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1.1 jtc 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: */
1.3 jtc 12:
1.7 jtc 13: #if defined(LIBM_SCCS) && !defined(lint)
1.8 ! jtc 14: static char rcsid[] = "$Id: e_jn.c,v 1.7 1994/09/22 16:39:45 jtc Exp $";
1.3 jtc 15: #endif
1.1 jtc 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:
1.5 jtc 43: #include "math.h"
44: #include "math_private.h"
1.1 jtc 45:
46: #ifdef __STDC__
47: static const double
48: #else
49: static double
50: #endif
51: invsqrtpi= 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
52: two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
53: one = 1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */
54:
1.5 jtc 55: #ifdef __STDC__
56: static const double zero = 0.00000000000000000000e+00;
57: #else
1.1 jtc 58: static double zero = 0.00000000000000000000e+00;
1.5 jtc 59: #endif
1.1 jtc 60:
61: #ifdef __STDC__
62: double __ieee754_jn(int n, double x)
63: #else
64: double __ieee754_jn(n,x)
65: int n; double x;
66: #endif
67: {
1.6 jtc 68: int32_t i,hx,ix,lx, sgn;
1.1 jtc 69: double a, b, temp, di;
70: double z, w;
71:
72: /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
73: * Thus, J(-n,x) = J(n,-x)
74: */
1.5 jtc 75: EXTRACT_WORDS(hx,lx,x);
1.1 jtc 76: ix = 0x7fffffff&hx;
77: /* if J(n,NaN) is NaN */
1.6 jtc 78: if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x;
1.1 jtc 79: if(n<0){
80: n = -n;
81: x = -x;
82: hx ^= 0x80000000;
83: }
84: if(n==0) return(__ieee754_j0(x));
85: if(n==1) return(__ieee754_j1(x));
86: sgn = (n&1)&(hx>>31); /* even n -- 0, odd n -- sign(x) */
87: x = fabs(x);
88: if((ix|lx)==0||ix>=0x7ff00000) /* if x is 0 or inf */
89: b = zero;
90: else if((double)n<=x) {
91: /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
92: if(ix>=0x52D00000) { /* x > 2**302 */
93: /* (x >> n**2)
94: * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
95: * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
96: * Let s=sin(x), c=cos(x),
97: * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
98: *
99: * n sin(xn)*sqt2 cos(xn)*sqt2
100: * ----------------------------------
101: * 0 s-c c+s
102: * 1 -s-c -c+s
103: * 2 -s+c -c-s
104: * 3 s+c c-s
105: */
106: switch(n&3) {
107: case 0: temp = cos(x)+sin(x); break;
108: case 1: temp = -cos(x)+sin(x); break;
109: case 2: temp = -cos(x)-sin(x); break;
110: case 3: temp = cos(x)-sin(x); break;
111: }
112: b = invsqrtpi*temp/sqrt(x);
113: } else {
114: a = __ieee754_j0(x);
115: b = __ieee754_j1(x);
116: for(i=1;i<n;i++){
117: temp = b;
118: b = b*((double)(i+i)/x) - a; /* avoid underflow */
119: a = temp;
120: }
121: }
122: } else {
123: if(ix<0x3e100000) { /* x < 2**-29 */
124: /* x is tiny, return the first Taylor expansion of J(n,x)
125: * J(n,x) = 1/n!*(x/2)^n - ...
126: */
127: if(n>33) /* underflow */
128: b = zero;
129: else {
130: temp = x*0.5; b = temp;
131: for (a=one,i=2;i<=n;i++) {
132: a *= (double)i; /* a = n! */
133: b *= temp; /* b = (x/2)^n */
134: }
135: b = b/a;
136: }
137: } else {
138: /* use backward recurrence */
139: /* x x^2 x^2
140: * J(n,x)/J(n-1,x) = ---- ------ ------ .....
141: * 2n - 2(n+1) - 2(n+2)
142: *
143: * 1 1 1
144: * (for large x) = ---- ------ ------ .....
145: * 2n 2(n+1) 2(n+2)
146: * -- - ------ - ------ -
147: * x x x
148: *
149: * Let w = 2n/x and h=2/x, then the above quotient
150: * is equal to the continued fraction:
151: * 1
152: * = -----------------------
153: * 1
154: * w - -----------------
155: * 1
156: * w+h - ---------
157: * w+2h - ...
