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e_jn.c (7109B)
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1 /* @(#)e_jn.c 5.1 93/09/24 */
2 /*
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5 *
6 * Developed at SunPro, a Sun Microsystems, Inc. business.
7 * Permission to use, copy, modify, and distribute this
8 * software is freely granted, provided that this notice
9 * is preserved.
10 * ====================================================
11 */
12
13 #ifndef lint
14 static char rcsid[] = "$FreeBSD: src/lib/msun/src/e_jn.c,v 1.8 2002/05/28 18:15:04 alfred Exp $";
15 #endif
16
17 /*
18 * __ieee754_jn(n, x), __ieee754_yn(n, x)
19 * floating point Bessel's function of the 1st and 2nd kind
20 * of order n
21 *
22 * Special cases:
23 * y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal;
24 * y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
25 * Note 2. About jn(n,x), yn(n,x)
26 * For n=0, j0(x) is called,
27 * for n=1, j1(x) is called,
28 * for n<x, forward recursion us used starting
29 * from values of j0(x) and j1(x).
30 * for n>x, a continued fraction approximation to
31 * j(n,x)/j(n-1,x) is evaluated and then backward
32 * recursion is used starting from a supposed value
33 * for j(n,x). The resulting value of j(0,x) is
34 * compared with the actual value to correct the
35 * supposed value of j(n,x).
36 *
37 * yn(n,x) is similar in all respects, except
38 * that forward recursion is used for all
39 * values of n>1.
40 *
41 */
42
43 #include "math.h"
44 #include "math_private.h"
45
46 static const double
47 invsqrtpi= 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
48 two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
49 one = 1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */
50
51 static const double zero = 0.00000000000000000000e+00;
52
53 double
54 __ieee754_jn(int n, double x)
55 {
56 int32_t i,hx,ix,lx, sgn;
57 double a, b, temp, di;
58 double z, w;
59
60 /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
61 * Thus, J(-n,x) = J(n,-x)
62 */
63 EXTRACT_WORDS(hx,lx,x);
64 ix = 0x7fffffff&hx;
65 /* if J(n,NaN) is NaN */
66 if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x;
67 if(n<0){
68 n = -n;
69 x = -x;
70 hx ^= 0x80000000;
71 }
72 if(n==0) return(__ieee754_j0(x));
73 if(n==1) return(__ieee754_j1(x));
74 sgn = (n&1)&(hx>>31); /* even n -- 0, odd n -- sign(x) */
75 x = fabs(x);
76 if((ix|lx)==0||ix>=0x7ff00000) /* if x is 0 or inf */
77 b = zero;
78 else if((double)n<=x) {
79 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
80 if(ix>=0x52D00000) { /* x > 2**302 */
81 /* (x >> n**2)
82 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
83 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
84 * Let s=sin(x), c=cos(x),
85 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
86 *
87 * n sin(xn)*sqt2 cos(xn)*sqt2
88 * ----------------------------------
89 * 0 s-c c+s
90 * 1 -s-c -c+s
91 * 2 -s+c -c-s
92 * 3 s+c c-s
93 */
94 switch(n&3) {
95 case 0: temp = cos(x)+sin(x); break;
96 case 1: temp = -cos(x)+sin(x); break;
97 case 2: temp = -cos(x)-sin(x); break;
98 case 3: temp = cos(x)-sin(x); break;
99 }
100 b = invsqrtpi*temp/sqrt(x);
101 } else {
102 a = __ieee754_j0(x);
103 b = __ieee754_j1(x);
104 for(i=1;i<n;i++){
105 temp = b;
106 b = b*((double)(i+i)/x) - a; /* avoid underflow */
107 a = temp;
108 }
109 }
110 } else {
111 if(ix<0x3e100000) { /* x < 2**-29 */
112 /* x is tiny, return the first Taylor expansion of J(n,x)
113 * J(n,x) = 1/n!*(x/2)^n - ...
114 */
115 if(n>33) /* underflow */
116 b = zero;
117 else {
118 temp = x*0.5; b = temp;
119 for (a=one,i=2;i<=n;i++) {
120 a *= (double)i; /* a = n! */
121 b *= temp; /* b = (x/2)^n */
122 }
123 b = b/a;
124 }
125 } else {
126 /* use backward recurrence */
127 /* x x^2 x^2
128 * J(n,x)/J(n-1,x) = ---- ------ ------ .....
