/* 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; }
double gsl_sf_psi_1piy(const double x) { EVAL_RESULT(gsl_sf_psi_1piy_e(x, &result)); }
/* 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); }