1 /* specfunc/bessel_i.c
2 *
3 * Copyright (C) 1996, 1997, 1998, 1999, 2000 Gerard Jungman
4 *
5 * This program is free software; you can redistribute it and/or modify
6 * it under the terms of the GNU General Public License as published by
7 * the Free Software Foundation; either version 3 of the License, or (at
8 * your option) any later version.
9 *
10 * This program is distributed in the hope that it will be useful, but
11 * WITHOUT ANY WARRANTY; without even the implied warranty of
12 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
13 * General Public License for more details.
14 *
15 * You should have received a copy of the GNU General Public License
16 * along with this program; if not, write to the Free Software
17 * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
18 */
19
20 /* Author: G. Jungman */
21
22 #include <config.h>
23 #include <gsl/gsl_math.h>
24 #include <gsl/gsl_errno.h>
25 #include <gsl/gsl_sf_pow_int.h>
26 #include <gsl/gsl_sf_bessel.h>
27
28 #include "error.h"
29
30 #include "bessel.h"
31
32
33 /* i_{l+1}/i_l
34 */
35 static
36 int
bessel_il_CF1(const int l,const double x,const double threshold,double * ratio)37 bessel_il_CF1(const int l, const double x, const double threshold, double * ratio)
38 {
39 const int kmax = 2000;
40 double tk = 1.0;
41 double sum = 1.0;
42 double rhok = 0.0;
43 int k;
44
45 for(k=1; k<=kmax; k++) {
46 double ak = (x/(2.0*l+1.0+2.0*k)) * (x/(2.0*l+3.0+2.0*k));
47 rhok = -ak*(1.0 + rhok)/(1.0 + ak*(1.0 + rhok));
48 tk *= rhok;
49 sum += tk;
50 if(fabs(tk/sum) < threshold) break;
51 }
52
53 *ratio = x/(2.0*l+3.0) * sum;
54
55 if(k == kmax)
56 GSL_ERROR ("error", GSL_EMAXITER);
57 else
58 return GSL_SUCCESS;
59 }
60
61
62 /*-*-*-*-*-*-*-*-*-*-*-* Functions with Error Codes *-*-*-*-*-*-*-*-*-*-*-*/
63
gsl_sf_bessel_i0_scaled_e(const double x,gsl_sf_result * result)64 int gsl_sf_bessel_i0_scaled_e(const double x, gsl_sf_result * result)
65 {
66 double ax = fabs(x);
67
68 /* CHECK_POINTER(result) */
69
70 if(x == 0.0) {
71 result->val = 1.0;
72 result->err = 0.0;
73 return GSL_SUCCESS;
74 }
75 else if(ax < 0.2) {
76 const double eax = exp(-ax);
77 const double y = ax*ax;
78 const double c1 = 1.0/6.0;
79 const double c2 = 1.0/120.0;
80 const double c3 = 1.0/5040.0;
81 const double c4 = 1.0/362880.0;
82 const double c5 = 1.0/39916800.0;
83 const double sum = 1.0 + y*(c1 + y*(c2 + y*(c3 + y*(c4 + y*c5))));
84 result->val = eax * sum;
85 result->err = 2.0 * GSL_DBL_EPSILON * result->val;
86 }
87 else if(ax < -0.5*GSL_LOG_DBL_EPSILON) {
88 result->val = (1.0 - exp(-2.0*ax))/(2.0*ax);
89 result->err = 2.0 * GSL_DBL_EPSILON * result->val;
90 }
91 else {
92 result->val = 1.0/(2.0*ax);
93 result->err = 2.0 * GSL_DBL_EPSILON * result->val;
94 }
95 return GSL_SUCCESS;
96 }
97
98
gsl_sf_bessel_i1_scaled_e(const double x,gsl_sf_result * result)99 int gsl_sf_bessel_i1_scaled_e(const double x, gsl_sf_result * result)
100 {
101 double ax = fabs(x);
102
103 /* CHECK_POINTER(result) */
104
105 if(x == 0.0) {
106 result->val = 0.0;
107 result->err = 0.0;
108 return GSL_SUCCESS;
109 }
110 else if(ax < 3.0*GSL_DBL_MIN) {
111 UNDERFLOW_ERROR(result);
112 }
113 else if(ax < 0.25) {
114 const double eax = exp(-ax);
115 const double y = x*x;
116 const double c1 = 1.0/10.0;
117 const double c2 = 1.0/280.0;
