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--- old/usr/src/lib/libm/common/C/jn.c
+++ new/usr/src/lib/libm/common/C/jn.c
1 1 /*
2 2 * CDDL HEADER START
3 3 *
4 4 * The contents of this file are subject to the terms of the
5 5 * Common Development and Distribution License (the "License").
6 6 * You may not use this file except in compliance with the License.
7 7 *
8 8 * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE
9 9 * or http://www.opensolaris.org/os/licensing.
10 10 * See the License for the specific language governing permissions
11 11 * and limitations under the License.
12 12 *
13 13 * When distributing Covered Code, include this CDDL HEADER in each
14 14 * file and include the License file at usr/src/OPENSOLARIS.LICENSE.
15 15 * If applicable, add the following below this CDDL HEADER, with the
16 16 * fields enclosed by brackets "[]" replaced with your own identifying
17 17 * information: Portions Copyright [yyyy] [name of copyright owner]
18 18 *
19 19 * CDDL HEADER END
20 20 */
21 21
22 22 /*
23 23 * Copyright 2011 Nexenta Systems, Inc. All rights reserved.
24 24 */
25 25 /*
26 26 * Copyright 2006 Sun Microsystems, Inc. All rights reserved.
27 27 * Use is subject to license terms.
28 28 */
29 29
30 30 #pragma weak jn = __jn
31 31 #pragma weak yn = __yn
32 32
33 33 /*
34 34 * floating point Bessel's function of the 1st and 2nd kind
35 35 * of order n: jn(n,x),yn(n,x);
36 36 *
37 37 * Special cases:
38 38 * y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal;
39 39 * y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
40 40 * Note 2. About jn(n,x), yn(n,x)
41 41 * For n=0, j0(x) is called,
42 42 * for n=1, j1(x) is called,
43 43 * for n<x, forward recursion us used starting
44 44 * from values of j0(x) and j1(x).
45 45 * for n>x, a continued fraction approximation to
46 46 * j(n,x)/j(n-1,x) is evaluated and then backward
47 47 * recursion is used starting from a supposed value
48 48 * for j(n,x). The resulting value of j(0,x) is
49 49 * compared with the actual value to correct the
50 50 * supposed value of j(n,x).
51 51 *
52 52 * yn(n,x) is similar in all respects, except
53 53 * that forward recursion is used for all
54 54 * values of n>1.
55 55 *
56 56 */
57 57
58 58 #include "libm.h"
59 59 #include <float.h> /* DBL_MIN */
60 60 #include <values.h> /* X_TLOSS */
61 61 #include "xpg6.h" /* __xpg6 */
62 62
63 63 #define GENERIC double
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64 64
65 65 static const GENERIC
66 66 invsqrtpi = 5.641895835477562869480794515607725858441e-0001,
67 67 two = 2.0,
68 68 zero = 0.0,
69 69 one = 1.0;
70 70
71 71 GENERIC
72 72 jn(int n, GENERIC x) {
73 73 int i, sgn;
74 - GENERIC a, b, temp;
74 + GENERIC a, b, temp = 0;
75 75 GENERIC z, w, ox, on;
76 76
77 77 /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
78 78 * Thus, J(-n,x) = J(n,-x)
79 79 */
80 80 ox = x; on = (GENERIC)n;
81 81 if(n<0){
82 82 n = -n;
83 83 x = -x;
84 84 }
85 85 if(isnan(x)) return x*x; /* + -> * for Cheetah */
86 86 if (!((int) _lib_version == libm_ieee ||
87 87 (__xpg6 & _C99SUSv3_math_errexcept) != 0)) {
88 88 if(fabs(x) > X_TLOSS) return _SVID_libm_err(on,ox,38);
89 89 }
90 90 if(n==0) return(j0(x));
91 91 if(n==1) return(j1(x));
92 92 if((n&1)==0)
93 93 sgn=0; /* even n */
94 94 else
95 95 sgn = signbit(x); /* old n */
96 96 x = fabs(x);
97 97 if(x == zero||!finite(x)) b = zero;
98 98 else if((GENERIC)n<=x) { /* Safe to use
99 99 J(n+1,x)=2n/x *J(n,x)-J(n-1,x)
100 100 */
101 101 if(x>1.0e91) { /* x >> n**2
102 102 Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
103 103 Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
104 104 Let s=sin(x), c=cos(x),
105 105 xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
106 106
107 107 n sin(xn)*sqt2 cos(xn)*sqt2
108 108 ----------------------------------
109 109 0 s-c c+s
110 110 1 -s-c -c+s
111 111 2 -s+c -c-s
112 112 3 s+c c-s
113 113 */
114 114 switch(n&3) {
115 115 case 0: temp = cos(x)+sin(x); break;
116 116 case 1: temp = -cos(x)+sin(x); break;
117 117 case 2: temp = -cos(x)-sin(x); break;
118 118 case 3: temp = cos(x)-sin(x); break;
119 119 }
120 120 b = invsqrtpi*temp/sqrt(x);
121 121 } else {
122 122 a = j0(x);
123 123 b = j1(x);
124 124 for(i=1;i<n;i++){
125 125 temp = b;
126 126 b = b*((GENERIC)(i+i)/x) - a; /* avoid underflow */
127 127 a = temp;
128 128 }
129 129 }
130 130 } else {
131 131 if(x<1e-9) { /* use J(n,x) = 1/n!*(x/2)^n */
132 132 b = pow(0.5*x,(GENERIC) n);
133 133 if (b!=zero) {
134 134 for(a=one,i=1;i<=n;i++) a *= (GENERIC)i;
135 135 b = b/a;
136 136 }
137 137 } else {
138 138 /* use backward recurrence */
139 139 /* x x^2 x^2
140 140 * J(n,x)/J(n-1,x) = ---- ------ ------ .....
