1 /*
2 * ====================================================
3 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
4 *
5 * Developed at SunSoft, a Sun Microsystems, Inc. business.
6 * Permission to use, copy, modify, and distribute this
7 * software is freely granted, provided that this notice
8 * is preserved.
9 * ====================================================
10 */
11
12 /*
13 * jn(n, x), yn(n, x)
14 * floating point Bessel's function of the 1st and 2nd kind
15 * of order n
16 *
17 * Special cases:
18 * y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal;
19 * y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
20 * Note 2. About jn(n,x), yn(n,x)
21 * For n=0, j0(x) is called.
22 * For n=1, j1(x) is called.
23 * For n<x, forward recursion is used starting
24 * from values of j0(x) and j1(x).
25 * For n>x, a continued fraction approximation to
26 * j(n,x)/j(n-1,x) is evaluated and then backward
27 * recursion is used starting from a supposed value
28 * for j(n,x). The resulting values of j(0,x) or j(1,x) are
29 * compared with the actual values to correct the
30 * supposed value of j(n,x).
31 *
32 * yn(n,x) is similar in all respects, except
33 * that forward recursion is used for all
34 * values of n>1.
35 */
36
37 #include "math.h"
38 #include "math_private.h"
39
40 static const volatile double vone = 1, vzero = 0;
41
42 static const double
43 invsqrtpi= 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
44 two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
45 one = 1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */
46
47 static const double zero = 0.00000000000000000000e+00;
48
49 double
jn(int n,double x)50 jn(int n, double x)
51 {
52 int32_t i,hx,ix,lx, sgn;
53 double a, b, c, s, temp, di;
54 double z, w;
55
56 /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
57 * Thus, J(-n,x) = J(n,-x)
58 */
59 EXTRACT_WORDS(hx,lx,x);
60 ix = 0x7fffffff&hx;
61 /* if J(n,NaN) is NaN */
62 if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x;
63 if(n<0){
64 n = -n;
65 x = -x;
66 hx ^= 0x80000000;
67 }
68 if(n==0) return(j0(x));
69 if(n==1) return(j1(x));
70 sgn = (n&1)&(hx>>31); /* even n -- 0, odd n -- sign(x) */
71 x = fabs(x);
72 if((ix|lx)==0||ix>=0x7ff00000) /* if x is 0 or inf */
73 b = zero;
74 else if((double)n<=x) {
75 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
76 if(ix>=0x52D00000) { /* x > 2**302 */
77 /* (x >> n**2)
78 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
79 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
80 * Let s=sin(x), c=cos(x),
81 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2), then
82 *
83 * n sin(xn)*sqt2 cos(xn)*sqt2
84 * ----------------------------------
85 * 0 s-c c+s
86 * 1 -s-c -c+s
87 * 2 -s+c -c-s
88 * 3 s+c c-s
89 */
90 sincos(x, &s, &c);
91 switch(n&3) {
92 case 0: temp = c+s; break;
93 case 1: temp = -c+s; break;
94 case 2: temp = -c-s; break;
95 case 3: temp = c-s; break;
96 }
97 b = invsqrtpi*temp/sqrt(x);
98 } else {
99 a = j0(x);
100 b = j1(x);
101 for(i=1;i<n;i++){
102 temp = b;
103 b = b*((double)(i+i)/x) - a; /* avoid underflow */
104 a = temp;
105 }
106 }
107 } else {
108 if(ix<0x3e100000) { /* x < 2**-29 */
109 /* x is tiny, return the first Taylor expansion of J(n,x)
110 * J(n,x) = 1/n!*(x/2)^n - ...
111 */
112 if(n>33) /* underflow */
113 b = zero;
114 else {
115 temp = x*0.5; b = temp;
116 for (a=one,i=2;i<=n;i++) {
117 a *= (double)i; /* a = n! */
118 b *= temp; /* b = (x/2)^n */
119 }
120 b = b/a;
121 }
122 } else {
123 /* use backward recurrence */
124 /* x x^2 x^2
125 * J(n,x)/J(n-1,x) = ---- ------ ------ .....
126 * 2n - 2(n+1) - 2(n+2)
127 *
128 * 1 1 1
129 * (for large x) = ---- ------ ------ .....