118 const double c3 = 1.0/15120.0;
119 const double c4 = 1.0/1330560.0;
120 const double c5 = 1.0/172972800.0;
121 const double sum = 1.0 + y*(c1 + y*(c2 + y*(c3 + y*(c4 + y*c5))));
122 result->val = eax * x/3.0 * sum;
123 result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
124 return GSL_SUCCESS;
125 }
126 else {
127 double ex = exp(-2.0*ax);
128 result->val = 0.5 * (ax*(1.0+ex) - (1.0-ex)) / (ax*ax);
129 result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
130 if(x < 0.0) result->val = -result->val;
131 return GSL_SUCCESS;
132 }
133 }
134
135
gsl_sf_bessel_i2_scaled_e(const double x,gsl_sf_result * result)136 int gsl_sf_bessel_i2_scaled_e(const double x, gsl_sf_result * result)
137 {
138 double ax = fabs(x);
139
140 /* CHECK_POINTER(result) */
141
142 if(x == 0.0) {
143 result->val = 0.0;
144 result->err = 0.0;
145 return GSL_SUCCESS;
146 }
147 else if(ax < 4.0*GSL_SQRT_DBL_MIN) {
148 UNDERFLOW_ERROR(result);
149 }
150 else if(ax < 0.25) {
151 const double y = x*x;
152 const double c1 = 1.0/14.0;
153 const double c2 = 1.0/504.0;
154 const double c3 = 1.0/33264.0;
155 const double c4 = 1.0/3459456.0;
156 const double c5 = 1.0/518918400.0;
157 const double sum = 1.0 + y*(c1 + y*(c2 + y*(c3 + y*(c4 + y*c5))));
158 const double pre = exp(-ax) * x*x/15.0;
159 result->val = pre * sum;
160 result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
161 return GSL_SUCCESS;
162 }
163 else {
164 double ex = exp(-2.0*ax);
165 double x2 = x*x;
166 result->val = 0.5 * ((3.0+x2)*(1.0-ex) - 3.0*ax*(1.0+ex))/(ax*ax*ax);
167 result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
168 return GSL_SUCCESS;
169 }
170 }
171
172
gsl_sf_bessel_il_scaled_e(const int l,double x,gsl_sf_result * result)173 int gsl_sf_bessel_il_scaled_e(const int l, double x, gsl_sf_result * result)
174 {
175 double sgn = 1.0;
176 double ax = fabs(x);
177
178 if(x < 0.0) {
179 /* i_l(-x) = (-1)^l i_l(x) */
180 sgn = ( GSL_IS_ODD(l) ? -1.0 : 1.0 );
181 x = -x;
182 }
183
184 if(l < 0) {
185 DOMAIN_ERROR(result);
186 }
187 else if(x == 0.0) {
188 result->val = ( l == 0 ? 1.0 : 0.0 );
189 result->err = 0.0;
190 return GSL_SUCCESS;
191 }
192 else if(l == 0) {
193 gsl_sf_result il;
194 int stat_il = gsl_sf_bessel_i0_scaled_e(x, &il);
195 result->val = sgn * il.val;
196 result->err = il.err;
197 return stat_il;
198 }
199 else if(l == 1) {
200 gsl_sf_result il;
201 int stat_il = gsl_sf_bessel_i1_scaled_e(x, &il);
202 result->val = sgn * il.val;
203 result->err = il.err;
204 return stat_il;
205 }
206 else if(l == 2) {
207 gsl_sf_result il;
208 int stat_il = gsl_sf_bessel_i2_scaled_e(x, &il);
209 result->val = sgn * il.val;
210 result->err = il.err;
211 return stat_il;
212 }
213 else if(x*x < 10.0*(l+1.5)/M_E) {
214 gsl_sf_result b;
215 int stat = gsl_sf_bessel_IJ_taylor_e(l+0.5, x, 1, 50, GSL_DBL_EPSILON, &b);
216 double pre = exp(-ax) * sqrt((0.5*M_PI)/x);
217 result->val = sgn * pre * b.val;
218 result->err = pre * b.err;
219 result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
220 return stat;
221 }
222 else if(l < 150) {
223 gsl_sf_result i0_scaled;
224 int stat_i0 = gsl_sf_bessel_i0_scaled_e(ax, &i0_scaled);
225 double rat;
226 int stat_CF1 = bessel_il_CF1(l, ax, GSL_DBL_EPSILON, &rat);
227 double iellp1 = rat * GSL_SQRT_DBL_MIN;
228 double iell = GSL_SQRT_DBL_MIN;
