/// Airy Ai function.
double
airy_ai(double x)
{
const gsl_mode_t mode = GSL_PREC_DOUBLE;
gsl_sf_result result;
int stat = gsl_sf_airy_Ai_e(x, mode, &result);
if (stat != GSL_SUCCESS)
{
std::ostringstream msg("Error in airy_ai:");
msg << " x=" << x;
throw std::runtime_error(msg.str());
}
else
return result.val;
}
/* uniform asymptotic, nu -> Inf, [Abramowitz+Stegun, 9.3.35]
*
* error:
* nu = 2: uniformly good to > 6D
* nu = 5: uniformly good to > 8D
* nu = 10: uniformly good to > 10D
* nu = 20: uniformly good to > 13D
*
*/
int gsl_sf_bessel_Jnu_asymp_Olver_e(double nu, double x, gsl_sf_result * result)
{
/* CHECK_POINTER(result) */
if(x <= 0.0 || nu <= 0.0) {
DOMAIN_ERROR(result);
}
else {
double zeta, abs_zeta;
double arg;
double pre;
double asum, bsum, asum_err;
gsl_sf_result ai;
gsl_sf_result aip;
double z = x/nu;
double crnu = pow(nu, 1.0/3.0);
double nu3 = nu*nu*nu;
double nu11 = nu3*nu3*nu3*nu*nu;
int stat_a, stat_ap;
if(fabs(1.0-z) < 0.02) {
const double a = 1.0-z;
const double c0 = 1.25992104989487316476721060728;
const double c1 = 0.37797631496846194943016318218;
const double c2 = 0.230385563409348235843147082474;
const double c3 = 0.165909603649648694839821892031;
const double c4 = 0.12931387086451008907;
const double c5 = 0.10568046188858133991;
const double c6 = 0.08916997952268186978;
const double c7 = 0.07700014900618802456;
pre = c0 + a*(c1 + a*(c2 + a*(c3 + a*(c4 + a*(c5 + a*(c6 + a*c7))))));
zeta = a * pre;
pre = sqrt(2.0*sqrt(pre/(1.0+z)));
abs_zeta = fabs(zeta);
}
else if(z < 1.0) {
double rt = sqrt(1.0 - z*z);
abs_zeta = pow(1.5*(log((1.0+rt)/z) - rt), 2.0/3.0);
zeta = abs_zeta;
pre = sqrt(2.0*sqrt(abs_zeta/(rt*rt)));
}
else {
/* z > 1 */
double rt = z * sqrt(1.0 - 1.0/(z*z));
abs_zeta = pow(1.5*(rt - acos(1.0/z)), 2.0/3.0);
zeta = -abs_zeta;
pre = sqrt(2.0*sqrt(abs_zeta/(rt*rt)));
}
asum = olver_Asum(nu, z, abs_zeta, &asum_err);
bsum = olver_Bsum(nu, z, abs_zeta);
arg = crnu*crnu * zeta;
stat_a = gsl_sf_airy_Ai_e(arg, GSL_MODE_DEFAULT, &ai);
stat_ap = gsl_sf_airy_Ai_deriv_e(arg, GSL_MODE_DEFAULT, &aip);
result->val = pre * (ai.val*asum/crnu + aip.val*bsum/(nu*crnu*crnu));
result->err = pre * (ai.err * fabs(asum/crnu));
result->err += pre * fabs(ai.val) * asum_err / crnu;
result->err += pre * fabs(ai.val * asum) / (crnu*nu11);
result->err += 8.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_ERROR_SELECT_2(stat_a, stat_ap);
}
}
static VALUE Airy_Ai_e(VALUE self, VALUE x, VALUE mode) {
int ret;
gsl_sf_result r;
ret = gsl_sf_airy_Ai_e(NUM2DBL(x), NUM2UINT(mode), &r);
return RESULT(&r);
}