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 "fdlibm.h"
40
41 #ifdef _NEED_FLOAT64
42
43 static const __float64
44 invsqrtpi = _F_64(5.64189583547756279280e-01), /* 0x3FE20DD7, 0x50429B6D */
45 two = _F_64(2.00000000000000000000e+00), /* 0x40000000, 0x00000000 */
46 one = _F_64(1.00000000000000000000e+00); /* 0x3FF00000, 0x00000000 */
47
48 static const __float64 zero = _F_64(0.00000000000000000000e+00);
49
50 __float64
jn64(int n,__float64 x)51 jn64(int n, __float64 x)
52 {
53 __int32_t i, hx, ix, lx, sgn;
54 __float64 a, b, temp, di;
55 __float64 z, w;
56
57 if (isnan(x))
58 return x + x;
59
60 if (isinf(x))
61 return _F_64(0.0);
62
63 /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
64 * Thus, J(-n,x) = J(n,-x)
65 */
66 EXTRACT_WORDS(hx, lx, x);
67 ix = 0x7fffffff & hx;
68
69 if (n < 0) {
70 n = -n;
71 x = -x;
72 hx ^= 0x80000000;
73 }
74 if (n == 0)
75 return (j064(x));
76 if (n == 1)
77 return (j164(x));
78 sgn = (n & 1) & (hx >> 31); /* even n -- 0, odd n -- sign(x) */
79 x = fabs64(x);
80 if ((ix | lx) == 0 || ix >= 0x7ff00000) /* if x is 0 or inf */
81 b = zero;
82 else if ((__float64)n <= x) {
83 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
84 if (ix >= 0x52D00000) { /* x > 2**302 */
85 /* (x >> n**2)
86 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
87 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
88 * Let s=sin(x), c=cos(x),
89 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
90 *
91 * n sin(xn)*sqt2 cos(xn)*sqt2
92 * ----------------------------------
93 * 0 s-c c+s
94 * 1 -s-c -c+s
95 * 2 -s+c -c-s
96 * 3 s+c c-s
97 */
98 switch (n & 3) {
99 case 0:
100 default:
101 temp = cos64(x) + sin64(x);
102 break;
103 case 1:
104 temp = -cos64(x) + sin64(x);
105 break;
106 case 2:
107 temp = -cos64(x) - sin64(x);
108 break;
109 case 3:
110 temp = cos64(x) - sin64(x);
111 break;
112 }
113 b = invsqrtpi * temp / sqrt64(x);
114 } else {
115 a = j064(x);
116 b = j164(x);
117 for (i = 1; i < n; i++) {
118 temp = b;
119 b = b * ((__float64)(i + i) / x) - a; /* avoid underflow */
120 a = temp;
121 }
122 }
123 } else {
124 if (ix < 0x3e100000) { /* x < 2**-29 */
125 /* x is tiny, return the first Taylor expansion of J(n,x)
126 * J(n,x) = 1/n!*(x/2)^n - ...
127 */
128 if (n > 33) /* underflow */
129 b = zero;
130 else {
131 temp = x * _F_64(0.5);
132 b = temp;
133 for (a = one, i = 2; i <= n; i++) {
134 a *= (__float64)i; /* a = n! */
135 b *= temp; /* b = (x/2)^n */
136 }
137 b = b / a;
138 }
139 } else {
140 /* use backward recurrence */
141 /* x x^2 x^2
142 * J(n,x)/J(n-1,x) = ---- ------ ------ .....
143 * 2n - 2(n+1) - 2(n+2)
144 *
145 * 1 1 1
146 * (for large x) = ---- ------ ------ .....
147 * 2n 2(n+1) 2(n+2)
148 * -- - ------ - ------ -
149 * x x x
150 *
151 * Let w = 2n/x and h=2/x, then the above quotient
152 * is equal to the continued fraction:
153 * 1
154 * = -----------------------
155 * 1
156 * w - -----------------
157 * 1
158 * w+h - ---------
159 * w+2h - ...
