5 * Common Development and Distribution License (the "License").
6 * You may not use this file except in compliance with the License.
7 *
8 * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE
9 * or http://www.opensolaris.org/os/licensing.
10 * See the License for the specific language governing permissions
11 * and limitations under the License.
12 *
13 * When distributing Covered Code, include this CDDL HEADER in each
14 * file and include the License file at usr/src/OPENSOLARIS.LICENSE.
15 * If applicable, add the following below this CDDL HEADER, with the
16 * fields enclosed by brackets "[]" replaced with your own identifying
17 * information: Portions Copyright [yyyy] [name of copyright owner]
18 *
19 * CDDL HEADER END
20 */
21
22 /*
23 * Copyright 2011 Nexenta Systems, Inc. All rights reserved.
24 */
25 /*
26 * Copyright 2006 Sun Microsystems, Inc. All rights reserved.
27 * Use is subject to license terms.
28 */
29
30 #pragma weak __jn = jn
31 #pragma weak __yn = yn
32
33 /*
34 * floating point Bessel's function of the 1st and 2nd kind
35 * of order n: jn(n,x),yn(n,x);
36 *
37 * Special cases:
38 * y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal;
39 * y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
40 * Note 2. About jn(n,x), yn(n,x)
41 * For n=0, j0(x) is called,
42 * for n=1, j1(x) is called,
43 * for n<x, forward recursion us used starting
44 * from values of j0(x) and j1(x).
69 one = 1.0;
70
71 GENERIC
72 jn(int n, GENERIC x)
73 {
74 int i, sgn;
75 GENERIC a, b, temp = 0;
76 GENERIC z, w, ox, on;
77
78 /*
79 * J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
80 * Thus, J(-n,x) = J(n,-x)
81 */
82 ox = x;
83 on = (GENERIC)n;
84
85 if (n < 0) {
86 n = -n;
87 x = -x;
88 }
89 if (isnan(x))
90 return (x*x); /* + -> * for Cheetah */
91 if (!((int)_lib_version == libm_ieee ||
92 (__xpg6 & _C99SUSv3_math_errexcept) != 0)) {
93 if (fabs(x) > X_TLOSS)
94 return (_SVID_libm_err(on, ox, 38));
95 }
96 if (n == 0)
97 return (j0(x));
98 if (n == 1)
99 return (j1(x));
100 if ((n&1) == 0)
101 sgn = 0; /* even n */
102 else
103 sgn = signbit(x); /* old n */
104 x = fabs(x);
105 if (x == zero||!finite(x)) b = zero;
106 else if ((GENERIC)n <= x) {
107 /*
108 * Safe to use
109 * J(n+1,x)=2n/x *J(n,x)-J(n-1,x)
110 */
111 if (x > 1.0e91) {
112 /*
113 * x >> n**2
114 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
115 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
116 * Let s=sin(x), c=cos(x),
117 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
118 *
119 * n sin(xn)*sqt2 cos(xn)*sqt2
120 * ----------------------------------
121 * 0 s-c c+s
122 * 1 -s-c -c+s
123 * 2 -s+c -c-s
124 * 3 s+c c-s
125 */
126 switch (n&3) {
127 case 0:
128 temp = cos(x)+sin(x);
129 break;
130 case 1:
131 temp = -cos(x)+sin(x);
132 break;
133 case 2:
134 temp = -cos(x)-sin(x);
135 break;
136 case 3:
137 temp = cos(x)-sin(x);
138 break;
139 }
140 b = invsqrtpi*temp/sqrt(x);
141 } else {
142 a = j0(x);
143 b = j1(x);
144 for (i = 1; i < n; i++) {
145 temp = b;
146 /* avoid underflow */
147 b = b*((GENERIC)(i+i)/x) - a;
148 a = temp;
149 }
150 }
151 } else {
152 if (x < 1e-9) { /* use J(n,x) = 1/n!*(x/2)^n */
153 b = pow(0.5*x, (GENERIC) n);
154 if (b != zero) {
155 for (a = one, i = 1; i <= n; i++)
156 a *= (GENERIC)i;
157 b = b/a;
158 }
159 } else {
160 /*
161 * use backward recurrence
162 * x x^2 x^2
163 * J(n,x)/J(n-1,x) = ---- ------ ------ .....
164 * 2n - 2(n+1) - 2(n+2)
165 *
166 * 1 1 1
167 * (for large x) = ---- ------ ------ .....
168 * 2n 2(n+1) 2(n+2)
169 * -- - ------ - ------ -
170 * x x x
171 *
172 * Let w = 2n/x and h = 2/x, then the above quotient
173 * is equal to the continued fraction:
174 * 1
175 * = -----------------------
176 * 1
177 * w - -----------------
178 * 1
179 * w+h - ---------
180 * w+2h - ...
