/* 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_jl_e(const int l, const double x, gsl_sf_result * result)
{
if(l < 0 || x < 0.0) {
DOMAIN_ERROR(result);
}
else if(x == 0.0) {
result->val = ( l > 0 ? 0.0 : 1.0 );
result->err = 0.0;
return GSL_SUCCESS;
}
else if(l == 0) {
return gsl_sf_bessel_j0_e(x, result);
}
else if(l == 1) {
return gsl_sf_bessel_j1_e(x, result);
}
else if(l == 2) {
return gsl_sf_bessel_j2_e(x, result);
}
else if(x*x < 10.0*(l+0.5)/M_E) {
gsl_sf_result b;
int status = gsl_sf_bessel_IJ_taylor_e(l+0.5, x, -1, 50, GSL_DBL_EPSILON, &b);
double pre = sqrt((0.5*M_PI)/x);
result->val = pre * b.val;
result->err = pre * b.err;
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return status;
}
else if(GSL_ROOT4_DBL_EPSILON * x > (l*l + l + 1.0)) {
gsl_sf_result b;
int status = gsl_sf_bessel_Jnu_asympx_e(l + 0.5, x, &b);
double pre = sqrt((0.5*M_PI)/x);
result->val = pre * b.val;
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val) + pre * b.err;
return status;
}
else if(l > 1.0/GSL_ROOT6_DBL_EPSILON) {
gsl_sf_result b;
int status = gsl_sf_bessel_Jnu_asymp_Olver_e(l + 0.5, x, &b);
double pre = sqrt((0.5*M_PI)/x);
result->val = pre * b.val;
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val) + pre * b.err;
return status;
}
else if(x > 1000.0 && x > l*l)
{
/* We need this path to avoid feeding large x to CF1 below; */
gsl_sf_result b;
int status = gsl_sf_bessel_Jnu_asympx_e(l + 0.5, x, &b);
double pre = sqrt((0.5*M_PI)/x);
result->val = pre * b.val;
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val) + pre * b.err;
return status;
}
else {
double sgn;
double ratio;
/* The CF1 call will hit 10000 iterations for x > 10000 + l */
int stat_CF1 = gsl_sf_bessel_J_CF1(l+0.5, x, &ratio, &sgn);
double jellp1 = GSL_SQRT_DBL_EPSILON * ratio;
double jell = GSL_SQRT_DBL_EPSILON;
double jellm1;
int ell;
for(ell = l; ell > 0; ell--) {
jellm1 = -jellp1 + (2*ell + 1)/x * jell;
jellp1 = jell;
jell = jellm1;
}
if(fabs(jell) > fabs(jellp1)) {
gsl_sf_result j0_result;
int stat_j0 = gsl_sf_bessel_j0_e(x, &j0_result);
double pre = GSL_SQRT_DBL_EPSILON / jell;
result->val = j0_result.val * pre;
result->err = j0_result.err * fabs(pre);
result->err += 4.0 * GSL_DBL_EPSILON * (0.5*l + 1.0) * fabs(result->val);
return GSL_ERROR_SELECT_2(stat_j0, stat_CF1);
}
else {
gsl_sf_result j1_result;
int stat_j1 = gsl_sf_bessel_j1_e(x, &j1_result);
double pre = GSL_SQRT_DBL_EPSILON / jellp1;
result->val = j1_result.val * pre;
result->err = j1_result.err * fabs(pre);
result->err += 4.0 * GSL_DBL_EPSILON * (0.5*l + 1.0) * fabs(result->val);
return GSL_ERROR_SELECT_2(stat_j1, stat_CF1);
}
}
}
int gsl_sf_bessel_Jn_e(int n, double x, gsl_sf_result * result)
{
int sign = 1;
if(n < 0) {
/* reduce to case n >= 0 */
n = -n;
if(GSL_IS_ODD(n)) sign = -sign;
}
if(x < 0.0) {
/* reduce to case x >= 0. */
x = -x;
if(GSL_IS_ODD(n)) sign = -sign;
}
/* CHECK_POINTER(result) */
if(n == 0) {
gsl_sf_result b0;
int stat_J0 = gsl_sf_bessel_J0_e(x, &b0);
result->val = sign * b0.val;
result->err = b0.err;
return stat_J0;
}
else if(n == 1) {
gsl_sf_result b1;
int stat_J1 = gsl_sf_bessel_J1_e(x, &b1);
result->val = sign * b1.val;
result->err = b1.err;
return stat_J1;
}
else {
if(x == 0.0) {
result->val = 0.0;
result->err = 0.0;
return GSL_SUCCESS;
}
else if(x*x < 10.0*(n+1.0)*GSL_ROOT5_DBL_EPSILON) {
gsl_sf_result b;
int status = gsl_sf_bessel_IJ_taylor_e((double)n, x, -1, 50, GSL_DBL_EPSILON, &b);
result->val = sign * b.val;
result->err = b.err;
result->err += GSL_DBL_EPSILON * fabs(result->val);
return status;
}
else if(GSL_ROOT3_DBL_EPSILON * x > (n*n+1.0)) {
int status = gsl_sf_bessel_Jnu_asympx_e((double)n, x, result);
result->val *= sign;
return status;
}
else if(n > 50) {
int status = gsl_sf_bessel_Jnu_asymp_Olver_e((double)n, x, result);
result->val *= sign;
return status;
}
else {
double ans;
double err;
double ratio;
double sgn;
int stat_b;
int stat_CF1 = gsl_sf_bessel_J_CF1((double)n, x, &ratio, &sgn);
/* backward recurrence */
double Jkp1 = GSL_SQRT_DBL_MIN * ratio;
double Jk = GSL_SQRT_DBL_MIN;
double Jkm1;
int k;
for(k=n; k>0; k--) {
Jkm1 = 2.0*k/x * Jk - Jkp1;
Jkp1 = Jk;
Jk = Jkm1;
}
if(fabs(Jkp1) > fabs(Jk)) {
gsl_sf_result b1;
stat_b = gsl_sf_bessel_J1_e(x, &b1);
ans = b1.val/Jkp1 * GSL_SQRT_DBL_MIN;
err = b1.err/Jkp1 * GSL_SQRT_DBL_MIN;
}
else {
gsl_sf_result b0;
stat_b = gsl_sf_bessel_J0_e(x, &b0);
ans = b0.val/Jk * GSL_SQRT_DBL_MIN;
err = b0.err/Jk * GSL_SQRT_DBL_MIN;
}
result->val = sign * ans;
result->err = fabs(err);
return GSL_ERROR_SELECT_2(stat_CF1, stat_b);
}
}
}
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);
}
}
}