229 double iellm1;
230 int ell;
231 for(ell = l; ell >= 1; ell--) {
232 iellm1 = iellp1 + (2*ell + 1)/x * iell;
233 iellp1 = iell;
234 iell = iellm1;
235 }
236 result->val = sgn * i0_scaled.val * (GSL_SQRT_DBL_MIN / iell);
237 result->err = i0_scaled.err * (GSL_SQRT_DBL_MIN / iell);
238 result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
239 return GSL_ERROR_SELECT_2(stat_i0, stat_CF1);
240 }
241 else if(GSL_MIN(0.29/(l*l+1.0), 0.5/(l*l+1.0+x*x)) < 0.5*GSL_ROOT3_DBL_EPSILON) {
242 int status = gsl_sf_bessel_Inu_scaled_asymp_unif_e(l + 0.5, x, result);
243 double pre = sqrt((0.5*M_PI)/x);
244 result->val *= sgn * pre;
245 result->err *= pre;
246 return status;
247 }
248 else {
249 /* recurse down from safe values */
250 double rt_term = sqrt((0.5*M_PI)/x);
251 const int LMAX = 2 + (int) (1.2 / GSL_ROOT6_DBL_EPSILON);
252 gsl_sf_result r_iellp1;
253 gsl_sf_result r_iell;
254 int stat_a1 = gsl_sf_bessel_Inu_scaled_asymp_unif_e(LMAX + 1 + 0.5, x, &r_iellp1);
255 int stat_a2 = gsl_sf_bessel_Inu_scaled_asymp_unif_e(LMAX + 0.5, x, &r_iell);
256 double iellp1 = r_iellp1.val;
257 double iell = r_iell.val;
258 double iellm1 = 0.0;
259 int ell;
260 iellp1 *= rt_term;
261 iell *= rt_term;
262 for(ell = LMAX; ell >= l+1; ell--) {
263 iellm1 = iellp1 + (2*ell + 1)/x * iell;
264 iellp1 = iell;
265 iell = iellm1;
266 }
267 result->val = sgn * iellm1;
268 result->err = fabs(result->val)*(GSL_DBL_EPSILON + fabs(r_iellp1.err/r_iellp1.val) + fabs(r_iell.err/r_iell.val));
269 result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
270
271 return GSL_ERROR_SELECT_2(stat_a1, stat_a2);
272 }
273 }
274
275
gsl_sf_bessel_il_scaled_array(const int lmax,const double x,double * result_array)276 int gsl_sf_bessel_il_scaled_array(const int lmax, const double x, double * result_array)
277 {
278 if(x == 0.0) {
279 int ell;
280 result_array[0] = 1.0;
281 for (ell = lmax; ell >= 1; ell--) {
282 result_array[ell] = 0.0;
283 };
284 return GSL_SUCCESS;
285 } else {
286 int ell;
287 gsl_sf_result r_iellp1;
288 gsl_sf_result r_iell;
289 int stat_0 = gsl_sf_bessel_il_scaled_e(lmax+1, x, &r_iellp1);
290 int stat_1 = gsl_sf_bessel_il_scaled_e(lmax, x, &r_iell);
291 double iellp1 = r_iellp1.val;
292 double iell = r_iell.val;
293 double iellm1;
294 result_array[lmax] = iell;
295 for(ell = lmax; ell >= 1; ell--) {
296 iellm1 = iellp1 + (2*ell + 1)/x * iell;
297 iellp1 = iell;
298 iell = iellm1;
299 result_array[ell-1] = iellm1;
300 }
301 return GSL_ERROR_SELECT_2(stat_0, stat_1);
302 }
303 }
304
305
306 /*-*-*-*-*-*-*-*-*-* Functions w/ Natural Prototypes *-*-*-*-*-*-*-*-*-*-*/
307
308 #include "eval.h"
309
gsl_sf_bessel_i0_scaled(const double x)310 double gsl_sf_bessel_i0_scaled(const double x)
311 {
312 EVAL_RESULT(gsl_sf_bessel_i0_scaled_e(x, &result));
313 }
314
gsl_sf_bessel_i1_scaled(const double x)315 double gsl_sf_bessel_i1_scaled(const double x)
316 {
317 EVAL_RESULT(gsl_sf_bessel_i1_scaled_e(x, &result));
318 }
319
gsl_sf_bessel_i2_scaled(const double x)320 double gsl_sf_bessel_i2_scaled(const double x)
321 {
322 EVAL_RESULT(gsl_sf_bessel_i2_scaled_e(x, &result));
323 }
324
gsl_sf_bessel_il_scaled(const int l,const double x)325 double gsl_sf_bessel_il_scaled(const int l, const double x)
326 {
327 EVAL_RESULT(gsl_sf_bessel_il_scaled_e(l, x, &result));
328 }
329
330