1 /* @(#)e_jn.c 1.4 95/01/18 */
2 /*
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5 *
6 * Developed at SunSoft, 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 #include <sys/cdefs.h>
14 /*
15 * jn(n, x), yn(n, x)
16 * floating point Bessel's function of the 1st and 2nd kind
17 * of order n
18 *
19 * Special cases:
20 * y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal;
21 * y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
22 * Note 2. About jn(n,x), yn(n,x)
23 * For n=0, j0(x) is called.
24 * For n=1, j1(x) is called.
25 * For n<x, forward recursion is used starting
26 * from values of j0(x) and j1(x).
27 * For n>x, a continued fraction approximation to
28 * j(n,x)/j(n-1,x) is evaluated and then backward
29 * recursion is used starting from a supposed value
30 * for j(n,x). The resulting values of j(0,x) or j(1,x) are
31 * compared with the actual values to correct the
32 * supposed value of j(n,x).
33 *
34 * yn(n,x) is similar in all respects, except
35 * that forward recursion is used for all
36 * values of n>1.
37 */
38
39 #include "math.h"
40 #include "math_private.h"
41
42 static const volatile double vone = 1, vzero = 0;
43
44 static const double
45 invsqrtpi= 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
46 two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
47 one = 1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */
48
49 static const double zero = 0.00000000000000000000e+00;
50
51 double
jn(int n,double x)52 jn(int n, double x)
53 {
54 int32_t i,hx,ix,lx, sgn;
55 double a, b, c, s, temp, di;
56 double z, w;
57
58 /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
59 * Thus, J(-n,x) = J(n,-x)
60 */
61 EXTRACT_WORDS(hx,lx,x);
62 ix = 0x7fffffff&hx;
63 /* if J(n,NaN) is NaN */
64 if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x;
65 if(n<0){
66 n = -n;
67 x = -x;
68 hx ^= 0x80000000;
69 }
70 if(n==0) return(j0(x));
71 if(n==1) return(j1(x));
72 sgn = (n&1)&(hx>>31); /* even n -- 0, odd n -- sign(x) */
73 x = fabs(x);
74 if((ix|lx)==0||ix>=0x7ff00000) /* if x is 0 or inf */
75 b = zero;
76 else if((double)n<=x) {
77 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
78 if(ix>=0x52D00000) { /* x > 2**302 */
79 /* (x >> n**2)
80 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
81 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
82 * Let s=sin(x), c=cos(x),
83 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2), then
84 *
85 * n sin(xn)*sqt2 cos(xn)*sqt2
86 * ----------------------------------
87 * 0 s-c c+s
88 * 1 -s-c -c+s
89 * 2 -s+c -c-s
90 * 3 s+c c-s
91 */
92 sincos(x, &s, &c);
93 switch(n&3) {
94 case 0: temp = c+s; break;
95 case 1: temp = -c+s; break;
96 case 2: temp = -c-s; break;
97 case 3: temp = c-s; break;
98 }
99 b = invsqrtpi*temp/sqrt(x);
100 } else {
101 a = j0(x);
102 b = j1(x);
103 for(i=1;i<n;i++){
104 temp = b;
105 b = b*((double)(i+i)/x) - a; /* avoid underflow */
106 a = temp;
107 }
108 }
109 } else {
110 if(ix<0x3e100000) { /* x < 2**-29 */
111 /* x is tiny, return the first Taylor expansion of J(n,x)
112 * J(n,x) = 1/n!*(x/2)^n - ...
113 */
114 if(n>33) /* underflow */
115 b = zero;
116 else {
117 temp = x*0.5; b = temp;
118 for (a=one,i=2;i<=n;i++) {
119 a *= (double)i; /* a = n! */
120 b *= temp; /* b = (x/2)^n */
121 }
122 b = b/a;
123 }
124 } else {
125 /* use backward recurrence */
126 /* x x^2 x^2
127 * J(n,x)/J(n-1,x) = ---- ------ ------ .....
128 * 2n - 2(n+1) - 2(n+2)
129 *
130 * 1 1 1
131 * (for large x) = ---- ------ ------ .....
132 * 2n 2(n+1) 2(n+2)
133 * -- - ------ - ------ -
134 * x x x
135 *
136 * Let w = 2n/x and h=2/x, then the above quotient
137 * is equal to the continued fraction:
138 * 1
139 * = -----------------------
140 * 1
141 * w - -----------------
142 * 1
143 * w+h - ---------
144 * w+2h - ...
