.0; /* uB sum is for F_{-lam-1} */ double uB_mm1 = -x*eta/lam; double uB_m; double A_sum = uA_mm2 + uA_mm1; double B_sum = uB_mm2 + uB_mm1; double A_abs_del_prev = fabs(A_sum); double B_abs_del_prev = fabs(B_sum); gsl_sf_result FA, FB; int m = 2; int stat_conn = coulomb_connection(lam, eta, &cos_phi_lam, &sin_phi_lam); if(stat_conn == GSL_EUNDRFLW) { F->val = 0.0; /* FIXME: should this be set to Inf too like G? */ F->err = 0.0; OVERFLOW_ERROR(G); } while(m < max_iter) { double abs_dA; double abs_dB; uA_m = x*(2.0*eta*uA_mm1 - x*uA_mm2)/(m*(m+tlp1)); uB_m = x*(2.0*eta*uB_mm1 - x*uB_mm2)/(m*(m-tlp1)); A_sum += uA_m; B_sum += uB_m; abs_dA = fabs(uA_m); abs_dB = fabs(uB_m); if(m > 15) { /* Don't bother checking until we have gone out a little ways; * a minor optimization. Also make sure to check both the * current and the previous increment because the odd and even * terms of the sum can have very different behaviour, depending * on the value of eta. */ double max_abs_dA = GSL_MAX(abs_dA, A_abs_del_prev); double max_abs_dB = GSL_MAX(abs_dB, B_abs_del_prev); double abs_A = fabs(A_sum); double abs_B = fabs(B_sum); if( max_abs_dA/(max_abs_dA + abs_A) < 4.0*GSL_DBL_EPSILON && max_abs_dB/(max_abs_dB + abs_B) < 4.0*GSL_DBL_EPSILON ) break; } A_abs_del_prev = abs_dA; B_abs_del_prev = abs_dB; uA_mm2 = uA_mm1; uA_mm1 = uA_m; uB_mm2 = uB_mm1; uB_mm1 = uB_m; m++; } FA.val = A_sum * ClamA.val * pow_x * x; FA.err = fabs(A_sum) * ClamA.err * pow_x * x + 2.0*GSL_DBL_EPSILON*fabs(FA.val); FB.val = B_sum * ClamB.val / pow_x; FB.err = fabs(B_sum) * ClamB.err / pow_x + 2.0*GSL_DBL_EPSILON*fabs(FB.val); F->val = FA.val; F->err = FA.err; G->val = (FA.val * cos_phi_lam - FB.val)/sin_phi_lam; G->err = (FA.err * fabs(cos_phi_lam) + FB.err)/fabs(sin_phi_lam); if(m >= max_iter) GSL_ERROR ("error", GSL_EMAXITER); else return GSL_ERROR_SELECT_2(stat_A, stat_B); } /* Evaluate the Frobenius series for F_0(eta,x) and G_0(eta,x). * See [Bardin et al., CPC 3, 73 (1972), (14)-(17)]; * note the misprint in (17): nu_0=1 is correct, not nu_0=0. */ static int coulomb_FG0_series(const double eta, const double x, gsl_sf_result * F, gsl_sf_result * G) { const int max_iter = 800; const double x2 = x*x; const double tex = 2.0*eta*x; gsl_sf_result C0; int stat_CL = CLeta(0.0, eta, &C0); gsl_sf_result r1pie; int psi_stat = gsl_sf_psi_1piy_e(eta, &r1pie); double u_mm2 = 0.0; /* u_0 */ double u_mm1 = x; /* u_1 */ double u_m; double v_mm2 = 1.0; /* nu_0 */ double v_mm1 = tex*(2.0*M_EULER-1.0+r1pie.val); /* nu_1 */ double v_m; double u_sum = u_mm2 + u_mm1; double v_sum = v_mm2 + v_mm1; double u_abs_del_prev = fabs(u_sum); double v_abs_del_prev = fabs(v_sum); int m = 2; double u_sum_err = 2.0 * GSL_DBL_EPSILON * fabs(u_sum); double v_sum_err = 2.0 * GSL_DBL_EPSILON * fabs(v_sum); double ln2x = log(2.0*x); while(m < max_iter) { double abs_du; double abs_dv; double m_mm1 = m*(m-1.0); u_m = (tex*u_mm1 - x2*u_mm2)/m_mm1; v_m = (tex*v_mm1 - x2*v_mm2 - 2.0*eta*(2*m-1)*u_m)/m_mm1; u_sum += u_m; v_sum += v_m; abs_du = fabs(u_m); abs_dv = fabs(v_m); u_sum_err += 2.0 * GSL_DBL_EPSILON * abs_du; v_sum_err += 2.0 * GSL_DBL_EPSILON * abs_dv; if(m > 15) { /* Don't bother checking until we have gone out a little ways; * a minor optimization. Also make sure to check both the * current and the previous increment because