340 lines
9.8 KiB
Rust
340 lines
9.8 KiB
Rust
/* 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.
|
|
*/
|
|
|
|
use super::{cos, fabs, get_high_word, get_low_word, j0, j1, log, sin, sqrt, y0, y1};
|
|
|
|
const INVSQRTPI: f64 = 5.64189583547756279280e-01; /* 0x3FE20DD7, 0x50429B6D */
|
|
|
|
/// Integer order of the [Bessel function](https://en.wikipedia.org/wiki/Bessel_function) of the first kind (f64).
|
|
#[cfg_attr(assert_no_panic, no_panic::no_panic)]
|
|
pub fn jn(n: i32, mut x: f64) -> f64 {
|
|
let mut ix: u32;
|
|
let lx: u32;
|
|
let nm1: i32;
|
|
let mut i: i32;
|
|
let mut sign: bool;
|
|
let mut a: f64;
|
|
let mut b: f64;
|
|
let mut temp: f64;
|
|
|
|
ix = get_high_word(x);
|
|
lx = get_low_word(x);
|
|
sign = (ix >> 31) != 0;
|
|
ix &= 0x7fffffff;
|
|
|
|
// -lx == !lx + 1
|
|
if ix | ((lx | (!lx).wrapping_add(1)) >> 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 = !sign;
|
|
} else {
|
|
nm1 = n - 1;
|
|
}
|
|
if nm1 == 0 {
|
|
return j1(x);
|
|
}
|
|
|
|
sign &= (n & 1) != 0; /* 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 as f64) < 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
|
|
*/
|
|
temp = match nm1 & 3 {
|
|
0 => -cos(x) + sin(x),
|
|
1 => -cos(x) - sin(x),
|
|
2 => cos(x) - sin(x),
|
|
// 3
|
|
_ => cos(x) + sin(x),
|
|
};
|
|
b = INVSQRTPI * temp / sqrt(x);
|
|
} else {
|
|
a = j0(x);
|
|
b = j1(x);
|
|
i = 0;
|
|
while i < nm1 {
|
|
i += 1;
|
|
temp = b;
|
|
b = b * (2.0 * (i as f64) / 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;
|
|
i = 2;
|
|
while i <= nm1 + 1 {
|
|
a *= i as f64; /* a = n! */
|
|
b *= temp; /* b = (x/2)^n */
|
|
i += 1;
|
|
}
|
|
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 */
|
|
let mut t: f64;
|
|
let mut q0: f64;
|
|
let mut q1: f64;
|
|
let mut w: f64;
|
|
let h: f64;
|
|
let mut z: f64;
|
|
let mut tmp: f64;
|
|
let nf: f64;
|
|
|
|
let mut k: i32;
|
|
|
|
nf = (nm1 as f64) + 1.0;
|
|
w = 2.0 * nf / x;
|
|
h = 2.0 / 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;
|
|
}
|
|
t = 0.0;
|
|
i = k;
|
|
while i >= 0 {
|
|
t = 1.0 / (2.0 * ((i as f64) + nf) / x - t);
|
|
i -= 1;
|
|
}
|
|
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 {
|
|
i = nm1;
|
|
while i > 0 {
|
|
temp = b;
|
|
b = b * (2.0 * (i as f64)) / x - a;
|
|
a = temp;
|
|
i -= 1;
|
|
}
|
|
} else {
|
|
i = nm1;
|
|
while i > 0 {
|
|
temp = b;
|
|
b = b * (2.0 * (i as f64)) / x - a;
|
|
a = temp;
|
|
/* scale b to avoid spurious overflow */
|
|
let x1p500 = f64::from_bits(0x5f30000000000000); // 0x1p500 == 2^500
|
|
if b > x1p500 {
|
|
a /= b;
|
|
t /= b;
|
|
b = 1.0;
|
|
}
|
|
i -= 1;
|
|
}
|
|
}
|
|
z = j0(x);
|
|
w = j1(x);
|
|
if fabs(z) >= fabs(w) {
|
|
b = t * z / b;
|
|
} else {
|
|
b = t * w / a;
|
|
}
|
|
}
|
|
|
|
if sign { -b } else { b }
|
|
}
|
|
|
|
/// Integer order of the [Bessel function](https://en.wikipedia.org/wiki/Bessel_function) of the second kind (f64).
|
|
#[cfg_attr(assert_no_panic, no_panic::no_panic)]
|
|
pub fn yn(n: i32, x: f64) -> f64 {
|
|
let mut ix: u32;
|
|
let lx: u32;
|
|
let mut ib: u32;
|
|
let nm1: i32;
|
|
let mut sign: bool;
|
|
let mut i: i32;
|
|
let mut a: f64;
|
|
let mut b: f64;
|
|
let mut temp: f64;
|
|
|
|
ix = get_high_word(x);
|
|
lx = get_low_word(x);
|
|
sign = (ix >> 31) != 0;
|
|
ix &= 0x7fffffff;
|
|
|
|
// -lx == !lx + 1
|
|
if ix | ((lx | (!lx).wrapping_add(1)) >> 31) > 0x7ff00000 {
|
|
/* nan */
|
|
return x;
|
|
}
|
|
if sign && (ix | lx) != 0 {
|
|
/* x < 0 */
|
|
return 0.0 / 0.0;
|
|
}
|
|
if ix == 0x7ff00000 {
|
|
return 0.0;
|
|
}
|
|
|
|
if n == 0 {
|
|
return y0(x);
|
|
}
|
|
if n < 0 {
|
|
nm1 = -(n + 1);
|
|
sign = (n & 1) != 0;
|
|
} else {
|
|
nm1 = n - 1;
|
|
sign = false;
|
|
}
|
|
if nm1 == 0 {
|
|
if sign {
|
|
return -y1(x);
|
|
} else {
|
|
return 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
|
|
*/
|
|
temp = match nm1 & 3 {
|
|
0 => -sin(x) - cos(x),
|
|
1 => -sin(x) + cos(x),
|
|
2 => sin(x) + cos(x),
|
|
// 3
|
|
_ => sin(x) - cos(x),
|
|
};
|
|
b = INVSQRTPI * temp / sqrt(x);
|
|
} else {
|
|
a = y0(x);
|
|
b = y1(x);
|
|
/* quit if b is -inf */
|
|
ib = get_high_word(b);
|
|
i = 0;
|
|
while i < nm1 && ib != 0xfff00000 {
|
|
i += 1;
|
|
temp = b;
|
|
b = (2.0 * (i as f64) / x) * b - a;
|
|
ib = get_high_word(b);
|
|
a = temp;
|
|
}
|
|
}
|
|
|
|
if sign { -b } else { b }
|
|
}
|