281 lines
7.0 KiB
C
281 lines
7.0 KiB
C
/* origin: FreeBSD /usr/src/lib/msun/src/e_jn.c */
|
|
/*
|
|
* ====================================================
|
|
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
|
*
|
|
* Developed at SunSoft, a Sun Microsystems, Inc. business.
|
|
* Permission to use, copy, modify, and distribute this
|
|
* software is freely granted, provided that this notice
|
|
* is preserved.
|
|
* ====================================================
|
|
*/
|
|
/*
|
|
* jn(n, x), yn(n, x)
|
|
* floating point Bessel's function of the 1st and 2nd kind
|
|
* of order n
|
|
*
|
|
* Special cases:
|
|
* y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal;
|
|
* y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
|
|
* Note 2. About jn(n,x), yn(n,x)
|
|
* For n=0, j0(x) is called,
|
|
* for n=1, j1(x) is called,
|
|
* for n<=x, forward recursion is used starting
|
|
* from values of j0(x) and j1(x).
|
|
* for n>x, a continued fraction approximation to
|
|
* j(n,x)/j(n-1,x) is evaluated and then backward
|
|
* recursion is used starting from a supposed value
|
|
* for j(n,x). The resulting value of j(0,x) is
|
|
* compared with the actual value to correct the
|
|
* supposed value of j(n,x).
|
|
*
|
|
* yn(n,x) is similar in all respects, except
|
|
* that forward recursion is used for all
|
|
* values of n>1.
|
|
*/
|
|
|
|
#include "libc/math/libm.h"
|
|
|
|
static const double invsqrtpi = 5.64189583547756279280e-01; /* 0x3FE20DD7, 0x50429B6D */
|
|
|
|
double jn(int n, double x)
|
|
{
|
|
uint32_t ix, lx;
|
|
int nm1, i, sign;
|
|
double a, b, temp;
|
|
|
|
EXTRACT_WORDS(ix, lx, x);
|
|
sign = ix>>31;
|
|
ix &= 0x7fffffff;
|
|
|
|
if ((ix | (lx|-lx)>>31) > 0x7ff00000) /* nan */
|
|
return x;
|
|
|
|
/* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
|
|
* Thus, J(-n,x) = J(n,-x)
|
|
*/
|
|
/* nm1 = |n|-1 is used instead of |n| to handle n==INT_MIN */
|
|
if (n == 0)
|
|
return j0(x);
|
|
if (n < 0) {
|
|
nm1 = -(n+1);
|
|
x = -x;
|
|
sign ^= 1;
|
|
} else
|
|
nm1 = n-1;
|
|
if (nm1 == 0)
|
|
return j1(x);
|
|
|
|
sign &= n; /* even n: 0, odd n: signbit(x) */
|
|
x = fabs(x);
|
|
if ((ix|lx) == 0 || ix == 0x7ff00000) /* if x is 0 or inf */
|
|
b = 0.0;
|
|
else if (nm1 < x) {
|
|
/* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
|
|
if (ix >= 0x52d00000) { /* x > 2**302 */
|
|
/* (x >> n**2)
|
|
* Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
|
|
* Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
|
|
* Let s=sin(x), c=cos(x),
|
|
* xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
|
|
*
|
|
* n sin(xn)*sqt2 cos(xn)*sqt2
|
|
* ----------------------------------
|
|
* 0 s-c c+s
|
|
* 1 -s-c -c+s
|
|
* 2 -s+c -c-s
|
|
* 3 s+c c-s
|
|
*/
|
|
switch(nm1&3) {
|
|
case 0: temp = -cos(x)+sin(x); break;
|
|
case 1: temp = -cos(x)-sin(x); break;
|
|
case 2: temp = cos(x)-sin(x); break;
|
|
default:
|
|
case 3: temp = cos(x)+sin(x); break;
|
|
}
|
|
b = invsqrtpi*temp/sqrt(x);
|
|
} else {
|
|
a = j0(x);
|
|
b = j1(x);
|
|
for (i=0; i<nm1; ) {
|
|
i++;
|
|
temp = b;
|
|
b = b*(2.0*i/x) - a; /* avoid underflow */
|
|
a = temp;
|
|
}
|
|
}
|
|
} else {
|
|
if (ix < 0x3e100000) { /* x < 2**-29 */
|
|
/* x is tiny, return the first Taylor expansion of J(n,x)
|
|
* J(n,x) = 1/n!*(x/2)^n - ...
|
|
*/
|
|
if (nm1 > 32) /* underflow */
|
|
b = 0.0;
|
|
else {
|
|
temp = x*0.5;
|
|
b = temp;
|
|
a = 1.0;
|
|
for (i=2; i<=nm1+1; i++) {
|
|
a *= (double)i; /* a = n! */
|
|
b *= temp; /* b = (x/2)^n */
|
|
}
|
|
b = b/a;
|
|
}
|
|
} else {
|
|
/* use backward recurrence */
|
|
/* x x^2 x^2
|
|
* J(n,x)/J(n-1,x) = ---- ------ ------ .....
|
|
* 2n - 2(n+1) - 2(n+2)
|
|
*
|
|
* 1 1 1
|
|
* (for large x) = ---- ------ ------ .....