181 *
182 * To determine how many terms needed, let
183 * Q(0) = w, Q(1) = w(w+h) - 1,
184 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
185 * When Q(k) > 1e4 good for single
186 * When Q(k) > 1e9 good for double
187 * When Q(k) > 1e17 good for quaduple
188 */
189 /* determine k */
190 GENERIC t, v;
191 double q0, q1, h, tmp;
192 int k, m;
193 w = (n+n)/(double)x;
194 h = 2.0/(double)x;
195 q0 = w;
196 z = w + h;
197 q1 = w*z - 1.0;
198 k = 1;
199
200 while (q1 < 1.0e9) {
201 k += 1;
202 z += h;
203 tmp = z*q1 - q0;
204 q0 = q1;
205 q1 = tmp;
206 }
207 m = n+n;
208 for (t = zero, i = 2*(n+k); i >= m; i -= 2)
209 t = one/(i/x-t);
210 a = t;
211 b = one;
212 /*
213 * estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
214 * hence, if n*(log(2n/x)) > ...
215 * single:
216 * 8.8722839355e+01
217 * double:
218 * 7.09782712893383973096e+02
219 * long double:
220 * 1.1356523406294143949491931077970765006170e+04
221 * then recurrent value may overflow and the result is
222 * likely underflow to zero
223 */
224 tmp = n;
225 v = two/x;
226 tmp = tmp*log(fabs(v*tmp));
227 if (tmp < 7.09782712893383973096e+02) {
228 for (i = n-1; i > 0; i--) {
229 temp = b;
230 b = ((i+i)/x)*b - a;
231 a = temp;
232 }
233 } else {
234 for (i = n-1; i > 0; i--) {
235 temp = b;
236 b = ((i+i)/x)*b - a;
237 a = temp;
238 if (b > 1e100) {
239 a /= b;
240 t /= b;
241 b = 1.0;
242 }
243 }
244 }
245 b = (t*j0(x)/b);
246 }
247 }
248 if (sgn != 0)
249 return (-b);
250 else
251 return (b);
252 }
253
254 GENERIC
255 yn(int n, GENERIC x)
256 {
257 int i;
258 int sign;
259 GENERIC a, b, temp = 0, ox, on;
260
261 ox = x;
262 on = (GENERIC)n;
263 if (isnan(x))
264 return (x*x); /* + -> * for Cheetah */
265 if (x <= zero) {
266 if (x == zero) {
267 /* return -one/zero; */
268 return (_SVID_libm_err((GENERIC)n, x, 12));
269 } else {
270 /* return zero/zero; */
271 return (_SVID_libm_err((GENERIC)n, x, 13));
272 }
273 }
274 if (!((int)_lib_version == libm_ieee ||
275 (__xpg6 & _C99SUSv3_math_errexcept) != 0)) {
276 if (x > X_TLOSS)
277 return (_SVID_libm_err(on, ox, 39));
278 }
279 sign = 1;
280 if (n < 0) {
281 n = -n;
282 if ((n&1) == 1) sign = -1;
283 }
284 if (n == 0)
285 return (y0(x));
286 if (n == 1)
287 return (sign*y1(x));
288 if (!finite(x))
289 return (zero);
290
291 if (x > 1.0e91) {
292 /*
293 * x >> n**2
294 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
295 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
296 * Let s = sin(x), c = cos(x),
297 * xn = x-(2n+1)*pi/4, sqt2 = sqrt(2), then
298 *
299 * n sin(xn)*sqt2 cos(xn)*sqt2
300 * ----------------------------------
301 * 0 s-c c+s
302 * 1 -s-c -c+s
303 * 2 -s+c -c-s
304 * 3 s+c c-s
305 */
306 switch (n&3) {
307 case 0:
308 temp = sin(x)-cos(x);
309 break;
310 case 1:
311 temp = -sin(x)-cos(x);
312 break;
313 case 2:
314 temp = -sin(x)+cos(x);
315 break;
316 case 3:
317 temp = sin(x)+cos(x);
318 break;
319 }
320 b = invsqrtpi*temp/sqrt(x);
321 } else {
322 a = y0(x);
323 b = y1(x);
324 /*
325 * fix 1262058 and take care of non-default rounding
326 */
327 for (i = 1; i < n; i++) {
328 temp = b;
329 b *= (GENERIC) (i + i) / x;
330 if (b <= -DBL_MAX)
331 break;
332 b -= a;
333 a = temp;
334 }
335 }
336 if (sign > 0)
337 return (b);
338 else
339 return (-b);
340 }
|
5 * Common Development and Distribution License (the "License").
6 * You may not use this file except in compliance with the License.
7 *
8 * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE
9 * or http://www.opensolaris.org/os/licensing.
