1
2 /* @(#)e_jn.c 1.4 95/01/18 */
3 /*
4 * ====================================================
5 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
6 *
7 * Developed at SunSoft, a Sun Microsystems, Inc. business.
8 * Permission to use, copy, modify, and distribute this
9 * software is freely granted, provided that this notice
10 * is preserved.
11 * ====================================================
12 */
13
14 #include <sys/cdefs.h>
15 __FBSDID("$FreeBSD: head/lib/msun/src/e_jn.c 279856 2015-03-10 17:10:54Z kargl $");
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 volatile double vone = 1, vzero = 0;
47
48 static const double
49 invsqrtpi= 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
50 two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
51 one = 1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */
52
53 static const double zero = 0.00000000000000000000e+00;
54
55 double
__ieee754_jn(int n,double x)56 __ieee754_jn(int n, double x)
57 {
58 int32_t i,hx,ix,lx, sgn;
59 double a, b, temp, di;
60 double z, w;
61
62 /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
63 * Thus, J(-n,x) = J(n,-x)
64 */
65 EXTRACT_WORDS(hx,lx,x);
66 ix = 0x7fffffff&hx;
67 /* if J(n,NaN) is NaN */
68 if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x;
69 if(n<0){
70 n = -n;
71 x = -x;
72 hx ^= 0x80000000;
73 }
74 if(n==0) return(__ieee754_j0(x));
75 if(n==1) return(__ieee754_j1(x));
76 sgn = (n&1)&(hx>>31); /* even n -- 0, odd n -- sign(x) */
77 x = fabs(x);
78 if((ix|lx)==0||ix>=0x7ff00000) /* if x is 0 or inf */
79 b = zero;
80 else if((double)n<=x) {
81 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
82 if(ix>=0x52D00000) { /* x > 2**302 */
83 /* (x >> n**2)
84 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
85 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
86 * Let s=sin(x), c=cos(x),
87 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
88 *
89 * n sin(xn)*sqt2 cos(xn)*sqt2
90 * ----------------------------------
91 * 0 s-c c+s
92 * 1 -s-c -c+s
93 * 2 -s+c -c-s
94 * 3 s+c c-s
95 */
96 switch(n&3) {
97 case 0: temp = cos(x)+sin(x); break;
98 case 1: temp = -cos(x)+sin(x); break;
99 case 2: temp = -cos(x)-sin(x); break;
100 case 3: temp = cos(x)-sin(x); break;
101 }
102 b = invsqrtpi*temp/sqrt(x);
103 } else {
104 a = __ieee754_j0(x);
105 b = __ieee754_j1(x);
106 for(i=1;i<n;i++){
107 temp = b;
108 b = b*((double)(i+i)/x) - a; /* avoid underflow */
109 a = temp;
110 }
111 }
112 } else {
113 if(ix<0x3e100000) { /* x < 2**-29 */
114 /* x is tiny, return the first Taylor expansion of J(n,x)
115 * J(n,x) = 1/n!*(x/2)^n - ...
116 */
117 if(n>33) /* underflow */
118 b = zero;
119 else {
120 temp = x*0.5; b = temp;
121 for (a=one,i=2;i<=n;i++) {
122 a *= (double)i; /* a = n! */
123 b *= temp; /* b = (x/2)^n */
124 }
125 b = b/a;
126 }
127 } else {
128 /* use backward recurrence */
129 /* x x^2 x^2
130 * J(n,x)/J(n-1,x) = ---- ------ ------ .....
131 * 2n - 2(n+1) - 2(n+2)
132 *
133 * 1 1 1
134 * (for large x) = ---- ------ ------ .....
135 * 2n 2(n+1) 2(n+2)
136 * -- - ------ - ------ -
137 * x x x
138 *
139 * Let w = 2n/x and h=2/x, then the above quotient
140 * is equal to the continued fraction:
141 * 1
142 * = -----------------------
143 * 1
144 * w - -----------------
145 * 1
146 * w+h - ---------
147 * w+2h - ...
