/* 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);
}
