int
gsl_sf_bessel_Ynu_e(double nu, double x, gsl_sf_result * result)
{
/* CHECK_POINTER(result) */
if(x <= 0.0 || nu < 0.0) {
DOMAIN_ERROR(result);
}
else if(nu > 50.0) {
return gsl_sf_bessel_Ynu_asymp_Olver_e(nu, x, result);
}
else {
/* -1/2 <= mu <= 1/2 */
int N = (int)(nu + 0.5);
double mu = nu - N;
gsl_sf_result Y_mu, Y_mup1;
int stat_mu;
double Ynm1;
double Yn;
double Ynp1;
int n;
if(x < 2.0) {
/* Determine Ymu, Ymup1 directly. This is really
* an optimization since this case could as well
* be handled by a call to gsl_sf_bessel_JY_mu_restricted(),
* as below.
*/
stat_mu = gsl_sf_bessel_Y_temme(mu, x, &Y_mu, &Y_mup1);
}
else {
/* Determine Ymu, Ymup1 and Jmu, Jmup1.
*/
gsl_sf_result J_mu, J_mup1;
stat_mu = gsl_sf_bessel_JY_mu_restricted(mu, x, &J_mu, &J_mup1, &Y_mu, &Y_mup1);
}
/* Forward recursion to get Ynu, Ynup1.
*/
Ynm1 = Y_mu.val;
Yn = Y_mup1.val;
for(n=1; n<=N; n++) {
Ynp1 = 2.0*(mu+n)/x * Yn - Ynm1;
Ynm1 = Yn;
Yn = Ynp1;
}
result->val = Ynm1; /* Y_nu */
result->err = (N + 1.0) * fabs(Ynm1) * (fabs(Y_mu.err/Y_mu.val) + fabs(Y_mup1.err/Y_mup1.val));
result->err += 2.0 * GSL_DBL_EPSILON * fabs(Ynm1);
return stat_mu;
}
}
/* Evaluate J_mu(x),J_{mu+1}(x) and Y_mu(x),Y_{mu+1}(x) for |mu| < 1/2
*/
int
gsl_sf_bessel_JY_mu_restricted(const double mu, const double x,
gsl_sf_result * Jmu, gsl_sf_result * Jmup1,
gsl_sf_result * Ymu, gsl_sf_result * Ymup1)
{
/* CHECK_POINTER(Jmu) */
/* CHECK_POINTER(Jmup1) */
/* CHECK_POINTER(Ymu) */
/* CHECK_POINTER(Ymup1) */
if(x < 0.0 || fabs(mu) > 0.5) {
Jmu->val = 0.0;
Jmu->err = 0.0;
Jmup1->val = 0.0;
Jmup1->err = 0.0;
Ymu->val = 0.0;
Ymu->err = 0.0;
Ymup1->val = 0.0;
Ymup1->err = 0.0;
GSL_ERROR ("error", GSL_EDOM);
}
else if(x == 0.0) {
if(mu == 0.0) {
Jmu->val = 1.0;
Jmu->err = 0.0;
}
else {
Jmu->val = 0.0;
Jmu->err = 0.0;
}
Jmup1->val = 0.0;
Jmup1->err = 0.0;
Ymu->val = 0.0;
Ymu->err = 0.0;
Ymup1->val = 0.0;
Ymup1->err = 0.0;
GSL_ERROR ("error", GSL_EDOM);
}
else {
int stat_Y;
int stat_J;
if(x < 2.0) {
/* Use Taylor series for J and the Temme series for Y.
* The Taylor series for J requires nu > 0, so we shift
* up one and use the recursion relation to get Jmu, in
* case mu < 0.
*/
gsl_sf_result Jmup2;
int stat_J1 = gsl_sf_bessel_IJ_taylor_e(mu+1.0, x, -1, 100, GSL_DBL_EPSILON, Jmup1);
int stat_J2 = gsl_sf_bessel_IJ_taylor_e(mu+2.0, x, -1, 100, GSL_DBL_EPSILON, &Jmup2);
double c = 2.0*(mu+1.0)/x;
Jmu->val = c * Jmup1->val - Jmup2.val;
Jmu->err = c * Jmup1->err + Jmup2.err;
Jmu->err += 2.0 * GSL_DBL_EPSILON * fabs(Jmu->val);
stat_J = GSL_ERROR_SELECT_2(stat_J1, stat_J2);
stat_Y = gsl_sf_bessel_Y_temme(mu, x, Ymu, Ymup1);
return GSL_ERROR_SELECT_2(stat_J, stat_Y);
}
else if(x < 1000.0) {
double P, Q;
double J_ratio;
double J_sgn;
const int stat_CF1 = gsl_sf_bessel_J_CF1(mu, x, &J_ratio, &J_sgn);
const int stat_CF2 = gsl_sf_bessel_JY_steed_CF2(mu, x, &P, &Q);
double Jprime_J_ratio = mu/x - J_ratio;
double gamma = (P - Jprime_J_ratio)/Q;
Jmu->val = J_sgn * sqrt(2.0/(M_PI*x) / (Q + gamma*(P-Jprime_J_ratio)));
Jmu->err = 4.0 * GSL_DBL_EPSILON * fabs(Jmu->val);
Jmup1->val = J_ratio * Jmu->val;
Jmup1->err = fabs(J_ratio) * Jmu->err;
Ymu->val = gamma * Jmu->val;
Ymu->err = fabs(gamma) * Jmu->err;
Ymup1->val = Ymu->val * (mu/x - P - Q/gamma);
Ymup1->err = Ymu->err * fabs(mu/x - P - Q/gamma) + 4.0*GSL_DBL_EPSILON*fabs(Ymup1->val);
return GSL_ERROR_SELECT_2(stat_CF1, stat_CF2);
}
else {
/* Use asymptotics for large argument.