the odd and even * terms of the sum can have very different behaviour, depending * on the value of eta. */ double max_abs_du = GSL_MAX(abs_du, u_abs_del_prev); double max_abs_dv = GSL_MAX(abs_dv, v_abs_del_prev); double abs_u = fabs(u_sum); double abs_v = fabs(v_sum); if( max_abs_du/(max_abs_du + abs_u) < 40.0*GSL_DBL_EPSILON && max_abs_dv/(max_abs_dv + abs_v) < 40.0*GSL_DBL_EPSILON ) break; } u_abs_del_prev = abs_du; v_abs_del_prev = abs_dv; u_mm2 = u_mm1; u_mm1 = u_m; v_mm2 = v_mm1; v_mm1 = v_m; m++; } F->val = C0.val * u_sum; F->err = C0.err * fabs(u_sum); F->err += fabs(C0.val) * u_sum_err; F->err += 2.0 * GSL_DBL_EPSILON * fabs(F->val); G->val = (v_sum + 2.0*eta*u_sum * ln2x) / C0.val; G->err = (fabs(v_sum) + fabs(2.0*eta*u_sum * ln2x)) / fabs(C0.val) * fabs(C0.err/C0.val); G->err += (v_sum_err + fabs(2.0*eta*u_sum_err*ln2x)) / fabs(C0.val); G->err += 2.0 * GSL_DBL_EPSILON * fabs(G->val); if(m == max_iter) GSL_ERROR ("error", GSL_EMAXITER); else return GSL_ERROR_SELECT_2(psi_stat, stat_CL); } /* Evaluate the Frobenius series for F_{-1/2}(eta,x) and G_{-1/2}(eta,x). * Homegrown algebra. */ static int coulomb_FGmhalf_series(const double eta, const double x, gsl_sf_result * F, gsl_sf_result * G) { const int max_iter = 800; const double rx = sqrt(x); const double x2 = x*x; const double tex = 2.0*eta*x; gsl_sf_result Cmhalf; int stat_CL = CLeta(-0.5, eta, &Cmhalf); double u_mm2 = 1.0; /* u_0 */ double u_mm1 = tex * u_mm2; /* u_1 */ double u_m; double v_mm2, v_mm1, v_m; double f_sum, g_sum; double tmp1; gsl_sf_result rpsi_1pe; gsl_sf_result rpsi_1p2e; int m = 2; gsl_sf_psi_1piy_e(eta, &rpsi_1pe); gsl_sf_psi_1piy_e(2.0*eta, &rpsi_1p2e); v_mm2 = 2.0*M_EULER - M_LN2 - rpsi_1pe.val + 2.0*rpsi_1p2e.val; v_mm1 = tex*(v_mm2 - 2.0*u_mm2); f_sum = u_mm2 + u_mm1; g_sum = v_mm2 + v_mm1; while(m < max_iter) { double m2 = m*m; u_m = (tex*u_mm1 - x2*u_mm2)/m2; v_m = (tex*v_mm1 - x2*v_mm2 - 2.0*m*u_m)/m2; f_sum += u_m; g_sum += v_m; if( f_sum != 0.0 && g_sum != 0.0 && (fabs(u_m/f_sum) + fabs(v_m/g_sum) < 10.0*GSL_DBL_EPSILON)) break; u_mm2 = u_mm1; u_mm1 = u_m; v_mm2 = v_mm1; v_mm1 = v_m; m++; } F->val = Cmhalf.val * rx * f_sum; F->err = Cmhalf.err * fabs(rx * f_sum) + 2.0*GSL_DBL_EPSILON*fabs(F->val); tmp1 = f_sum*log(x); G->val = -rx*(tmp1 + g_sum)/Cmhalf.val; G->err = fabs(rx)*(fabs(tmp1) + fabs(g_sum))/fabs(Cmhalf.val) * fabs(Cmhalf.err/Cmhalf.val); if(m == max_iter) GSL_ERROR ("error", GSL_EMAXITER); else return stat_CL; } /* Evolve the backwards recurrence for F,F'. * * F_{lam-1} = (S_lam F_lam + F_lam') / R_lam * F_{lam-1}' = (S_lam F_{lam-1} - R_lam F_lam) * where * R_lam = sqrt(1 + (eta/lam)^2) * S_lam = lam/x + eta/lam * */ static int coulomb_F_recur(double lam_min, int kmax, double eta, double x, double F_lam_max, double Fp_lam_max, double * F_lam_min, double * Fp_lam_min ) { double x_inv = 1.0/x; double fcl = F_lam_max; double fpl = Fp_lam_max; double lam_max = lam_min + kmax; double lam = lam_max; int k; for(k=kmax-1; k>=0; k--) { double el = eta/lam; double rl = hypot(1.0, el); double sl = el + lam*x_inv; double fc_lm1; fc_lm1 = (fcl*sl + fpl)/rl; fpl = fc_lm1*sl - fcl*rl; fcl = fc_lm1; lam -= 1.0; } *F_lam_min = fcl; *Fp_lam_min = fpl; return GSL_SUCCESS; } /* Evolve the forward recurrence for G,G'. * * G_{lam+1} = (S_lam G_lam - G_lam')/R_lam * G_{lam+1}' = R_{lam+1} G_lam - S_lam G_{lam+1} * * where S_lam and R_lam are as above in the F recursion. */ static int coulomb_G_recur(const double lam_min, const int kmax, const double eta, const double x, const double G_lam_min, const double Gp_lam_min, double * G_lam_max, double * Gp_lam_max ) { double x_inv = 1.0/x; double gcl = G_lam_min; double gpl = Gp_lam_min; double lam = lam_min + 1.0; int k; for(k=1; k<=kmax; k++) { double el = eta/lam; double rl = hypot(1.0, el); double sl = el + lam*x_inv; double gcl1 = (sl*gcl - gpl)/rl; gpl = rl*gcl - sl*gcl1; gcl = gcl1; lam += 1.0; } *G_lam_max = gcl; *Gp_lam_max = gpl; return GSL_SUCCESS; } /* Evaluate the first continued fraction, giving * the ratio F'/F at the upper lambda value. * We also determine the sign of F at that point, * since it is the sign of the last denominator * in the continued fraction. */ static int coulomb_CF1(double lambda, double eta, double x, double * fcl_sign, double * result, int * count ) { const double CF1_small = 1.e-30; const double CF1_abort = 1.0e+05; const double CF1_acc = 2.0*GSL_DBL_EPSILON; const double x_inv = 1.0/x; const double px = lambda + 1.0 + CF1_abort; double pk = lambda + 1.0; double F = eta/pk + pk*x_inv; double D, C; double df; *fcl_sign = 1.0; *count = 0; if(fabs(F) < CF1_small) F = CF1_small; D = 0.0; C = F; do { double pk1 = pk + 1.0; double ek = eta / pk; double rk2 = 1.0 + ek*ek; double tk = (pk + pk1)*(x_inv + ek/pk1); D = tk - rk2 * D; C = tk - rk2 / C; if(fabs(C) < CF1_small) C = CF1_small; if(fabs(D) < CF1_small) D = CF1_small; D = 1.0/D; df = D * C; F = F * df; if(D < 0.0) { /* sign of result depends on sign of denominator */ *fcl_sign = - *fcl_sign; } pk = pk1; if( pk > px ) { *result = F; GSL_ERROR ("error", GSL_ERUNAWAY); } ++(*count); } while(fabs(df-1.0) > CF1_acc); *result = F; return GSL_SUCCESS; } #if 0 static int old_coulomb_CF1(const double lambda, double eta, double x, double * fcl_sign, double * result ) { const double CF1_abort = 1.e5; const double CF1_acc = 10.0*GSL_DBL_EPSILON; const double x_inv = 1.0/x; const double px = lambda + 1.0 + CF1_abort; double pk = lambda + 1.0; double D; double df; double F; double p; double pk1; double ek; double fcl = 1.0; double tk; while(1) { ek = eta/pk; F = (ek + pk*x_inv)*fcl + (fcl - 1.0)*x_inv; pk1 = pk + 1.0; if(fabs(eta*x + pk*pk1) > CF1_acc) break; fcl = (1.0 + ek*ek)/(1.0 + eta*eta/(pk1*pk1)); pk = 2.0 + pk; } D = 1.0/((pk + pk1)*(x_inv + ek/pk1)); df = -fcl*(1.0 + ek*ek)*D; if(fcl != 1.0) fcl = -1.0; if(D < 0.0) fcl = -fcl; F = F + df; p = 1.0; do { pk = pk1; pk1 = pk + 1.0; ek = eta / pk; tk = (pk + pk1)*(x_inv + ek/pk1); D = tk - D*(1.0+ek*ek); if(fabs(D) < sqrt(CF1_acc)) { p += 1.0; if(p > 2.0) { printf("HELP............\n"); } } D = 1.0/D; if(D < 0.0) { /* sign of result depends on sign of denominator */ fcl = -fcl; } df = df*(D*tk - 1.0); F = F + df; if( pk > px ) { GSL_ERROR ("error", GSL_ERUNAWAY); } } while(fabs(df) > fabs(F)*CF1_acc); *fcl_sign = fcl; *result = F; return GSL_SUCCESS; } #endif /* 0 */ /* Evaluate the second continued fraction to * obtain the ratio * (G' + i F')/(G + i F) := P + i Q * at the specified lambda value. */ static int coulomb_CF2(const double lambda, const double eta, const double x, double * result_P, double * result_Q, int * count ) { int status = GSL_SUCCESS; const double CF2_acc = 4.0*GSL_DBL_EPSILON; const double CF2_abort = 2.0e+05; const double wi = 2.0*eta; const double x_inv = 1.0/x; const double e2mm1 = eta*eta + lambda*(lambda + 1.0); double ar = -e2mm1; double ai = eta; double br = 2.0*(x - eta); double bi = 2.0; double dr = br/(br*br + bi*bi); double di = -bi/(br*br + bi*bi); double dp = -x_inv*(ar*di + ai*dr); double dq = x_inv*(ar*dr - ai*di); double A, B, C, D; double pk = 0.0; double P = 0.0; double Q = 1.0 - eta*x_inv; *count = 0; do { P += dp; Q += dq; pk += 2.0; ar += pk; ai += wi; bi += 2.0; D = ar*dr - ai*di + br; di = ai*dr + ar*di + bi; C = 1.0/(D*D + di*di); dr = C*D; di = -C*di; A = br*dr - bi*di - 1.; B = bi*dr + br*di; C = dp*A - dq*B; dq = dp*B + dq*A; dp = C; if(pk > CF2_abort) { status = GSL_ERUNAWAY; break; } ++(*count); } while(fabs(dp)+fabs(dq) > (fabs(P)+fabs(Q))*CF2_acc); if(Q < CF2_abort*GSL_DBL_EPSILON*fabs(P)) { status = GSL_ELOSS; } *result_P = P; *result_Q = Q; return status; } /* WKB evaluation of F, G. Assumes 0 < x < turning point. * Overflows are trapped, GSL_EOVRFLW is signalled, * and an exponent is returned such that: * * result_F = fjwkb * exp(-exponent) * result_G = gjwkb * exp( exponent) * * See [Biedenharn et al. Phys. Rev. 97, 542-554 (1955), Section IV] * * Unfortunately, this is not very accurate in general. The * test cases typically have 3-4 digits of precision. One could * argue that this is ok for general use because, for instance, * F is exponentially small in this region and so the absolute * accuracy is still roughly acceptable. But it would be better * to have a systematic method for improving the precision. See * the Abad+Sesma method discussion below. */ static int coulomb_jwkb(const double lam, const double eta, const double x, gsl_sf_result * fjwkb, gsl_sf_result * gjwkb, double * exponent) { const double llp1 = lam*(lam+1.0) + 6.0/35.0; const double llp1_eff = GSL_MAX(llp1, 0.0); const double rho_ghalf = sqrt(x*(2.0*eta - x) + llp1_eff); const double sinh_arg = sqrt(llp1_eff/(eta*eta+llp1_eff)) * rho_ghalf / x; const double sinh_inv = log(sinh_arg + hypot(1.0,sinh_arg)); const double phi = fabs(rho_ghalf - eta*atan2(rho_ghalf,x-eta) - sqrt(llp1_eff) * sinh_inv); const double zeta_half = pow(3.0*phi/2.0, 1.0/3.0); const double prefactor = sqrt(M_PI*phi*x/(6.0 * rho_ghalf)); double F = prefactor * 3.0/zeta_half; double G = prefactor * 3.0/zeta_half; /* Note the sqrt(3) from Bi normalization */ double F_exp; double G_exp; const double airy_scale_exp = phi; gsl_sf_result ai; gsl_sf_result bi; gsl_sf_airy_Ai_scaled_e(zeta_half*zeta_half, GSL_MODE_DEFAULT, &ai); gsl_sf_airy_Bi_scaled_e(zeta_half*zeta_half, GSL_MODE_DEFAULT, &bi); F *= ai.val; G *= bi.val; F_exp = log(F) - airy_scale_exp; G_exp = log(G) + airy_scale_exp; if(G_exp >= GSL_LOG_DBL_MAX) { fjwkb->val = F; gjwkb->val = G; fjwkb->err = 1.0e-3 * fabs(F); /* FIXME: real error here ... could be smaller */ gjwkb->err = 1.0e-3 * fabs(G); *exponent = airy_scale_exp; GSL_ERROR ("error", GSL_EOVRFLW); } else { fjwkb->val = exp(F_exp); gjwkb->val = exp(G_exp); fjwkb->err = 1.0e-3 * fabs(fjwkb->val); gjwkb->err = 1.0e-3 * fabs(gjwkb->val); *exponent = 0.0; return GSL_SUCCESS; } } /* Asymptotic evaluation of F and G below the minimal turning point. * * This is meant to be a drop-in