129 * 2n - 2(n+1) - 2(n+2)
130 *
131 * 1 1 1
132 * (for large x) = ---- ------ ------ .....
133 * 2n 2(n+1) 2(n+2)
134 * -- - ------ - ------ -
135 * x x x
136 *
137 * Let w = 2n/x and h=2/x, then the above quotient
138 * is equal to the continued fraction:
139 * 1
140 * = -----------------------
141 * 1
142 * w - -----------------
143 * 1
144 * w+h - ---------
145 * w+2h - ...
146 *
147 * To determine how many terms needed, let
148 * Q(0) = w, Q(1) = w(w+h) - 1,
149 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
150 * When Q(k) > 1e4 good for single
151 * When Q(k) > 1e9 good for double
152 * When Q(k) > 1e17 good for quadruple
153 */
154 /* determine k */
155 double t,v;
156 double q0,q1,h,tmp; int32_t k,m;
157 w = (n+n)/(double)x; h = 2.0/(double)x;
158 q0 = w; z = w+h; q1 = w*z - 1.0; k=1;
159 while(q1<1.0e9) {
160 k += 1; z += h;
161 tmp = z*q1 - q0;
162 q0 = q1;
163 q1 = tmp;
164 }
165 m = n+n;
166 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t);
167 a = t;
168 b = one;
169 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
170 * Hence, if n*(log(2n/x)) > ...
171 * single 8.8722839355e+01
172 * double 7.09782712893383973096e+02
173 * long double 1.1356523406294143949491931077970765006170e+04
174 * then recurrent value may overflow and the result is
175 * likely underflow to zero
176 */
177 tmp = n;
178 v = two/x;
179 tmp = tmp*__ieee754_log(fabs(v*tmp));
180 if(tmp<7.09782712893383973096e+02) {
181 for(i=n-1,di=(double)(i+i);i>0;i--){
182 temp = b;
183 b *= di;
184 b = b/x - a;
185 a = temp;
186 di -= two;
187 }
188 } else {
189 for(i=n-1,di=(double)(i+i);i>0;i--){
190 temp = b;
191 b *= di;
192 b = b/x - a;
193 a = temp;
194 di -= two;
195 /* scale b to avoid spurious overflow */
196 if(b>1e100) {
197 a /= b;
198 t /= b;
199 b = one;
200 }
201 }
202 }
203 b = (t*__ieee754_j0(x)/b);
204 }
205 }
206 if(sgn==1) return -b; else return b;
207 }
208
209 double
210 __ieee754_yn(int n, double x)
211 {
212 int32_t i,hx,ix,lx;
213 int32_t sign;
214 double a, b, temp;
215
216 EXTRACT_WORDS(hx,lx,x);
217 ix = 0x7fffffff&hx;
218 /* if Y(n,NaN) is NaN */
219 if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x;
220 if((ix|lx)==0) return -one/zero;
221 if(hx<0) return zero/zero;
222 sign = 1;
223 if(n<0){
224 n = -n;
225 sign = 1 - ((n&1)<<1);
226 }
227 if(n==0) return(__ieee754_y0(x));
228 if(n==1) return(sign*__ieee754_y1(x));
229 if(ix==0x7ff00000) return zero;
230 if(ix>=0x52D00000) { /* x > 2**302 */
231 /* (x >> n**2)
232 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
233 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
234 * Let s=sin(x), c=cos(x),
235 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
236 *
237 * n sin(xn)*sqt2 cos(xn)*sqt2
238 * ----------------------------------
239 * 0 s-c c+s
240 * 1 -s-c -c+s
241 * 2 -s+c -c-s
242 * 3 s+c c-s
243 */
244 switch(n&3) {
245 case 0: temp = sin(x)-cos(x); break;
246 case 1: temp = -sin(x)-cos(x); break;
247 case 2: temp = -sin(x)+cos(x); break;
248 case 3: temp = sin(x)+cos(x); break;
249 }
250 b = invsqrtpi*temp/sqrt(x);
251 } else {
252 u_int32_t high;
253 a = __ieee754_y0(x);
254 b = __ieee754_y1(x);
255 /* quit if b is -inf */
256 GET_HIGH_WORD(high,b);
257 for(i=1;i<n&&high!=0xfff00000;i++){
258 temp = b;
259 b = ((double)(i+i)/x)*b - a;
260 GET_HIGH_WORD(high,b);
261 a = temp;
262 }
263 }
264 if(sign>0) return b; else return -b;
265 }