148 *
149 * To determine how many terms needed, let
150 * Q(0) = w, Q(1) = w(w+h) - 1,
151 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
152 * When Q(k) > 1e4 good for single
153 * When Q(k) > 1e9 good for double
154 * When Q(k) > 1e17 good for quadruple
155 */
156 /* determine k */
157 double t,v;
158 double q0,q1,h,tmp; int32_t k,m;
159 w = (n+n)/(double)x; h = 2.0/(double)x;
160 q0 = w; z = w+h; q1 = w*z - 1.0; k=1;
161 while(q1<1.0e9) {
162 k += 1; z += h;
163 tmp = z*q1 - q0;
164 q0 = q1;
165 q1 = tmp;
166 }
167 m = n+n;
168 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t);
169 a = t;
170 b = one;
171 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
172 * Hence, if n*(log(2n/x)) > ...
173 * single 8.8722839355e+01
174 * double 7.09782712893383973096e+02
175 * long double 1.1356523406294143949491931077970765006170e+04
176 * then recurrent value may overflow and the result is
177 * likely underflow to zero
178 */
179 tmp = n;
180 v = two/x;
181 tmp = tmp*__ieee754_log(fabs(v*tmp));
182 if(tmp<7.09782712893383973096e+02) {
183 for(i=n-1,di=(double)(i+i);i>0;i--){
184 temp = b;
185 b *= di;
186 b = b/x - a;
187 a = temp;
188 di -= two;
189 }
190 } else {
191 for(i=n-1,di=(double)(i+i);i>0;i--){
192 temp = b;
193 b *= di;
194 b = b/x - a;
195 a = temp;
196 di -= two;
197 /* scale b to avoid spurious overflow */
198 if(b>1e100) {
199 a /= b;
200 t /= b;
201 b = one;
202 }
203 }
204 }
205 z = __ieee754_j0(x);
206 w = __ieee754_j1(x);
207 if (fabs(z) >= fabs(w))
208 b = (t*z/b);
209 else
210 b = (t*w/a);
211 }
212 }
213 if(sgn==1) return -b; else return b;
214 }
215
216 double
__ieee754_yn(int n,double x)217 __ieee754_yn(int n, double x)
218 {
219 int32_t i,hx,ix,lx;
220 int32_t sign;
221 double a, b, temp;
222
223 EXTRACT_WORDS(hx,lx,x);
224 ix = 0x7fffffff&hx;
225 /* yn(n,NaN) = NaN */
226 if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x;
227 /* yn(n,+-0) = -inf and raise divide-by-zero exception. */
228 if((ix|lx)==0) return -one/vzero;
229 /* yn(n,x<0) = NaN and raise invalid exception. */
230 if(hx<0) return vzero/vzero;
231 sign = 1;
232 if(n<0){
233 n = -n;
234 sign = 1 - ((n&1)<<1);
235 }
236 if(n==0) return(__ieee754_y0(x));
237 if(n==1) return(sign*__ieee754_y1(x));
238 if(ix==0x7ff00000) return zero;
239 if(ix>=0x52D00000) { /* x > 2**302 */
240 /* (x >> n**2)
241 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
242 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
243 * Let s=sin(x), c=cos(x),
244 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
245 *
246 * n sin(xn)*sqt2 cos(xn)*sqt2
247 * ----------------------------------
248 * 0 s-c c+s
249 * 1 -s-c -c+s
250 * 2 -s+c -c-s
251 * 3 s+c c-s
252 */
253 switch(n&3) {
254 case 0: temp = sin(x)-cos(x); break;
255 case 1: temp = -sin(x)-cos(x); break;
256 case 2: temp = -sin(x)+cos(x); break;
257 case 3: temp = sin(x)+cos(x); break;
258 }
259 b = invsqrtpi*temp/sqrt(x);
260 } else {
261 u_int32_t high;
262 a = __ieee754_y0(x);
263 b = __ieee754_y1(x);
264 /* quit if b is -inf */
265 GET_HIGH_WORD(high,b);
266 for(i=1;i<n&&high!=0xfff00000;i++){
267 temp = b;
268 b = ((double)(i+i)/x)*b - a;
269 GET_HIGH_WORD(high,b);
270 a = temp;
271 }
272 }
273 if(sign>0) return b; else return -b;
274 }
275