10 * See the License for the specific language governing permissions
11 * and limitations under the License.
12 *
13 * When distributing Covered Code, include this CDDL HEADER in each
14 * file and include the License file at usr/src/OPENSOLARIS.LICENSE.
15 * If applicable, add the following below this CDDL HEADER, with the
16 * fields enclosed by brackets "[]" replaced with your own identifying
17 * information: Portions Copyright [yyyy] [name of copyright owner]
18 *
19 * CDDL HEADER END
20 */
21
22 /*
23 * Copyright 2011 Nexenta Systems, Inc. All rights reserved.
24 */
25
26 /*
27 * Copyright 2006 Sun Microsystems, Inc. All rights reserved.
28 * Use is subject to license terms.
29 */
30
31 #pragma weak __jn = jn
32 #pragma weak __yn = yn
33
34 /*
35 * floating point Bessel's function of the 1st and 2nd kind
36 * of order n: jn(n,x),yn(n,x);
37 *
38 * Special cases:
39 * y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal;
40 * y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
41 * Note 2. About jn(n,x), yn(n,x)
42 * For n=0, j0(x) is called,
43 * for n=1, j1(x) is called,
44 * for n<x, forward recursion us used starting
45 * from values of j0(x) and j1(x).
70 one = 1.0;
71
72 GENERIC
73 jn(int n, GENERIC x)
74 {
75 int i, sgn;
76 GENERIC a, b, temp = 0;
77 GENERIC z, w, ox, on;
78
79 /*
80 * J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
81 * Thus, J(-n,x) = J(n,-x)
82 */
83 ox = x;
84 on = (GENERIC)n;
85
86 if (n < 0) {
87 n = -n;
88 x = -x;
89 }
90
91 if (isnan(x))
92 return (x * x); /* + -> * for Cheetah */
93
94 if (!((int)_lib_version == libm_ieee || (__xpg6 &
95 _C99SUSv3_math_errexcept) != 0)) {
96 if (fabs(x) > X_TLOSS)
97 return (_SVID_libm_err(on, ox, 38));
98 }
99
100 if (n == 0)
101 return (j0(x));
102
103 if (n == 1)
104 return (j1(x));
105
106 if ((n & 1) == 0)
107 sgn = 0; /* even n */
108 else
109 sgn = signbit(x); /* old n */
110
111 x = fabs(x);
112
113 if (x == zero || !finite(x)) {
114 b = zero;
115 } else if ((GENERIC)n <= x) {
116 /*
117 * Safe to use
118 * J(n+1,x)=2n/x *J(n,x)-J(n-1,x)
119 */
120 if (x > 1.0e91) {
121 /*
122 * x >> n**2
123 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
124 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
125 * Let s=sin(x), c=cos(x),
126 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
127 *
128 * n sin(xn)*sqt2 cos(xn)*sqt2
129 * ----------------------------------
130 * 0 s-c c+s
131 * 1 -s-c -c+s
132 * 2 -s+c -c-s
133 * 3 s+c c-s
134 */
135 switch (n & 3) {
136 case 0:
137 temp = cos(x) + sin(x);
138 break;
139 case 1:
140 temp = -cos(x) + sin(x);
141 break;
142 case 2:
143 temp = -cos(x) - sin(x);
144 break;
145 case 3:
146 temp = cos(x) - sin(x);
147 break;
148 }
149
150 b = invsqrtpi * temp / sqrt(x);
151 } else {
152 a = j0(x);
153 b = j1(x);
154
155 for (i = 1; i < n; i++) {
156 temp = b;
157 /* avoid underflow */
158 b = b * ((GENERIC)(i + i) / x) - a;
159 a = temp;
160 }
161 }
162 } else {
163 if (x < 1e-9) { /* use J(n,x) = 1/n!*(x/2)^n */
164 b = pow(0.5 * x, (GENERIC)n);
165
166 if (b != zero) {
167 for (a = one, i = 1; i <= n; i++)
168 a *= (GENERIC)i;
169
170 b = b / a;
171 }
172 } else {
173 /*
174 * use backward recurrence
175 * x x^2 x^2
176 * J(n,x)/J(n-1,x) = ---- ------ ------ .....
177 * 2n - 2(n+1) - 2(n+2)
178 *
179 * 1 1 1
180 * (for large x) = ---- ------ ------ .....
181 * 2n 2(n+1) 2(n+2)
182 * -- - ------ - ------ -
183 * x x x
184 *
185 * Let w = 2n/x and h = 2/x, then the above quotient
186 * is equal to the continued fraction:
187 * 1
188 * = -----------------------
189 * 1
190 * w - -----------------
191 * 1
192 * w+h - ---------
193 * w+2h - ...
