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 /*
14 * jn(n, x), yn(n, x)
15 * floating point Bessel's function of the 1st and 2nd kind
16 * of order n
17 *
18 * Special cases:
19 * y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal;
20 * y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
21 * Note 2. About jn(n,x), yn(n,x)
22 * For n=0, j0(x) is called,
23 * for n=1, j1(x) is called,
24 * for n<x, forward recursion us used starting
25 * from values of j0(x) and j1(x).
26 * for n>x, a continued fraction approximation to
27 * j(n,x)/j(n-1,x) is evaluated and then backward
28 * recursion is used starting from a supposed value
29 * for j(n,x). The resulting value of j(0,x) is
30 * compared with the actual value to correct the
31 * supposed value of j(n,x).
32 *
33 * yn(n,x) is similar in all respects, except
34 * that forward recursion is used for all
35 * values of n>1.
36 *
37 */
38
39 #include "math.h"
40 #include "math_private.h"
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, 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 switch(n&3) {
91 case 0: temp = cos(x)+sin(x); break;
92 case 1: temp = -cos(x)+sin(x); break;
93 case 2: temp = -cos(x)-sin(x); break;
94 case 3: temp = cos(x)-sin(x); break;
95 }
96 b = invsqrtpi*temp/sqrt(x);
97 } else {
98 a = j0(x);
99 b = j1(x);
100 for(i=1;i<n;i++){
101 temp = b;
102 b = b*((double)(i+i)/x) - a; /* avoid underflow */
103 a = temp;
104 }
105 }
106 } else {
107 if(ix<0x3e100000) { /* x < 2**-29 */
108 /* x is tiny, return the first Taylor expansion of J(n,x)
109 * J(n,x) = 1/n!*(x/2)^n - ...
110 */
111 if(n>33) /* underflow */
112 b = zero;
113 else {
114 temp = x*0.5; b = temp;
115 for (a=one,i=2;i<=n;i++) {
116 a *= (double)i; /* a = n! */
117 b *= temp; /* b = (x/2)^n */
118 }
119 b = b/a;
120 }
121 } else {
122 /* use backward recurrence */
123 /* x x^2 x^2
124 * J(n,x)/J(n-1,x) = ---- ------ ------ .....
125 * 2n - 2(n+1) - 2(n+2)
126 *
127 * 1 1 1
128 * (for large x) = ---- ------ ------ .....
129 * 2n 2(n+1) 2(n+2)
130 * -- - ------ - ------ -
131 * x x x
132 *
133 * Let w = 2n/x and h=2/x, then the above quotient
134 * is equal to the continued fraction:
135 * 1
136 * = -----------------------
137 * 1
138 * w - -----------------
139 * 1
140 * w+h - ---------
141 * w+2h - ...
142 *
143 * To determine how many terms needed, let
144 * Q(0) = w, Q(1) = w(w+h) - 1,
145 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
146 * When Q(k) > 1e4 good for single
147 * When Q(k) > 1e9 good for double
148 * When Q(k) > 1e17 good for quadruple
149 */
150 /* determine k */
151 double t,v;
152 double q0,q1,h,tmp; int32_t k,m;
153 w = (n+n)/(double)x; h = 2.0/(double)x;
154 q0 = w; z = w+h; q1 = w*z - 1.0; k=1;
155 while(q1<1.0e9) {
156 k += 1; z += h;
157 tmp = z*q1 - q0;
158 q0 = q1;
159 q1 = tmp;
160 }
161 m = n+n;
162 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t);
163 a = t;
164 b = one;
165 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
166 * Hence, if n*(log(2n/x)) > ...
167 * single 8.8722839355e+01
168 * double 7.09782712893383973096e+02
169 * long double 1.1356523406294143949491931077970765006170e+04
170 * then recurrent value may overflow and the result is
171 * likely underflow to zero
172 */
173 tmp = n;
174 v = two/x;
175 tmp = tmp*log(fabs(v*tmp));
176 if(tmp<7.09782712893383973096e+02) {
177 for(i=n-1,di=(double)(i+i);i>0;i--){
178 temp = b;
179 b *= di;
180 b = b/x - a;
181 a = temp;
182 di -= two;
183 }
184 } else {
185 for(i=n-1,di=(double)(i+i);i>0;i--){
186 temp = b;
187 b *= di;
188 b = b/x - a;
189 a = temp;
190 di -= two;
191 /* scale b to avoid spurious overflow */
192 if(b>1e100) {
193 a /= b;
194 t /= b;
195 b = one;
196 }
197 }
198 }
199 b = (t*j0(x)/b);
200 }
201 }
202 if(sgn==1) return -b; else return b;
203 }
204
205 double
yn(int n,double x)206 yn(int n, double x)
207 {
208 int32_t i,hx,ix,lx;
209 int32_t sign;
210 double a, b, temp;
211
212 EXTRACT_WORDS(hx,lx,x);
213 ix = 0x7fffffff&hx;
214 /* if Y(n,NaN) is NaN */
215 if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x;
216 if((ix|lx)==0) return -one/zero;
217 if(hx<0) return zero/zero;
218 sign = 1;
219 if(n<0){
220 n = -n;
221 sign = 1 - ((n&1)<<1);
222 }
223 if(n==0) return(y0(x));
224 if(n==1) return(sign*y1(x));
225 if(ix==0x7ff00000) return zero;
226 if(ix>=0x52D00000) { /* x > 2**302 */
227 /* (x >> n**2)
228 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
229 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
230 * Let s=sin(x), c=cos(x),
231 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
232 *
233 * n sin(xn)*sqt2 cos(xn)*sqt2
234 * ----------------------------------
235 * 0 s-c c+s
236 * 1 -s-c -c+s
237 * 2 -s+c -c-s
238 * 3 s+c c-s
239 */
240 switch(n&3) {
241 case 0: temp = sin(x)-cos(x); break;
242 case 1: temp = -sin(x)-cos(x); break;
243 case 2: temp = -sin(x)+cos(x); break;
244 case 3: temp = sin(x)+cos(x); break;
245 }
246 b = invsqrtpi*temp/sqrt(x);
247 } else {
248 u_int32_t high;
249 a = y0(x);
250 b = y1(x);
251 /* quit if b is -inf */
252 GET_HIGH_WORD(high,b);
253 for(i=1;i<n&&high!=0xfff00000;i++){
254 temp = b;
255 b = ((double)(i+i)/x)*b - a;
256 GET_HIGH_WORD(high,b);
257 a = temp;
258 }
259 }
260 if(sign>0) return b; else return -b;
261 }
262