2 * ====================================================
3 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5 * Developed at SunPro, a Sun Microsystems, Inc. business.
6 * Permission to use, copy, modify, and distribute this
7 * software is freely granted, provided that this notice
9 * ====================================================
12 /* Modifications for 128-bit long double are
13 Copyright (C) 2001 Stephen L. Moshier <moshier@na-net.ornl.gov>
14 and are incorporated herein by permission of the author. The author
15 reserves the right to distribute this material elsewhere under different
16 copying permissions. These modifications are distributed here under
19 This library is free software; you can redistribute it and/or
20 modify it under the terms of the GNU Lesser General Public
21 License as published by the Free Software Foundation; either
22 version 2.1 of the License, or (at your option) any later version.
24 This library is distributed in the hope that it will be useful,
25 but WITHOUT ANY WARRANTY; without even the implied warranty of
26 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
27 Lesser General Public License for more details.
29 You should have received a copy of the GNU Lesser General Public
30 License along with this library; if not, see
31 <http://www.gnu.org/licenses/>. */
34 * __ieee754_jn(n, x), __ieee754_yn(n, x)
35 * floating point Bessel's function of the 1st and 2nd kind
39 * y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal;
40 * y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
41 * Note 2. About jn(n,x), yn(n,x)
42 * For n=0, j0(x) is called,
43 * for n=1, j1(x) is called,
44 * for n<x, forward recursion us used starting
45 * from values of j0(x) and j1(x).
46 * for n>x, a continued fraction approximation to
47 * j(n,x)/j(n-1,x) is evaluated and then backward
48 * recursion is used starting from a supposed value
49 * for j(n,x). The resulting value of j(0,x) is
50 * compared with the actual value to correct the
51 * supposed value of j(n,x).
53 * yn(n,x) is similar in all respects, except
54 * that forward recursion is used for all
62 #include <math_private.h>
64 static const long double
65 invsqrtpi
= 5.6418958354775628694807945156077258584405E-1L,
72 __ieee754_jnl (int n
, long double x
)
76 long double a
, b
, temp
, di
;
81 /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
82 * Thus, J(-n,x) = J(n,-x)
86 EXTRACT_WORDS (se
, lx
, xhi
);
89 /* if J(n,NaN) is NaN */
92 if (((ix
- 0x7ff00000) | lx
) != 0)
103 return (__ieee754_j0l (x
));
105 return (__ieee754_j1l (x
));
106 sgn
= (n
& 1) & (se
>> 31); /* even n -- 0, odd n -- sign(x) */
109 if (x
== 0.0L || ix
>= 0x7ff00000) /* if x is 0 or inf */
111 else if ((long double) n
<= x
)
113 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
114 if (ix
>= 0x52d00000)
117 /* ??? Could use an expansion for large x here. */
120 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
121 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
122 * Let s=sin(x), c=cos(x),
123 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
125 * n sin(xn)*sqt2 cos(xn)*sqt2
126 * ----------------------------------
134 __sincosl (x
, &s
, &c
);
150 b
= invsqrtpi
* temp
/ __ieee754_sqrtl (x
);
154 a
= __ieee754_j0l (x
);
155 b
= __ieee754_j1l (x
);
156 for (i
= 1; i
< n
; i
++)
159 b
= b
* ((long double) (i
+ i
) / x
) - a
; /* avoid underflow */
168 /* x is tiny, return the first Taylor expansion of J(n,x)
169 * J(n,x) = 1/n!*(x/2)^n - ...
171 if (n
>= 33) /* underflow, result < 10^-300 */
177 for (a
= one
, i
= 2; i
<= n
; i
++)
179 a
*= (long double) i
; /* a = n! */
180 b
*= temp
; /* b = (x/2)^n */
187 /* use backward recurrence */
189 * J(n,x)/J(n-1,x) = ---- ------ ------ .....