130 * 2n 2(n+1) 2(n+2)
131 * -- - ------ - ------ -
132 * x x x
133 *
134 * Let w = 2n/x and h=2/x, then the above quotient
135 * is equal to the continued fraction:
136 * 1
137 * = -----------------------
138 * 1
139 * w - -----------------
140 * 1
141 * w+h - ---------
142 * w+2h - ...
143 *
144 * To determine how many terms needed, let
145 * Q(0) = w, Q(1) = w(w+h) - 1,
146 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
147 * When Q(k) > 1e4 good for single
148 * When Q(k) > 1e9 good for double
149 * When Q(k) > 1e17 good for quadruple
150 */
151 /* determine k */
152 double t,v;
153 double q0,q1,h,tmp; int32_t k,m;
154 w = (n+n)/(double)x; h = 2.0/(double)x;
155 q0 = w; z = w+h; q1 = w*z - 1.0; k=1;
156 while(q1<1.0e9) {
157 k += 1; z += h;
158 tmp = z*q1 - q0;
159 q0 = q1;
160 q1 = tmp;
161 }
162 m = n+n;
163 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t);
164 a = t;
165 b = one;
166 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
167 * Hence, if n*(log(2n/x)) > ...
168 * single 8.8722839355e+01
169 * double 7.09782712893383973096e+02
170 * long double 1.1356523406294143949491931077970765006170e+04
171 * then recurrent value may overflow and the result is
172 * likely underflow to zero
173 */
174 tmp = n;
175 v = two/x;
176 tmp = tmp*log(fabs(v*tmp));
177 if(tmp<7.09782712893383973096e+02) {
178 for(i=n-1,di=(double)(i+i);i>0;i--){
179 temp = b;
180 b *= di;
181 b = b/x - a;
182 a = temp;
183 di -= two;
184 }
185 } else {
186 for(i=n-1,di=(double)(i+i);i>0;i--){
187 temp = b;
188 b *= di;
189 b = b/x - a;
190 a = temp;
191 di -= two;
192 /* scale b to avoid spurious overflow */
193 if(b>1e100) {
194 a /= b;
195 t /= b;
196 b = one;
197 }
198 }
199 }
200 z = j0(x);
201 w = j1(x);
202 if (fabs(z) >= fabs(w))
203 b = (t*z/b);
204 else
205 b = (t*w/a);
206 }
207 }
208 if(sgn==1) return -b; else return b;
209 }
210
211 double
yn(int n,double x)212 yn(int n, double x)
213 {
214 int32_t i,hx,ix,lx;
215 int32_t sign;
216 double a, b, c, s, temp;
217
218 EXTRACT_WORDS(hx,lx,x);
219 ix = 0x7fffffff&hx;
220 /* yn(n,NaN) = NaN */
221 if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x;
222 /* yn(n,+-0) = -inf and raise divide-by-zero exception. */
223 if((ix|lx)==0) return -one/vzero;
224 /* yn(n,x<0) = NaN and raise invalid exception. */
225 if(hx<0) return vzero/vzero;
226 sign = 1;
227 if(n<0){
228 n = -n;
229 sign = 1 - ((n&1)<<1);
230 }
231 if(n==0) return(y0(x));
232 if(n==1) return(sign*y1(x));
233 if(ix==0x7ff00000) return zero;
234 if(ix>=0x52D00000) { /* x > 2**302 */
235 /* (x >> n**2)
236 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
237 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
238 * Let s=sin(x), c=cos(x),
239 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2), then
240 *
241 * n sin(xn)*sqt2 cos(xn)*sqt2
242 * ----------------------------------
243 * 0 s-c c+s
244 * 1 -s-c -c+s
245 * 2 -s+c -c-s
246 * 3 s+c c-s
247 */
248 sincos(x, &s, &c);
249 switch(n&3) {
250 case 0: temp = s-c; break;
251 case 1: temp = -s-c; break;
252 case 2: temp = -s+c; break;
253 case 3: temp = s+c; break;
254 }
255 b = invsqrtpi*temp/sqrt(x);
256 } else {
257 u_int32_t high;
258 a = y0(x);
259 b = y1(x);
260 /* quit if b is -inf */
261 GET_HIGH_WORD(high,b);
262 for(i=1;i<n&&high!=0xfff00000;i++){
263 temp = b;
264 b = ((double)(i+i)/x)*b - a;
265 GET_HIGH_WORD(high,b);
266 a = temp;
267 }
268 }
269 if(sign>0) return b; else return -b;
270 }
271