158: *
159: * To determine how many terms needed, let
160: * Q(0) = w, Q(1) = w(w+h) - 1,
161: * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
162: * When Q(k) > 1e4 good for single
163: * When Q(k) > 1e9 good for double
164: * When Q(k) > 1e17 good for quadruple
165: */
166: /* determine k */
167: double t,v;
1.6 jtc 168: double q0,q1,h,tmp; int32_t k,m;
1.1 jtc 169: w = (n+n)/(double)x; h = 2.0/(double)x;
170: q0 = w; z = w+h; q1 = w*z - 1.0; k=1;
171: while(q1<1.0e9) {
172: k += 1; z += h;
173: tmp = z*q1 - q0;
174: q0 = q1;
175: q1 = tmp;
176: }
177: m = n+n;
178: for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t);
179: a = t;
180: b = one;
181: /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
182: * Hence, if n*(log(2n/x)) > ...
183: * single 8.8722839355e+01
184: * double 7.09782712893383973096e+02
185: * long double 1.1356523406294143949491931077970765006170e+04
186: * then recurrent value may overflow and the result is
187: * likely underflow to zero
188: */
189: tmp = n;
190: v = two/x;
191: tmp = tmp*__ieee754_log(fabs(v*tmp));
192: if(tmp<7.09782712893383973096e+02) {
193: for(i=n-1,di=(double)(i+i);i>0;i--){
194: temp = b;
195: b *= di;
196: b = b/x - a;
197: a = temp;
198: di -= two;
199: }
200: } else {
201: for(i=n-1,di=(double)(i+i);i>0;i--){
202: temp = b;
203: b *= di;
204: b = b/x - a;
205: a = temp;
206: di -= two;
207: /* scale b to avoid spurious overflow */
208: if(b>1e100) {
209: a /= b;
210: t /= b;
211: b = one;
212: }
213: }
214: }
215: b = (t*__ieee754_j0(x)/b);
216: }
217: }
218: if(sgn==1) return -b; else return b;
219: }
220:
221: #ifdef __STDC__
222: double __ieee754_yn(int n, double x)
223: #else
224: double __ieee754_yn(n,x)
225: int n; double x;
226: #endif
227: {
1.6 jtc 228: int32_t i,hx,ix,lx;
229: int32_t sign;
1.1 jtc 230: double a, b, temp;
231:
1.5 jtc 232: EXTRACT_WORDS(hx,lx,x);
1.1 jtc 233: ix = 0x7fffffff&hx;
234: /* if Y(n,NaN) is NaN */
1.6 jtc 235: if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x;
1.1 jtc 236: if((ix|lx)==0) return -one/zero;
237: if(hx<0) return zero/zero;
238: sign = 1;
239: if(n<0){
240: n = -n;
1.8 ! jtc 241: sign = 1 - ((n&1)<<1);
1.1 jtc 242: }
243: if(n==0) return(__ieee754_y0(x));
244: if(n==1) return(sign*__ieee754_y1(x));
245: if(ix==0x7ff00000) return zero;
246: if(ix>=0x52D00000) { /* x > 2**302 */
247: /* (x >> n**2)
248: * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
249: * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
250: * Let s=sin(x), c=cos(x),
251: * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
252: *
253: * n sin(xn)*sqt2 cos(xn)*sqt2
254: * ----------------------------------
255: * 0 s-c c+s
256: * 1 -s-c -c+s
257: * 2 -s+c -c-s
258: * 3 s+c c-s
259: */
260: switch(n&3) {
261: case 0: temp = sin(x)-cos(x); break;
262: case 1: temp = -sin(x)-cos(x); break;
263: case 2: temp = -sin(x)+cos(x); break;
264: case 3: temp = sin(x)+cos(x); break;
265: }
266: b = invsqrtpi*temp/sqrt(x);
267: } else {
1.6 jtc 268: u_int32_t high;
1.1 jtc 269: a = __ieee754_y0(x);
270: b = __ieee754_y1(x);
271: /* quit if b is -inf */
1.5 jtc 272: GET_HIGH_WORD(high,b);
273: for(i=1;i<n&&high!=0xfff00000;i++){
1.1 jtc 274: temp = b;
275: b = ((double)(i+i)/x)*b - a;
1.5 jtc 276: GET_HIGH_WORD(high,b);
1.1 jtc 277: a = temp;
278: }
279: }
280: if(sign>0) return b; else return -b;
281: }