*/
const int stat_J0 = gsl_sf_bessel_Jnu_asympx_e(mu, x, Jmu);
const int stat_J1 = gsl_sf_bessel_Jnu_asympx_e(mu+1.0, x, Jmup1);
const int stat_Y0 = gsl_sf_bessel_Ynu_asympx_e(mu, x, Ymu);
const int stat_Y1 = gsl_sf_bessel_Ynu_asympx_e(mu+1.0, x, Ymup1);
stat_J = GSL_ERROR_SELECT_2(stat_J0, stat_J1);
stat_Y = GSL_ERROR_SELECT_2(stat_Y0, stat_Y1);
return GSL_ERROR_SELECT_2(stat_J, stat_Y);
}
}
}
int
gsl_sf_bessel_Jnu_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(x == 0.0) {
if(nu == 0.0) {
result->val = 1.0;
result->err = 0.0;
}
else {
result->val = 0.0;
result->err = 0.0;
}
return GSL_SUCCESS;
}
else if(x*x < 10.0*(nu+1.0)) {
return gsl_sf_bessel_IJ_taylor_e(nu, x, -1, 100, GSL_DBL_EPSILON, result);
}
else if(nu > 50.0) {
return gsl_sf_bessel_Jnu_asymp_Olver_e(nu, x, result);
}
else {
/* -1/2 <= mu <= 1/2 */
int N = (int)(nu + 0.5);
double mu = nu - N;
/* Determine the J ratio at nu.
*/
double Jnup1_Jnu;
double sgn_Jnu;
const int stat_CF1 = gsl_sf_bessel_J_CF1(nu, x, &Jnup1_Jnu, &sgn_Jnu);
if(x < 2.0) {
/* Determine Y_mu, Y_mup1 directly and recurse forward to nu.
* Then use the CF1 information to solve for J_nu and J_nup1.
*/
gsl_sf_result Y_mu, Y_mup1;
const int stat_mu = gsl_sf_bessel_Y_temme(mu, x, &Y_mu, &Y_mup1);
double Ynm1 = Y_mu.val;
double Yn = Y_mup1.val;
double Ynp1 = 0.0;
int n;
for(n=1; n<N; n++) {
Ynp1 = 2.0*(mu+n)/x * Yn - Ynm1;
Ynm1 = Yn;
Yn = Ynp1;
}
result->val = 2.0/(M_PI*x) / (Jnup1_Jnu*Yn - Ynp1);
result->err = GSL_DBL_EPSILON * (N + 2.0) * fabs(result->val);
return GSL_ERROR_SELECT_2(stat_mu, stat_CF1);
}
else {
/* Recurse backward from nu to mu, determining the J ratio
* at mu. Use this together with a Steed method CF2 to
* determine the actual J_mu, and thus obtain the normalization.
*/
double Jmu;
double Jmup1_Jmu;
double sgn_Jmu;
double Jmuprime_Jmu;
double P, Q;
const int stat_CF2 = gsl_sf_bessel_JY_steed_CF2(mu, x, &P, &Q);
double gamma;
double Jnp1 = sgn_Jnu * GSL_SQRT_DBL_MIN * Jnup1_Jnu;
double Jn = sgn_Jnu * GSL_SQRT_DBL_MIN;
double Jnm1;
int n;
for(n=N; n>0; n--) {
Jnm1 = 2.0*(mu+n)/x * Jn - Jnp1;
Jnp1 = Jn;
Jn = Jnm1;
}
Jmup1_Jmu = Jnp1/Jn;
sgn_Jmu = GSL_SIGN(Jn);
Jmuprime_Jmu = mu/x - Jmup1_Jmu;
gamma = (P - Jmuprime_Jmu)/Q;
Jmu = sgn_Jmu * sqrt(2.0/(M_PI*x) / (Q + gamma*(P-Jmuprime_Jmu)));
result->val = Jmu * (sgn_Jnu * GSL_SQRT_DBL_MIN) / Jn;
result->err = 2.0 * GSL_DBL_EPSILON * (N + 2.0) * fabs(result->val);
return GSL_ERROR_SELECT_2(stat_CF2, stat_CF1);
}
}
}
