/* [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 ); }