1 /* origin: FreeBSD /usr/src/lib/msun/src/e_jn.c */
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 * 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 value of j(0,x) is
29 * compared with the actual value 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 "libm.h"
38
39 static const double invsqrtpi = 5.64189583547756279280e-01; /* 0x3FE20DD7, 0x50429B6D */
40
41 double jn(int n, double x)
42 {
43 uint32_t ix, lx;
44 int nm1, i, sign;
45 double a, b, temp;
46
47 EXTRACT_WORDS(ix, lx, x);
48 sign = ix>>31;
49 ix &= 0x7fffffff;
50
51 if ((ix | (lx|-lx)>>31) > 0x7ff00000) /* nan */
52 return x;
53
54 /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
55 * Thus, J(-n,x) = J(n,-x)
56 */
57 /* nm1 = |n|-1 is used instead of |n| to handle n==INT_MIN */
58 if (n == 0)
59 return j0(x);
60 if (n < 0) {
61 nm1 = -(n+1);
62 x = -x;
63 sign ^= 1;
64 } else
65 nm1 = n-1;
66 if (nm1 == 0)
67 return j1(x);
68
69 sign &= n; /* even n: 0, odd n: signbit(x) */
70 x = fabs(x);
71 if ((ix|lx) == 0 || ix == 0x7ff00000) /* if x is 0 or inf */
72 b = 0.0;
73 else if (nm1 < x) {
74 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
75 if (ix >= 0x52d00000) { /* x > 2**302 */
76 /* (x >> n**2)
77 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
78 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
79 * Let s=sin(x), c=cos(x),
80 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
81 *
82 * n sin(xn)*sqt2 cos(xn)*sqt2
83 * ----------------------------------
84 * 0 s-c c+s
85 * 1 -s-c -c+s
86 * 2 -s+c -c-s
87 * 3 s+c c-s
88 */
89 switch(nm1&3) {
90 case 0: temp = -cos(x)+sin(x); break;
91 case 1: temp = -cos(x)-sin(x); break;
92 case 2: temp = cos(x)-sin(x); break;
93 default:
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=0; i<nm1; ) {
101 i++;
102 temp = b;
103 b = b*(2.0*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 (nm1 > 32) /* underflow */
113 b = 0.0;
114 else {
115 temp = x*0.5;
116 b = temp;
117 a = 1.0;
118 for (i=2; i<=nm1+1; 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,q0,q1,w,h,z,tmp,nf;
155 int k;
156
157 nf = nm1 + 1.0;
158 w = 2*nf/x;
159 h = 2/x;
160 z = w+h;
161 q0 = w;
162 q1 = w*z - 1.0;
163 k = 1;
164 while (q1 < 1.0e9) {
165 k += 1;
166 z += h;
167 tmp = z*q1 - q0;
168 q0 = q1;
169 q1 = tmp;
170 }
171 for (t=0.0, i=k; i>=0; i--)
172 t = 1/(2*(i+nf)/x - t);
173 a = t;
174 b = 1.0;
175 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
176 * Hence, if n*(log(2n/x)) > ...
177 * single 8.8722839355e+01
178 * double 7.09782712893383973096e+02
179 * long double 1.1356523406294143949491931077970765006170e+04
180 * then recurrent value may overflow and the result is
181 * likely underflow to zero
182 */
183 tmp = nf*log(fabs(w));
184 if (tmp < 7.09782712893383973096e+02) {
185 for (i=nm1; i>0; i--) {
186 temp = b;
187 b = b*(2.0*i)/x - a;
188 a = temp;
189 }
190 } else {
191 for (i=nm1; i>0; i--) {
192 temp = b;
193 b = b*(2.0*i)/x - a;
194 a = temp;
195 /* scale b to avoid spurious overflow */
196 if (b > 0x1p500) {
197 a /= b;
198 t /= b;
199 b = 1.0;
200 }
201 }
202 }
203 z = j0(x);
204 w = j1(x);
205 if (fabs(z) >= fabs(w))
206 b = t*z/b;
207 else
208 b = t*w/a;
209 }
210 }
211 return sign ? -b : b;
212 }
213
214
215 double yn(int n, double x)
216 {
217 uint32_t ix, lx, ib;
218 int nm1, sign, i;
219 double a, b, temp;
220
221 EXTRACT_WORDS(ix, lx, x);
222 sign = ix>>31;
223 ix &= 0x7fffffff;
224
225 if ((ix | (lx|-lx)>>31) > 0x7ff00000) /* nan */
226 return x;
227 if (sign && (ix|lx)!=0) /* x < 0 */
228 return 0/0.0;
229 if (ix == 0x7ff00000)
230 return 0.0;
231
232 if (n == 0)
233 return y0(x);
234 if (n < 0) {
235 nm1 = -(n+1);
236 sign = n&1;
237 } else {
238 nm1 = n-1;
239 sign = 0;
240 }
241 if (nm1 == 0)
242 return sign ? -y1(x) : y1(x);
243
244 if (ix >= 0x52d00000) { /* x > 2**302 */
245 /* (x >> n**2)
246 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
247 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
248 * Let s=sin(x), c=cos(x),
249 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
250 *
251 * n sin(xn)*sqt2 cos(xn)*sqt2
252 * ----------------------------------
253 * 0 s-c c+s
254 * 1 -s-c -c+s
255 * 2 -s+c -c-s
256 * 3 s+c c-s
257 */
258 switch(nm1&3) {
259 case 0: temp = -sin(x)-cos(x); break;
260 case 1: temp = -sin(x)+cos(x); break;
261 case 2: temp = sin(x)+cos(x); break;
262 default:
263 case 3: temp = sin(x)-cos(x); break;
264 }
265 b = invsqrtpi*temp/sqrt(x);
266 } else {
267 a = y0(x);
268 b = y1(x);
269 /* quit if b is -inf */
270 GET_HIGH_WORD(ib, b);
271 for (i=0; i<nm1 && ib!=0xfff00000; ){
272 i++;
273 temp = b;
274 b = (2.0*i/x)*b - a;
275 GET_HIGH_WORD(ib, b);
276 a = temp;
277 }
278 }
279 return sign ? -b : b;
280 }
281