replacement for coulomb_jwkb(). * It uses the expressions in [Abad+Sesma]. This requires some * work because I am not sure where it is valid. They mumble * something about |x| < |lam|^(-1/2) or 8|eta x| > lam when |x| < 1. * This seems true, but I thought the result was based on a uniform * expansion and could be controlled by simply using more terms. */ #if 0 static int coulomb_AS_xlt2eta(const double lam, const double eta, const double x, gsl_sf_result * f_AS, gsl_sf_result * g_AS, double * exponent) { /* no time to do this now... */ } #endif /* 0 */ /*-*-*-*-*-*-*-*-*-*-*-* Functions with Error Codes *-*-*-*-*-*-*-*-*-*-*-*/ int gsl_sf_coulomb_wave_FG_e(const double eta, const double x, const double lam_F, const int k_lam_G, /* lam_G = lam_F - k_lam_G */ gsl_sf_result * F, gsl_sf_result * Fp, gsl_sf_result * G, gsl_sf_result * Gp, double * exp_F, double * exp_G) { const double lam_G = lam_F - k_lam_G; if(x < 0.0 || lam_F <= -0.5 || lam_G <= -0.5) { GSL_SF_RESULT_SET(F, 0.0, 0.0); GSL_SF_RESULT_SET(Fp, 0.0, 0.0); GSL_SF_RESULT_SET(G, 0.0, 0.0); GSL_SF_RESULT_SET(Gp, 0.0, 0.0); *exp_F = 0.0; *exp_G = 0.0; GSL_ERROR ("domain error", GSL_EDOM); } else if(x == 0.0) { gsl_sf_result C0; CLeta(0.0, eta, &C0); GSL_SF_RESULT_SET(F, 0.0, 0.0); GSL_SF_RESULT_SET(Fp, 0.0, 0.0); GSL_SF_RESULT_SET(G, 0.0, 0.0); /* FIXME: should be Inf */ GSL_SF_RESULT_SET(Gp, 0.0, 0.0); /* FIXME: should be Inf */ *exp_F = 0.0; *exp_G = 0.0; if(lam_F == 0.0){ GSL_SF_RESULT_SET(Fp, C0.val, C0.err); } if(lam_G == 0.0) { GSL_SF_RESULT_SET(Gp, 1.0/C0.val, fabs(C0.err/C0.val)/fabs(C0.val)); } GSL_ERROR ("domain error", GSL_EDOM); /* After all, since we are asking for G, this is a domain error... */ } else if(x < 1.2 && 2.0*M_PI*eta < 0.9*(-GSL_LOG_DBL_MIN) && fabs(eta*x) < 10.0) { /* Reduce to a small lambda value and use the series * representations for F and G. We cannot allow eta to * be large and positive because the connection formula * for G_lam is badly behaved due to an underflow in sin(phi_lam) * [see coulomb_FG_series() and coulomb_connection() above]. * Note that large negative eta is ok however. */ const double SMALL = GSL_SQRT_DBL_EPSILON; const int N = (int)(lam_F + 0.5); const int span = GSL_MAX(k_lam_G, N); const double lam_min = lam_F - N; /* -1/2 <= lam_min < 1/2 */ double F_lam_F, Fp_lam_F; double G_lam_G = 0.0, Gp_lam_G = 0.0; double F_lam_F_err, Fp_lam_F_err; double Fp_over_F_lam_F; double F_sign_lam_F; double F_lam_min_unnorm, Fp_lam_min_unnorm; double Fp_over_F_lam_min; gsl_sf_result F_lam_min; gsl_sf_result G_lam_min, Gp_lam_min; double F_scale; double Gerr_frac; double F_scale_frac_err; double F_unnorm_frac_err; /* Determine F'/F at lam_F. */ int CF1_count; int stat_CF1 = coulomb_CF1(lam_F, eta, x, &F_sign_lam_F, &Fp_over_F_lam_F, &CF1_count); int stat_ser; int stat_Fr; int stat_Gr; /* Recurse down with unnormalized F,F' values. */ F_lam_F = SMALL; Fp_lam_F = Fp_over_F_lam_F * F_lam_F; if(span != 0) { stat_Fr = coulomb_F_recur(lam_min, span, eta, x, F_lam_F, Fp_lam_F, &F_lam_min_unnorm, &Fp_lam_min_unnorm ); } else { F_lam_min_unnorm = F_lam_F; Fp_lam_min_unnorm = Fp_lam_F; stat_Fr = GSL_SUCCESS; } /* Determine F and G at lam_min. */ if(lam_min == -0.5) { stat_ser = coulomb_FGmhalf_series(eta, x, &F_lam_min, &G_lam_min); } else if(lam_min == 0.0) { stat_ser = coulomb_FG0_series(eta, x, &F_lam_min, &G_lam_min); } else if(lam_min == 0.5) { /* This cannot happen. */ F->val = F_lam_F; F->err = 2.0 * GSL_DBL_EPSILON * fabs(F->val); Fp->val = Fp_lam_F; Fp->err = 2.0 * GSL_DBL_EPSILON * fabs(Fp->val); G->val = G_lam_G; G->err = 2.0 * GSL_DBL_EPSILON * fabs(G->val); Gp->val = Gp_lam_G; Gp->err = 2.0 * GSL_DBL_EPSILON * fabs(Gp->val); *exp_F = 0.0; *exp_G = 0.0; GSL_ERROR ("error", GSL_ESANITY); } else { stat_ser = coulomb_FG_series(lam_min, eta, x, &F_lam_min, &G_lam_min); } /* Determine remaining quantities. */ Fp_over_F_lam_min = Fp_lam_min_unnorm / F_lam_min_unnorm; Gp_lam_min.val = Fp_over_F_lam_min*G_lam_min.val - 1.0/F_lam_min.val; Gp_lam_min.err = fabs(Fp_over_F_lam_min)*G_lam_min.err; Gp_lam_min.err += fabs(1.0/F_lam_min.val) * fabs(F_lam_min.err/F_lam_min.val); F_scale = F_lam_min.val / F_lam_min_unnorm; /* Apply scale to the original F,F' values. */ F_scale_frac_err = fabs(F_lam_min.err/F_lam_min.val); F_unnorm_frac_err = 2.0*GSL_DBL_EPSILON*(CF1_count+span+1); F_lam_F *= F_scale; F_lam_F_err = fabs(F_lam_F) * (F_unnorm_frac_err + F_scale_frac_err); Fp_lam_F *= F_scale; Fp_lam_F_err = fabs(Fp_lam_F) * (F_unnorm_frac_err + F_scale_frac_err); /* Recurse up to get the required G,G' values. */ stat_Gr = coulomb_G_recur(lam_min, GSL_MAX(N-k_lam_G,0), eta, x, G_lam_min.val, Gp_lam_min.val, &G_lam_G, &Gp_lam_G ); F->val = F_lam_F; F->err = F_lam_F_err; F->err += 2.0 * GSL_DBL_EPSILON * fabs(F_lam_F); Fp->val = Fp_lam_F; Fp->err = Fp_lam_F_err; Fp->err += 2.0 * GSL_DBL_EPSILON * fabs(Fp_lam_F); Gerr_frac = fabs(G_lam_min.err/G_lam_min.val) + fabs(Gp_lam_min.err/Gp_lam_min.val); G->val = G_lam_G; G->err = Gerr_frac * fabs(G_lam_G); G->err += 2.0 * (CF1_count+1) * GSL_DBL_EPSILON * fabs(G->val); Gp->val = Gp_lam_G; Gp->err = Gerr_frac * fabs(Gp->val); Gp->err += 2.0 * (CF1_count+1) * GSL_DBL_EPSILON * fabs(Gp->val); *exp_F = 0.0; *exp_G = 0.0; return GSL_ERROR_SELECT_4(stat_ser, stat_CF1, stat_Fr, stat_Gr); } else if(x < 2.0*eta) { /* Use WKB approximation to obtain F and G at the two * lambda values, and use the Wronskian and the * continued fractions for F'/F to obtain F' and G'. */ gsl_sf_result F_lam_F, G_lam_F; gsl_sf_result F_lam_G, G_lam_G; double exp_lam_F, exp_lam_G; int stat_lam_F; int stat_lam_G; int stat_CF1_lam_F; int stat_CF1_lam_G; int CF1_count; double Fp_over_F_lam_F; double Fp_over_F_lam_G; double F_sign_lam_F; double F_sign_lam_G; stat_lam_F = coulomb_jwkb(lam_F, eta, x, &F_lam_F, &G_lam_F, &exp_lam_F); if(k_lam_G == 0) { stat_lam_G = stat_lam_F; F_lam_G = F_lam_F; G_lam_G = G_lam_F; exp_lam_G = exp_lam_F; } else { stat_lam_G = coulomb_jwkb(lam_G, eta, x, &F_lam_G, &G_lam_G, &exp_lam_G); } stat_CF1_lam_F = coulomb_CF1(lam_F, eta, x, &F_sign_lam_F, &Fp_over_F_lam_F, &CF1_count); if(k_lam_G == 0) { stat_CF1_lam_G = stat_CF1_lam_F; F_sign_lam_G = F_sign_lam_F; Fp_over_F_lam_G = Fp_over_F_lam_F; } else { stat_CF1_lam_G = coulomb_CF1(lam_G, eta, x, &F_sign_lam_G, &Fp_over_F_lam_G, &CF1_count); } F->val = F_lam_F.val; F->err = F_lam_F.err; G->val = G_lam_G.val; G->err = G_lam_G.err; Fp->val = Fp_over_F_lam_F * F_lam_F.val; Fp->err = fabs(Fp_over_F_lam_F) * F_lam_F.err; Fp->err += 2.0*GSL_DBL_EPSILON*fabs(Fp->val); Gp->val = Fp_over_F_lam_G * G_lam_G.val - 1.0/F_lam_G.val; Gp->err = fabs(Fp_over_F_lam_G) * G_lam_G.err; Gp->err += fabs(1.0/F_lam_G.val) * fabs(F_lam_G.err/F_lam_G.val); *exp_F = exp_lam_F; *exp_G = exp_lam_G; if(stat_lam_F == GSL_EOVRFLW || stat_lam_G == GSL_EOVRFLW) { GSL_ERROR ("overflow", GSL_EOVRFLW); } else { return GSL_ERROR_SELECT_2(stat_lam_F, stat_lam_G); } } else { /* x > 2 eta, so we know that we can find a lambda value such * that x is above the turning point. We do this, evaluate * using Steed's method at that oscillatory point, then * use recursion on F and G to obtain the required values. * * lam_0 = a value of lambda such that x is below the turning point * lam_min = minimum of lam_0 and the requested lam_G, since * we must go at least as low as lam_G */ const double SMALL = GSL_SQRT_DBL_EPSILON; const double C = sqrt(1.0 + 4.0*x*(x-2.0*eta)); const int N = ceil(lam_F - C + 0.5); const double lam_0 = lam_F - GSL_MAX(N, 0); const double lam_min = GSL_MIN(lam_0, lam_G); double F_lam_F, Fp_lam_F; double G_lam_G, Gp_lam_G; double F_lam_min_unnorm, Fp_lam_min_unnorm; double F_lam_min, Fp_lam_min; double G_lam_min, Gp_lam_min; double Fp_over_F_lam_F; double Fp_over_F_lam_min; double F_sign_lam_F, F_sign_lam_min; double P_lam_min, Q_lam_min; double alpha; double gamma; double F_scale; int CF1_count; int CF2_count; int stat_CF1 = coulomb_CF1(lam_F, eta, x, &F_sign_lam_F, &Fp_over_F_lam_F, &CF1_count); int stat_CF2; int stat_Fr; int stat_Gr; int F_recur_count; int G_recur_count; double err_amplify; F_lam_F = F_sign_lam_F * SMALL; /* unnormalized */ Fp_lam_F = Fp_over_F_lam_F * F_lam_F; /* Backward recurrence to get F,Fp at lam_min */ F_recur_count = GSL_MAX(k_lam_G, N); stat_Fr = coulomb_F_recur(lam_min, F_recur_count, eta, x, F_lam_F, Fp_lam_F, &F_lam_min_unnorm, &Fp_lam_min_unnorm ); Fp_over_F_lam_min = Fp_lam_min_unnorm / F_lam_min_unnorm; /* Steed evaluation to complete evaluation of F,Fp,G,Gp at lam_min */ stat_CF2 = coulomb_CF2(lam_min, eta, x, &P_lam_min, &Q_lam_min, &CF2_count); alpha = Fp_over_F_lam_min - P_lam_min; gamma = alpha/Q_lam_min; F_sign_lam_min = GSL_SIGN(F_lam_min_unnorm) ; F_lam_min = F_sign_lam_min / sqrt(alpha*alpha/Q_lam_min + Q_lam_min); Fp_lam_min = Fp_over_F_lam_min * F_lam_min; G_lam_min = gamma * F_lam_min; Gp_lam_min = (P_lam_min * gamma - Q_lam_min) * F_lam_min; /* Apply scale to values of F,Fp at lam_F (the top). */ F_scale = F_lam_min / F_lam_min_unnorm; F_lam_F *= F_scale; Fp_lam_F *= F_scale; /* Forward recurrence to get G,Gp at lam_G (the top). */ G_recur_count = GSL_MAX(N-k_lam_G,0); stat_Gr = coulomb_G_recur(lam_min, G_recur_count, eta, x, G_lam_min, Gp_lam_min, &G_lam_G, &Gp_lam_G ); err_amplify = CF1_count + CF2_count + F_recur_count + G_recur_count + 1; F->val = F_lam_F; F->err = 8.0*err_amplify*GSL_DBL_EPSILON * fabs(F->val); Fp->val = Fp_lam_F; Fp->err = 8.0*err_amplify*GSL_DBL_EPSILON * fabs(Fp->val); G->val = G_lam_G; G->err = 8.0*err_amplify*GSL_DBL_EPSILON * fabs(G->val); Gp->val = Gp_lam_G; Gp->err = 8.0*err_amplify*GSL_DBL_EPSILON * fabs(Gp->val); *exp_F = 0.0; *exp_G = 0.0; return GSL_ERROR_SELECT_4(stat_CF1, stat_CF2, stat_Fr, stat_Gr); } } int gsl_sf_coulomb_wave_F_array(double lam_min, int kmax, double eta, double x, double * fc_array, double * F_exp) { if(x == 0.0) { int k; *F_exp = 0.0; for(k=0; k<=kmax; k++) { fc_array[k] = 0.0; } if(lam_min == 0.0){ gsl_sf_result