194 *
195 * To determine how many terms needed, let
196 * Q(0) = w, Q(1) = w(w+h) - 1,
197 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
198 * When Q(k) > 1e4 good for single
199 * When Q(k) > 1e9 good for double
200 * When Q(k) > 1e17 good for quaduple
201 */
202 /* determine k */
203 GENERIC t, v;
204 double q0, q1, h, tmp;
205 int k, m;
206
207 w = (n + n) / (double)x;
208 h = 2.0 / (double)x;
209 q0 = w;
210 z = w + h;
211 q1 = w * z - 1.0;
212 k = 1;
213
214 while (q1 < 1.0e9) {
215 k += 1;
216 z += h;
217 tmp = z * q1 - q0;
218 q0 = q1;
219 q1 = tmp;
220 }
221
222 m = n + n;
223
224 for (t = zero, i = 2 * (n + k); i >= m; i -= 2)
225 t = one / (i / x - t);
226
227 a = t;
228 b = one;
229
230 /* BEGIN CSTYLED */
231 /*
232 * estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
233 * hence, if n*(log(2n/x)) > ...
234 * single:
235 * 8.8722839355e+01
236 * double:
237 * 7.09782712893383973096e+02
238 * long double:
239 * 1.1356523406294143949491931077970765006170e+04
240 * then recurrent value may overflow and the result is
241 * likely underflow to zero
242 */
243 /* END CSTYLED */
244 tmp = n;
245 v = two / x;
246 tmp = tmp * log(fabs(v * tmp));
247
248 if (tmp < 7.09782712893383973096e+02) {
249 for (i = n - 1; i > 0; i--) {
250 temp = b;
251 b = ((i + i) / x) * b - a;
252 a = temp;
253 }
254 } else {
255 for (i = n - 1; i > 0; i--) {
256 temp = b;
257 b = ((i + i) / x) * b - a;
258 a = temp;
259
260 if (b > 1e100) {
261 a /= b;
262 t /= b;
263 b = 1.0;
264 }
265 }
266 }
267
268 b = (t * j0(x) / b);
269 }
270 }
271
272 if (sgn != 0)
273 return (-b);
274 else
275 return (b);
276 }
277
278 GENERIC
279 yn(int n, GENERIC x)
280 {
281 int i;
282 int sign;
283 GENERIC a, b, temp = 0, ox, on;
284
285 ox = x;
286 on = (GENERIC)n;
287
288 if (isnan(x))
289 return (x * x); /* + -> * for Cheetah */
290
291 if (x <= zero) {
292 if (x == zero) {
293 /* return -one/zero; */
294 return (_SVID_libm_err((GENERIC)n, x, 12));
295 } else {
296 /* return zero/zero; */
297 return (_SVID_libm_err((GENERIC)n, x, 13));
298 }
299 }
300
301 if (!((int)_lib_version == libm_ieee || (__xpg6 &
302 _C99SUSv3_math_errexcept) != 0)) {
303 if (x > X_TLOSS)
304 return (_SVID_libm_err(on, ox, 39));
305 }
306
307 sign = 1;
308
309 if (n < 0) {
310 n = -n;
311
312 if ((n & 1) == 1)
313 sign = -1;
314 }
315
316 if (n == 0)
317 return (y0(x));
318
319 if (n == 1)
320 return (sign * y1(x));
321
322 if (!finite(x))
323 return (zero);
324
325 if (x > 1.0e91) {
326 /*
327 * x >> n**2
328 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
329 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
330 * Let s = sin(x), c = cos(x),
331 * xn = x-(2n+1)*pi/4, sqt2 = sqrt(2), then
332 *
333 * n sin(xn)*sqt2 cos(xn)*sqt2
334 * ----------------------------------
335 * 0 s-c c+s
336 * 1 -s-c -c+s
337 * 2 -s+c -c-s
338 * 3 s+c c-s
339 */
340 switch (n & 3) {
341 case 0:
342 temp = sin(x) - cos(x);
343 break;
344 case 1:
345 temp = -sin(x) - cos(x);
346 break;
347 case 2:
348 temp = -sin(x) + cos(x);
349 break;
350 case 3:
351 temp = sin(x) + cos(x);
352 break;
353 }
354
355 b = invsqrtpi * temp / sqrt(x);
356 } else {
357 a = y0(x);
358 b = y1(x);
359
360 /*
361 * fix 1262058 and take care of non-default rounding
362 */
363 for (i = 1; i < n; i++) {
364 temp = b;
365 b *= (GENERIC)(i + i) / x;
366
367 if (b <= -DBL_MAX)
368 break;
369
370 b -= a;
371 a = temp;
372 }
373 }
374
375 if (sign > 0)
376 return (b);
377 else
378 return (-b);
379 }
|