/* [Carlson, Numer. Math. 33 (1979) 1, (4.6)] */
int
gsl_sf_ellint_Ecomp_e(double k, gsl_mode_t mode, gsl_sf_result * result)
{
if(k*k >= 1.0) {
DOMAIN_ERROR(result);
}
else if(k*k >= 1.0 - GSL_SQRT_DBL_EPSILON) {
/* [Abramowitz+Stegun, 17.3.36] */
const double y = 1.0 - k*k;
const double a[] = { 0.44325141463, 0.06260601220, 0.04757383546 };
const double b[] = { 0.24998368310, 0.09200180037, 0.04069697526 };
const double ta = 1.0 + y*(a[0] + y*(a[1] + a[2]*y));
const double tb = -y*log(y) * (b[0] + y*(b[1] + b[2]*y));
result->val = ta + tb;
result->err = 2.0 * GSL_DBL_EPSILON * result->val;
return GSL_SUCCESS;
}
else {
gsl_sf_result rf;
gsl_sf_result rd;
const double y = 1.0 - k*k;
const int rfstatus = gsl_sf_ellint_RF_e(0.0, y, 1.0, mode, &rf);
const int rdstatus = gsl_sf_ellint_RD_e(0.0, y, 1.0, mode, &rd);
result->val = rf.val - k*k/3.0 * rd.val;
result->err = rf.err + k*k/3.0 * rd.err;
return GSL_ERROR_SELECT_2(rfstatus, rdstatus);
}
}
/* [Carlson, Numer. Math. 33 (1979) 1, (4.4)] */
int
gsl_sf_ellint_D_e(double phi, double k, double n, gsl_mode_t mode, gsl_sf_result * result)
{
const double sin_phi = sin(phi);
const double sin2_phi = sin_phi * sin_phi;
const double sin3_phi = sin2_phi * sin_phi;
const double x = 1.0 - sin2_phi;
const double y = 1.0 - k*k*sin2_phi;
gsl_sf_result rd;
const int status = gsl_sf_ellint_RD_e(x, y, 1.0, mode, &rd);
result->val = sin3_phi/3.0 * rd.val;
result->err = GSL_DBL_EPSILON * fabs(result->val) + fabs(sin3_phi/3.0 * rd.err);
return status;
}
/// Carlson elliptic integrals R_D.
double
ellint_rd(double x, double y, double z)
{
if (x == 0.0 && y == 0.0 && z == 0.0)
return 0.0;
const gsl_mode_t mode = GSL_PREC_DOUBLE;
gsl_sf_result result;
int stat = gsl_sf_ellint_RD_e(x, y, z, mode, &result);
if (stat != GSL_SUCCESS)
{
std::ostringstream msg("Error in ellint_rd:");
msg << " x=" << x << " y=" << y << " z=" << z;
throw std::runtime_error(msg.str());
}
else
return result.val;
}
/* [Carlson, Numer. Math. 33 (1979) 1, (4.2)] */
int
gsl_sf_ellint_E_e(double phi, double k, gsl_mode_t mode, gsl_sf_result * result)
{
const double sin_phi = sin(phi);
const double sin2_phi = sin_phi * sin_phi;
const double x = 1.0 - sin2_phi;
const double y = 1.0 - k*k*sin2_phi;
if(x < GSL_DBL_EPSILON) {
return gsl_sf_ellint_Ecomp_e(k, mode, result);
}
else {
gsl_sf_result rf;
gsl_sf_result rd;
const double sin3_phi = sin2_phi * sin_phi;
const int rfstatus = gsl_sf_ellint_RF_e(x, y, 1.0, mode, &rf);
const int rdstatus = gsl_sf_ellint_RD_e(x, y, 1.0, mode, &rd);
result->val = sin_phi * rf.val - k*k/3.0 * sin3_phi * rd.val;
result->err = GSL_DBL_EPSILON * fabs(sin_phi * rf.val);
result->err += fabs(sin_phi*rf.err);
result->err += k*k/3.0 * GSL_DBL_EPSILON * fabs(sin3_phi * rd.val);
result->err += k*k/3.0 * fabs(sin3_phi*rd.err);
return GSL_ERROR_SELECT_2(rfstatus, rdstatus);
}
}
double gsl_sf_ellint_RD(double x, double y, double z, gsl_mode_t mode)
{
EVAL_RESULT(gsl_sf_ellint_RD_e(x, y, z, mode, &result));
}
/**
* C++ version of gsl_sf_ellint_RD_e().
* Carlson's symmetric basis of functions
*
* RD(x,y,z) = 3/2 Integral[(t+x)^(-1/2) (t+y)^(-1/2) (t+z)^(-3/2), {t,0,Inf}]
* @param x A real number
* @param y A real number
* @param z A real number
* @param mode The mode
* @param result The result as a @c gsl::sf::result object
* @return GSL_SUCCESS or GSL_EDOM
*/
inline int RD_e( double x, double y, double z, mode_t mode, result& result ){
return gsl_sf_ellint_RD_e( x, y, z, mode, &result ); }
