/* 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 + sqrt(1.0 + sinh_arg*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;
}
}
static VALUE Airy_Ai_scaled_e(VALUE self, VALUE x, VALUE mode) {
int ret;
gsl_sf_result r;
ret = gsl_sf_airy_Ai_scaled_e(NUM2DBL(x), NUM2UINT(mode), &r);
return RESULT(&r);
}
