1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
|
/* specfunc/bessel_Yn.c
*
* Copyright (C) 1996, 1997, 1998, 1999, 2000 Gerard Jungman
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation; either version 3 of the License, or (at
* your option) any later version.
*
* This program is distributed in the hope that it will be useful, but
* WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
* General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with this program; if not, write to the Free Software
* Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
*/
/* Author: G. Jungman */
#include <config.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_sf_gamma.h>
#include <gsl/gsl_sf_psi.h>
#include <gsl/gsl_sf_bessel.h>
#include "error.h"
#include "bessel.h"
#include "bessel_amp_phase.h"
#include "bessel_olver.h"
/*-*-*-*-*-*-*-*-*-*-*-* Private Section *-*-*-*-*-*-*-*-*-*-*-*/
/* assumes n >= 1 */
static int bessel_Yn_small_x(const int n, const double x, gsl_sf_result * result)
{
int k;
double y = 0.25 * x * x;
double ln_x_2 = log(0.5*x);
gsl_sf_result ln_nm1_fact;
double k_term;
double term1, sum1, ln_pre1;
double term2, sum2, pre2;
gsl_sf_lnfact_e((unsigned int)(n-1), &ln_nm1_fact);
ln_pre1 = -n*ln_x_2 + ln_nm1_fact.val;
if(ln_pre1 > GSL_LOG_DBL_MAX - 3.0) GSL_ERROR ("error", GSL_EOVRFLW);
sum1 = 1.0;
k_term = 1.0;
for(k=1; k<=n-1; k++) {
k_term *= y/(k * (n-k));
sum1 += k_term;
}
term1 = -exp(ln_pre1) * sum1 / M_PI;
pre2 = -exp(n*ln_x_2) / M_PI;
if(fabs(pre2) > 0.0) {
const int KMAX = 20;
gsl_sf_result psi_n;
gsl_sf_result npk_fact;
double yk = 1.0;
double k_fact = 1.0;
double psi_kp1 = -M_EULER;
double psi_npkp1;
gsl_sf_psi_int_e(n, &psi_n);
gsl_sf_fact_e((unsigned int)n, &npk_fact);
psi_npkp1 = psi_n.val + 1.0/n;
sum2 = (psi_kp1 + psi_npkp1 - 2.0*ln_x_2)/npk_fact.val;
for(k=1; k<KMAX; k++) {
psi_kp1 += 1./k;
psi_npkp1 += 1./(n+k);
k_fact *= k;
npk_fact.val *= n+k;
yk *= -y;
k_term = yk*(psi_kp1 + psi_npkp1 - 2.0*ln_x_2)/(k_fact*npk_fact.val);
sum2 += k_term;
}
term2 = pre2 * sum2;
}
else {
term2 = 0.0;
}
result->val = term1 + term2;
result->err = GSL_DBL_EPSILON * (fabs(ln_pre1)*fabs(term1) + fabs(term2));
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
/*-*-*-*-*-*-*-*-*-*-*-* Functions with Error Codes *-*-*-*-*-*-*-*-*-*-*-*/
int
gsl_sf_bessel_Yn_e(int n, const 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 = -1;
}
/* CHECK_POINTER(result) */
if(n == 0) {
int status = gsl_sf_bessel_Y0_e(x, result);
result->val *= sign;
return status;
}
else if(n == 1) {
int status = gsl_sf_bessel_Y1_e(x, result);
result->val *= sign;
return status;
}
else {
if(x <= 0.0) {
DOMAIN_ERROR(result);
}
if(x < 5.0) {
int status = bessel_Yn_small_x(n, x, result);
result->val *= sign;
return status;
}
else if(GSL_ROOT3_DBL_EPSILON * x > (n*n + 1.0)) {
int status = gsl_sf_bessel_Ynu_asympx_e((double)n, x, result);
result->val *= sign;
return status;
}
else if(n > 50) {
int status = gsl_sf_bessel_Ynu_asymp_Olver_e((double)n, x, result);
result->val *= sign;
return status;
}
else {
double two_over_x = 2.0/x;
gsl_sf_result r_by;
gsl_sf_result r_bym;
int stat_1 = gsl_sf_bessel_Y1_e(x, &r_by);
int stat_0 = gsl_sf_bessel_Y0_e(x, &r_bym);
double bym = r_bym.val;
double by = r_by.val;
double byp;
int j;
for(j=1; j<n; j++) {
byp = j*two_over_x*by - bym;
bym = by;
by = byp;
}
result->val = sign * by;
result->err = fabs(result->val) * (fabs(r_by.err/r_by.val) + fabs(r_bym.err/r_bym.val));
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_ERROR_SELECT_2(stat_1, stat_0);
}
}
}
int
gsl_sf_bessel_Yn_array(const int nmin, const int nmax, const double x, double * result_array)
{
/* CHECK_POINTER(result_array) */
if(nmin < 0 || nmax < nmin || x <= 0.0) {
int j;
for(j=0; j<=nmax-nmin; j++) result_array[j] = 0.0;
GSL_ERROR ("error", GSL_EDOM);
}
else {
gsl_sf_result r_Ynm1;
gsl_sf_result r_Yn;
int stat_nm1 = gsl_sf_bessel_Yn_e(nmin, x, &r_Ynm1);
int stat_n = gsl_sf_bessel_Yn_e(nmin+1, x, &r_Yn);
double Ynp1;
double Yn = r_Yn.val;
double Ynm1 = r_Ynm1.val;
int n;
int stat = GSL_ERROR_SELECT_2(stat_nm1, stat_n);
if(stat == GSL_SUCCESS) {
for(n=nmin+1; n<=nmax+1; n++) {
result_array[n-nmin-1] = Ynm1;
Ynp1 = -Ynm1 + 2.0*n/x * Yn;
Ynm1 = Yn;
Yn = Ynp1;
}
}
else {
for(n=nmin; n<=nmax; n++) {
result_array[n-nmin] = 0.0;
}
}
return stat;
}
}
/*-*-*-*-*-*-*-*-*-* Functions w/ Natural Prototypes *-*-*-*-*-*-*-*-*-*-*/
#include "eval.h"
double gsl_sf_bessel_Yn(const int n, const double x)
{
EVAL_RESULT(gsl_sf_bessel_Yn_e(n, x, &result));
}
|