f_0; CLeta(0.0, eta, &f_0); fc_array[0] = f_0.val; } return GSL_SUCCESS; } else { const double x_inv = 1.0/x; const double lam_max = lam_min + kmax; gsl_sf_result F, Fp; gsl_sf_result G, Gp; double G_exp; int stat_FG = gsl_sf_coulomb_wave_FG_e(eta, x, lam_max, 0, &F, &Fp, &G, &Gp, F_exp, &G_exp); double fcl = F.val; double fpl = Fp.val; double lam = lam_max; int k; fc_array[kmax] = F.val; for(k=kmax-1; k>=0; k--) { double el = eta/lam; double rl = hypot(1.0, el); double sl = el + lam*x_inv; double fc_lm1 = (fcl*sl + fpl)/rl; fc_array[k] = fc_lm1; fpl = fc_lm1*sl - fcl*rl; fcl = fc_lm1; lam -= 1.0; } return stat_FG; } } int gsl_sf_coulomb_wave_FG_array(double lam_min, int kmax, double eta, double x, double * fc_array, double * gc_array, double * F_exp, double * G_exp) { const double x_inv = 1.0/x; const double lam_max = lam_min + kmax; gsl_sf_result F, Fp; gsl_sf_result G, Gp; int stat_FG = gsl_sf_coulomb_wave_FG_e(eta, x, lam_max, kmax, &F, &Fp, &G, &Gp, F_exp, G_exp); double fcl = F.val; double fpl = Fp.val; double lam = lam_max; int k; double gcl, gpl; fc_array[kmax] = F.val; for(k=kmax-1; k>=0; k--) { double el = eta/lam; double rl = hypot(1.0, el); double sl = el + lam*x_inv; double fc_lm1; fc_lm1 = (fcl*sl + fpl)/rl; fc_array[k] = fc_lm1; fpl = fc_lm1*sl - fcl*rl; fcl = fc_lm1; lam -= 1.0; } gcl = G.val; gpl = Gp.val; lam = lam_min + 1.0; gc_array[0] = G.val; for(k=1; k<=kmax; k++) { double el = eta/lam; double rl = hypot(1.0, el); double sl = el + lam*x_inv; double gcl1 = (sl*gcl - gpl)/rl; gc_array[k] = gcl1; gpl = rl*gcl - sl*gcl1; gcl = gcl1; lam += 1.0; } return stat_FG; } int gsl_sf_coulomb_wave_FGp_array(double lam_min, int kmax, double eta, double x, double * fc_array, double * fcp_array, double * gc_array, double * gcp_array, double * F_exp, double * G_exp) { const double x_inv = 1.0/x; const double lam_max = lam_min + kmax; gsl_sf_result F, Fp; gsl_sf_result G, Gp; int stat_FG = gsl_sf_coulomb_wave_FG_e(eta, x, lam_max, kmax, &F, &Fp, &G, &Gp, F_exp, G_exp); double fcl = F.val; double fpl = Fp.val; double lam = lam_max; int k; double gcl, gpl; fc_array[kmax] = F.val; fcp_array[kmax] = Fp.val; for(k=kmax-1; k>=0; k--) { double el = eta/lam; double rl = hypot(1.0, el); double sl = el + lam*x_inv; double fc_lm1; fc_lm1 = (fcl*sl + fpl)/rl; fc_array[k] = fc_lm1; fpl = fc_lm1*sl - fcl*rl; fcp_array[k] = fpl; fcl = fc_lm1; lam -= 1.0; } gcl = G.val; gpl = Gp.val; lam = lam_min + 1.0; gc_array[0] = G.val; gcp_array[0] = Gp.val; for(k=1; k<=kmax; k++) { double el = eta/lam; double rl = hypot(1.0, el); double sl = el + lam*x_inv; double gcl1 = (sl*gcl - gpl)/rl; gc_array[k] = gcl1; gpl = rl*gcl - sl*gcl1; gcp_array[k] = gpl; gcl = gcl1; lam += 1.0; } return stat_FG; } int gsl_sf_coulomb_wave_sphF_array(double lam_min, int kmax, double eta, double x, double * fc_array, double * F_exp) { if(x < 0.0 || lam_min < -0.5) { GSL_ERROR ("domain error", GSL_EDOM); } else if(x < 10.0/GSL_DBL_MAX) { int k; for(k=0; k<=kmax; k++) { fc_array[k] = 0.0; } if(lam_min == 0.0) { fc_array[0] = sqrt(C0sq(eta)); } *F_exp = 0.0; if(x == 0.0) return GSL_SUCCESS; else GSL_ERROR ("underflow", GSL_EUNDRFLW); } else { int k; int stat_F = gsl_sf_coulomb_wave_F_array(lam_min, kmax, eta, x, fc_array, F_exp); for(k=0; k<=kmax; k++) { fc_array[k] = fc_array[k] / x; } return stat_F; } }