190 * 2n - 2(n+1) - 2(n+2)
193 * (for large x) = ---- ------ ------ .....
195 * -- - ------ - ------ -
198 * Let w = 2n/x and h=2/x, then the above quotient
199 * is equal to the continued fraction:
201 * = -----------------------
203 * w - -----------------
208 * To determine how many terms needed, let
209 * Q(0) = w, Q(1) = w(w+h) - 1,
210 * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
211 * When Q(k) > 1e4 good for single
212 * When Q(k) > 1e9 good for double
213 * When Q(k) > 1e17 good for quadruple
217 long double q0
, q1
, h
, tmp
;
219 w
= (n
+ n
) / (long double) x
;
220 h
= 2.0L / (long double) x
;
234 for (t
= zero
, i
= 2 * (n
+ k
); i
>= m
; i
-= 2)
235 t
= one
/ (i
/ x
- t
);
238 /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
239 * Hence, if n*(log(2n/x)) > ...
240 * single 8.8722839355e+01
241 * double 7.09782712893383973096e+02
242 * long double 1.1356523406294143949491931077970765006170e+04
243 * then recurrent value may overflow and the result is
244 * likely underflow to zero
248 tmp
= tmp
* __ieee754_logl (fabsl (v
* tmp
));
250 if (tmp
< 1.1356523406294143949491931077970765006170e+04L)
252 for (i
= n
- 1, di
= (long double) (i
+ i
); i
> 0; i
--)
263 for (i
= n
- 1, di
= (long double) (i
+ i
); i
> 0; i
--)
270 /* scale b to avoid spurious overflow */
279 /* j0() and j1() suffer enormous loss of precision at and
280 * near zero; however, we know that their zero points never
281 * coincide, so just choose the one further away from zero.
283 z
= __ieee754_j0l (x
);
284 w
= __ieee754_j1l (x
);
285 if (fabsl (z
) >= fabsl (w
))
296 strong_alias (__ieee754_jnl
, __jnl_finite
)
299 __ieee754_ynl (int n
, long double x
)
304 long double a
, b
, temp
, ret
;
308 EXTRACT_WORDS (se
, lx
, xhi
);
309 ix
= se
& 0x7fffffff;
311 /* if Y(n,NaN) is NaN */
312 if (ix
>= 0x7ff00000)
314 if (((ix
- 0x7ff00000) | lx
) != 0)
320 return ((n
< 0 && (n
& 1) != 0) ? 1.0L : -1.0L) / 0.0L;
322 return zero
/ (zero
* x
);
328 sign
= 1 - ((n
& 1) << 1);
331 return (__ieee754_y0l (x
));
333 SET_RESTORE_ROUNDL (FE_TONEAREST
);
336 ret
= sign
* __ieee754_y1l (x
);
339 if (ix
>= 0x7ff00000)
341 if (ix
>= 0x52D00000)
344 /* ??? See comment above on the possible futility of this. */
347 * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
348 * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
349 * Let s=sin(x), c=cos(x),
350 * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
352 * n sin(xn)*sqt2 cos(xn)*sqt2
353 * ----------------------------------
361 __sincosl (x
, &s
, &c
);
377 b
= invsqrtpi
* temp
/ __ieee754_sqrtl (x
);
381 a
= __ieee754_y0l (x
);
382 b
= __ieee754_y1l (x
);
383 /* quit if b is -inf */
385 GET_HIGH_WORD (se
, xhi
);
387 for (i
= 1; i
< n
&& se
!= 0xfff00000; i
++)
390 b
= ((long double) (i
+ i
) / x
) * b
- a
;
392 GET_HIGH_WORD (se
, xhi
);
397 /* If B is +-Inf, set up errno accordingly. */
399 __set_errno (ERANGE
);
407 ret
= __copysignl (LDBL_MAX
, ret
) * LDBL_MAX
;
410 strong_alias (__ieee754_ynl
, __ynl_finite
)