141 141 * 2n - 2(n+1) - 2(n+2)
142 142 *
143 143 * 1 1 1
144 144 * (for large x) = ---- ------ ------ .....
145 145 * 2n 2(n+1) 2(n+2)
146 146 * -- - ------ - ------ -
147 147 * x x x
148 148 *
149 149 * Let w = 2n/x and h=2/x, then the above quotient
150 150 * is equal to the continued fraction:
151 151 * 1
152 152 * = -----------------------
153 153 * 1
154 154 * w - -----------------
155 155 * 1
156 156 * w+h - ---------
157 157 * w+2h - ...
158 158 *
159 159 * To determine how many terms needed, let
160 160 * Q(0) = w, Q(1) = w(w+h) - 1,
161 161 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
162 162 * When Q(k) > 1e4 good for single
163 163 * When Q(k) > 1e9 good for double
164 164 * When Q(k) > 1e17 good for quaduple
165 165 */
166 166 /* determin k */
167 167 GENERIC t,v;
168 168 double q0,q1,h,tmp; int k,m;
169 169 w = (n+n)/(double)x; h = 2.0/(double)x;
170 170 q0 = w; z = w+h; q1 = w*z - 1.0; k=1;
171 171 while(q1<1.0e9) {
172 172 k += 1; z += h;
173 173 tmp = z*q1 - q0;
174 174 q0 = q1;
175 175 q1 = tmp;
176 176 }
177 177 m = n+n;
178 178 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t);
179 179 a = t;
180 180 b = one;
181 181 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
182 182 hence, if n*(log(2n/x)) > ...
183 183 single 8.8722839355e+01
184 184 double 7.09782712893383973096e+02
185 185 long double 1.1356523406294143949491931077970765006170e+04
186 186 then recurrent value may overflow and the result is
187 187 likely underflow to zero
188 188 */
189 189 tmp = n;
190 190 v = two/x;
191 191 tmp = tmp*log(fabs(v*tmp));
192 192 if(tmp<7.09782712893383973096e+02) {
193 193 for(i=n-1;i>0;i--){
194 194 temp = b;
195 195 b = ((i+i)/x)*b - a;
196 196 a = temp;
197 197 }
198 198 } else {
199 199 for(i=n-1;i>0;i--){
200 200 temp = b;
201 201 b = ((i+i)/x)*b - a;
202 202 a = temp;
203 203 if(b>1e100) {
204 204 a /= b;
205 205 t /= b;
206 206 b = 1.0;
207 207 }
208 208 }
209 209 }
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210 210 b = (t*j0(x)/b);
211 211 }
212 212 }
213 213 if(sgn==1) return -b; else return b;
214 214 }
215 215
216 216 GENERIC
217 217 yn(int n, GENERIC x) {
218 218 int i;
219 219 int sign;
220 - GENERIC a, b, temp, ox, on;
220 + GENERIC a, b, temp = 0, ox, on;
221 221
222 222 ox = x; on = (GENERIC)n;
223 223 if(isnan(x)) return x*x; /* + -> * for Cheetah */
224 - if (x <= zero)
225 - if(x==zero)
224 + if (x <= zero) {
225 + if(x==zero) {
226 226 /* return -one/zero; */
227 227 return _SVID_libm_err((GENERIC)n,x,12);
228 - else
228 + } else {
229 229 /* return zero/zero; */
230 230 return _SVID_libm_err((GENERIC)n,x,13);
231 + }
232 + }
231 233 if (!((int) _lib_version == libm_ieee ||
232 234 (__xpg6 & _C99SUSv3_math_errexcept) != 0)) {
233 235 if(x > X_TLOSS) return _SVID_libm_err(on,ox,39);
234 236 }
235 237 sign = 1;
236 238 if(n<0){
237 239 n = -n;
238 240 if((n&1) == 1) sign = -1;
239 241 }
240 242 if(n==0) return(y0(x));
241 243 if(n==1) return(sign*y1(x));
242 244 if(!finite(x)) return zero;
243 245
244 246 if(x>1.0e91) { /* x >> n**2
245 247 Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
246 248 Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
247 249 Let s=sin(x), c=cos(x),
248 250 xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
249 251
250 252 n sin(xn)*sqt2 cos(xn)*sqt2
251 253 ----------------------------------
252 254 0 s-c c+s
253 255 1 -s-c -c+s
254 256 2 -s+c -c-s
255 257 3 s+c c-s
256 258 */
257 259 switch(n&3) {
258 260 case 0: temp = sin(x)-cos(x); break;
259 261 case 1: temp = -sin(x)-cos(x); break;
260 262 case 2: temp = -sin(x)+cos(x); break;
261 263 case 3: temp = sin(x)+cos(x); break;
262 264 }
263 265 b = invsqrtpi*temp/sqrt(x);
264 266 } else {
265 267 a = y0(x);
266 268 b = y1(x);
267 269 /*
268 270 * fix 1262058 and take care of non-default rounding
269 271 */
270 272 for (i = 1; i < n; i++) {
271 273 temp = b;
272 274 b *= (GENERIC) (i + i) / x;
273 275 if (b <= -DBL_MAX)
274 276 break;
275 277 b -= a;
276 278 a = temp;
277 279 }
278 280 }
279 281 if(sign>0) return b; else return -b;
280 282 }
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