|
|
* 2n 2(n+1) 2(n+2)
|
|
* -- - ------ - ------ -
|
|
* x x x
|
|
*
|
|
* Let w = 2n/x and h=2/x, then the above quotient
|
|
* is equal to the continued fraction:
|
|
* 1
|
|
* = -----------------------
|
|
* 1
|
|
* w - -----------------
|
|
* 1
|
|
* w+h - ---------
|
|
* w+2h - ...
|
|
*
|
|
* To determine how many terms needed, let
|
|
* Q(0) = w, Q(1) = w(w+h) - 1,
|
|
* Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
|
|
* When Q(k) > 1e4 good for single
|
|
* When Q(k) > 1e9 good for double
|
|
* When Q(k) > 1e17 good for quadruple
|
|
*/
|
|
/* determine k */
|
|
double t,q0,q1,w,h,z,tmp,nf;
|
|
int k;
|
|
|
|
nf = nm1 + 1.0;
|
|
w = 2*nf/x;
|
|
h = 2/x;
|
|
z = w+h;
|
|
q0 = w;
|
|
q1 = w*z - 1.0;
|
|
k = 1;
|
|
while (q1 < 1.0e9) {
|
|
k += 1;
|
|
z += h;
|
|
tmp = z*q1 - q0;
|
|
q0 = q1;
|
|
q1 = tmp;
|
|
}
|
|
for (t=0.0, i=k; i>=0; i--)
|
|
t = 1/(2*(i+nf)/x - t);
|
|
a = t;
|
|
b = 1.0;
|
|
/* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
|
|
* Hence, if n*(log(2n/x)) > ...
|
|
* single 8.8722839355e+01
|
|
* double 7.09782712893383973096e+02
|
|
* long double 1.1356523406294143949491931077970765006170e+04
|
|
* then recurrent value may overflow and the result is
|
|
* likely underflow to zero
|
|
*/
|
|
tmp = nf*log(fabs(w));
|
|
if (tmp < 7.09782712893383973096e+02) {
|
|
for (i=nm1; i>0; i--) {
|
|
temp = b;
|
|
b = b*(2.0*i)/x - a;
|
|
a = temp;
|
|
}
|
|
} else {
|
|
for (i=nm1; i>0; i--) {
|
|
temp = b;
|
|
b = b*(2.0*i)/x - a;
|
|
a = temp;
|
|
/* scale b to avoid spurious overflow */
|
|
if (b > 0x1p500) {
|
|
a /= b;
|
|
t /= b;
|
|
b = 1.0;
|
|
}
|
|
}
|
|
}
|
|
z = j0(x);
|
|
w = j1(x);
|
|
if (fabs(z) >= fabs(w))
|
|
b = t*z/b;
|
|
else
|
|
b = t*w/a;
|
|
}
|
|
}
|
|
return sign ? -b : b;
|
|
}
|
|
|
|
|
|
double yn(int n, double x)
|
|
{
|
|
uint32_t ix, lx, ib;
|
|
int nm1, sign, i;
|
|
double a, b, temp;
|
|
|
|
EXTRACT_WORDS(ix, lx, x);
|
|
sign = ix>>31;
|
|
ix &= 0x7fffffff;
|
|
|
|
if ((ix | (lx|-lx)>>31) > 0x7ff00000) /* nan */
|
|
return x;
|
|
if (sign && (ix|lx)!=0) /* x < 0 */
|
|
return 0/0.0;
|
|
if (ix == 0x7ff00000)
|
|
return 0.0;
|
|
|
|
if (n == 0)
|
|
return y0(x);
|
|
if (n < 0) {
|
|
nm1 = -(n+1);
|
|
sign = n&1;
|
|
} else {
|
|
nm1 = n-1;
|
|
sign = 0;
|
|
}
|
|
if (nm1 == 0)
|
|
return sign ? -y1(x) : y1(x);
|
|
|
|
if (ix >= 0x52d00000) { /* x > 2**302 */
|
|
/* (x >> n**2)
|
|
* Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
|
|
* Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
|
|
* Let s=sin(x), c=cos(x),
|
|
* xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
|
|
*
|
|
* n sin(xn)*sqt2 cos(xn)*sqt2
|
|
* ----------------------------------
|
|
* 0 s-c c+s
|
|
* 1 -s-c -c+s
|
|
* 2 -s+c -c-s
|
|
* 3 s+c c-s
|
|
*/
|
|
switch(nm1&3) {
|
|
case 0: temp = -sin(x)-cos(x); break;
|
|
case 1: temp = -sin(x)+cos(x); break;
|
|
case 2: temp = sin(x)+cos(x); break;
|
|
default:
|
|
case 3: temp = sin(x)-cos(x); break;
|
|
}
|
|
b = invsqrtpi*temp/sqrt(x);
|
|
} else {
|
|
a = y0(x);
|
|
b = y1(x);
|
|
/* quit if b is -inf */
|
|
GET_HIGH_WORD(ib, b);
|
|
for (i=0; i<nm1 && ib!=0xfff00000; ){
|
|
i++;
|
|
temp = b;
|
|
b = (2.0*i/x)*b - a;
|
|
GET_HIGH_WORD(ib, b);
|
|
a = temp;
|
|
}
|
|
}
|
|
return sign ? -b : b;
|
|
}
|