/* Evaluate bessel_I(nu, x), allowing nu < 0.
* This is fine here because we do not not allow
* nu to be a negative integer.
* x > 0.
*/
static
int
hyperg_0F1_bessel_I(const double nu, const double x, gsl_sf_result * result)
{
if(x > GSL_LOG_DBL_MAX) {
OVERFLOW_ERROR(result);
}
if(nu < 0.0) {
const double anu = -nu;
const double s = 2.0/M_PI * sin(anu*M_PI);
const double ex = exp(x);
gsl_sf_result I;
gsl_sf_result K;
int stat_I = gsl_sf_bessel_Inu_scaled_e(anu, x, &I);
int stat_K = gsl_sf_bessel_Knu_scaled_e(anu, x, &K);
result->val = ex * I.val + s * (K.val / ex);
result->err = ex * I.err + fabs(s * K.err/ex);
result->err += fabs(s * (K.val/ex)) * GSL_DBL_EPSILON * anu * M_PI;
return GSL_ERROR_SELECT_2(stat_K, stat_I);
}
else {
const double ex = exp(x);
gsl_sf_result I;
int stat_I = gsl_sf_bessel_Inu_scaled_e(nu, x, &I);
result->val = ex * I.val;
result->err = ex * I.err + GSL_DBL_EPSILON * fabs(result->val);
return stat_I;
}
}
int
gsl_sf_bessel_lnKnu_e(const double nu, const double x, gsl_sf_result * result)
{
/* CHECK_POINTER(result) */
if(x <= 0.0 || nu < 0.0) {
DOMAIN_ERROR(result);
}
else if(nu == 0.0) {
gsl_sf_result K_scaled;
/* This cannot underflow, and
* it will not throw GSL_EDOM
* since that is already checked.
*/
gsl_sf_bessel_K0_scaled_e(x, &K_scaled);
result->val = -x + log(fabs(K_scaled.val));
result->err = GSL_DBL_EPSILON * fabs(x) + fabs(K_scaled.err/K_scaled.val);
result->err += GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
else if(x < 2.0 && nu > 1.0) {
/* Make use of the inequality
* Knu(x) <= 1/2 (2/x)^nu Gamma(nu),
* which follows from the integral representation
* [Abramowitz+Stegun, 9.6.23 (2)]. With this
* we decide whether or not there is an overflow
* problem because x is small.
*/
double ln_bound;
gsl_sf_result lg_nu;
gsl_sf_lngamma_e(nu, &lg_nu);
ln_bound = -M_LN2 - nu*log(0.5*x) + lg_nu.val;
if(ln_bound > GSL_LOG_DBL_MAX - 20.0) {
/* x must be very small or nu very large (or both).
*/
double xi = 0.25*x*x;
double sum = 1.0 - xi/(nu-1.0);
if(nu > 2.0) sum += (xi/(nu-1.0)) * (xi/(nu-2.0));
result->val = ln_bound + log(sum);
result->err = lg_nu.err;
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
/* can drop-through here */
}
{
/* We passed the above tests, so no problem.
* Evaluate as usual. Note the possible drop-through
* in the above code!
*/
gsl_sf_result K_scaled;
gsl_sf_bessel_Knu_scaled_e(nu, x, &K_scaled);
result->val = -x + log(fabs(K_scaled.val));
result->err = GSL_DBL_EPSILON * fabs(x) + fabs(K_scaled.err/K_scaled.val);
result->err += GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
}
int
gsl_sf_bessel_Knu_e(const double nu, const double x, gsl_sf_result * result)
{
gsl_sf_result b;
int stat_K = gsl_sf_bessel_Knu_scaled_e(nu, x, &b);
int stat_e = gsl_sf_exp_mult_err_e(-x, 0.0, b.val, b.err, result);
return GSL_ERROR_SELECT_2(stat_e, stat_K);
}
double gsl_sf_bessel_Knu_scaled(const double nu, const double x)
{
EVAL_RESULT(gsl_sf_bessel_Knu_scaled_e(nu, x, &result));
}
