/* 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 <config.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_sf_exp.h>
#include <gsl/gsl_sf_psi.h>
#include <gsl/gsl_sf_airy.h>
#include <gsl/gsl_sf_pow_int.h>
#include <gsl/gsl_sf_gamma.h>
#include <gsl/gsl_sf_coulomb.h>
#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<kmax; k++) {
double del;
double del_p;
double abs_del;
double abs_del_p;
Ak = (2.0*eta*Akm1 - Akm2)/(k*(2.0*lam + 1.0 + k));
xpow *= x;
del = Ak*xpow;
del_p = (k+lam+1.0)*del;
sum += del;
sump += del_p;
abs_del = fabs(del);
abs_del_p = fabs(del_p);
if( abs_del/(fabs(sum)+abs_del) < GSL_DBL_EPSILON
&& prev_abs_del/(fabs(sum)+prev_abs_del) < GSL_DBL_EPSILON
&& abs_del_p/(fabs(sump)+abs_del_p) < GSL_DBL_EPSILON
&& prev_abs_del_p/(fabs(sump)+prev_abs_del_p) < GSL_DBL_EPSILON
&& 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, Gp_lam_G;
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;
}
}