145 *
146 * To determine how many terms needed, let
147 * Q(0) = w, Q(1) = w(w+h) - 1,
148 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
149 * When Q(k) > 1e4 good for single
150 * When Q(k) > 1e9 good for double
151 * When Q(k) > 1e17 good for quadruple
152 */
153 /* determine k */
154 double t,v;
155 double q0,q1,h,tmp; int32_t k,m;
156 w = (n+n)/(double)x; h = 2.0/(double)x;
157 q0 = w; z = w+h; q1 = w*z - 1.0; k=1;
158 while(q1<1.0e9) {
159 k += 1; z += h;
160 tmp = z*q1 - q0;
161 q0 = q1;
162 q1 = tmp;
163 }
164 m = n+n;
165 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t);
166 a = t;
167 b = one;
168 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
169 * Hence, if n*(log(2n/x)) > ...
170 * single 8.8722839355e+01
171 * double 7.09782712893383973096e+02
172 * long double 1.1356523406294143949491931077970765006170e+04
173 * then recurrent value may overflow and the result is
174 * likely underflow to zero
175 */
176 tmp = n;
177 v = two/x;
178 tmp = tmp*log(fabs(v*tmp));
179 if(tmp<7.09782712893383973096e+02) {
180 for(i=n-1,di=(double)(i+i);i>0;i--){
181 temp = b;
182 b *= di;
183 b = b/x - a;
184 a = temp;
185 di -= two;
186 }
187 } else {
188 for(i=n-1,di=(double)(i+i);i>0;i--){
189 temp = b;
190 b *= di;
191 b = b/x - a;
192 a = temp;
193 di -= two;
194 /* scale b to avoid spurious overflow */
195 if(b>1e100) {
196 a /= b;
197 t /= b;
198 b = one;
199 }
200 }
201 }
202 z = j0(x);
203 w = j1(x);
204 if (fabs(z) >= fabs(w))
205 b = (t*z/b);
206 else
207 b = (t*w/a);
208 }
209 }
210 if(sgn==1) return -b; else return b;
211 }
212
213 double
yn(int n,double x)214 yn(int n, double x)
215 {
216 int32_t i,hx,ix,lx;
217 int32_t sign;
218 double a, b, c, s, temp;
219
220 EXTRACT_WORDS(hx,lx,x);
221 ix = 0x7fffffff&hx;
222 /* yn(n,NaN) = NaN */
223 if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x;
224 /* yn(n,+-0) = -inf and raise divide-by-zero exception. */
225 if((ix|lx)==0) return -one/vzero;
226 /* yn(n,x<0) = NaN and raise invalid exception. */
227 if(hx<0) return vzero/vzero;
228 sign = 1;
229 if(n<0){
230 n = -n;
231 sign = 1 - ((n&1)<<1);
232 }
233 if(n==0) return(y0(x));
234 if(n==1) return(sign*y1(x));
235 if(ix==0x7ff00000) return zero;
236 if(ix>=0x52D00000) { /* x > 2**302 */
237 /* (x >> n**2)
238 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
239 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
240 * Let s=sin(x), c=cos(x),
241 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2), then
242 *
243 * n sin(xn)*sqt2 cos(xn)*sqt2
244 * ----------------------------------
245 * 0 s-c c+s
246 * 1 -s-c -c+s
247 * 2 -s+c -c-s
248 * 3 s+c c-s
249 */
250 sincos(x, &s, &c);
251 switch(n&3) {
252 case 0: temp = s-c; break;
253 case 1: temp = -s-c; break;
254 case 2: temp = -s+c; break;
255 case 3: temp = s+c; break;
256 }
257 b = invsqrtpi*temp/sqrt(x);
258 } else {
259 u_int32_t high;
260 a = y0(x);
261 b = y1(x);
262 /* quit if b is -inf */
263 GET_HIGH_WORD(high,b);
264 for(i=1;i<n&&high!=0xfff00000;i++){
265 temp = b;
266 b = ((double)(i+i)/x)*b - a;
267 GET_HIGH_WORD(high,b);
268 a = temp;
269 }
270 }
271 if(sign>0) return b; else return -b;
272 }
273