160 *
161 * To determine how many terms needed, let
162 * Q(0) = w, Q(1) = w(w+h) - 1,
163 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
164 * When Q(k) > 1e4 good for single
165 * When Q(k) > 1e9 good for double
166 * When Q(k) > 1e17 good for quadruple
167 */
168 /* determine k */
169 __float64 t, v;
170 __float64 q0, q1, h, tmp;
171 __int32_t k, m;
172 w = (n + n) / (__float64)x;
173 h = _F_64(2.0) / (__float64)x;
174 q0 = w;
175 z = w + h;
176 q1 = w * z - _F_64(1.0);
177 k = 1;
178 while (q1 < _F_64(1.0e9)) {
179 k += 1;
180 z += h;
181 tmp = z * q1 - q0;
182 q0 = q1;
183 q1 = tmp;
184 }
185 m = n + n;
186 for (t = zero, i = 2 * (n + k); i >= m; i -= 2)
187 t = one / (i / x - t);
188 a = t;
189 b = one;
190 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
191 * Hence, if n*(log(2n/x)) > ...
192 * single 8.8722839355e+01
193 * double 7.09782712893383973096e+02
194 * long double 1.1356523406294143949491931077970765006170e+04
195 * then recurrent value may overflow and the result is
196 * likely underflow to zero
197 */
198 tmp = n;
199 v = two / x;
200 tmp = tmp * log(fabs64(v * tmp));
201 if (tmp < _F_64(7.09782712893383973096e+02)) {
202 for (i = n - 1, di = (__float64)(i + i); i > 0; i--) {
203 temp = b;
204 b *= di;
205 b = b / x - a;
206 a = temp;
207 di -= two;
208 }
209 } else {
210 for (i = n - 1, di = (__float64)(i + i); i > 0; i--) {
211 temp = b;
212 b *= di;
213 b = b / x - a;
214 a = temp;
215 di -= two;
216 /* scale b to avoid spurious overflow */
217 if (b > _F_64(1e100)) {
218 a /= b;
219 t /= b;
220 b = one;
221 }
222 }
223 }
224 b = (t * j064(x) / b);
225 }
226 }
227 if (sgn == 1)
228 return -b;
229 else
230 return b;
231 }
232
_MATH_ALIAS_d_id(jn)233 _MATH_ALIAS_d_id(jn)
234
235 __float64
236 yn64(int n, __float64 x)
237 {
238 __int32_t i, hx, ix, lx;
239 __int32_t sign;
240 __float64 a, b, temp;
241
242 EXTRACT_WORDS(hx, lx, x);
243 ix = 0x7fffffff & hx;
244 /* if Y(n,NaN) is NaN */
245
246 if ((ix | lx) == 0)
247 return __math_divzero(1);
248
249 if (isnan(x))
250 return x + x;
251
252 if (hx < 0)
253 return __math_invalid(x);
254
255 if (ix == 0x7ff00000)
256 return _F_64(0.0);
257
258 sign = 1;
259 if (n < 0) {
260 n = -n;
261 sign = 1 - ((n & 1) << 1);
262 }
263 if (n == 0)
264 return (y064(x));
265 if (n == 1)
266 return (sign * y164(x));
267
268 if (ix >= 0x52D00000) { /* x > 2**302 */
269 /* (x >> n**2)
270 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
271 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
272 * Let s=sin(x), c=cos(x),
273 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
274 *
275 * n sin(xn)*sqt2 cos(xn)*sqt2
276 * ----------------------------------
277 * 0 s-c c+s
278 * 1 -s-c -c+s
279 * 2 -s+c -c-s
280 * 3 s+c c-s
281 */
282 switch (n & 3) {
283 case 0:
284 default:
285 temp = sin64(x) - cos64(x);
286 break;
287 case 1:
288 temp = -sin64(x) - cos64(x);
289 break;
290 case 2:
291 temp = -sin64(x) + cos64(x);
292 break;
293 case 3:
294 temp = sin64(x) + cos64(x);
295 break;
296 }
297 b = invsqrtpi * temp / sqrt64(x);
298 } else {
299 __uint32_t high;
300 a = y064(x);
301 b = y164(x);
302 /* quit if b is -inf */
303 GET_HIGH_WORD(high, b);
304 for (i = 1; i < n && high != 0xfff00000; i++) {
305 temp = b;
306 b = ((__float64)(i + i) / x) * b - a;
307 GET_HIGH_WORD(high, b);
308 a = temp;
309 }
310 }
311 if (sign > 0)
312 return b;
313 else
314 return -b;
315 }
316
317 _MATH_ALIAS_d_id(yn)
318
319 #endif /* _NEED_FLOAT64 */
320
