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/* kn.c
*
* Modified Bessel function, third kind, integer order
*
*
*
* SYNOPSIS:
*
* double x, y, kn();
* int n;
*
* y = kn( n, x );
*
*
*
* DESCRIPTION:
*
* Returns modified Bessel function of the third kind
* of order n of the argument.
*
* The range is partitioned into the two intervals [0,9.55] and
* (9.55, infinity). An ascending power series is used in the
* low range, and an asymptotic expansion in the high range.
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE 0,30 90000 1.8e-8 3.0e-10
*
* Error is high only near the crossover point x = 9.55
* between the two expansions used.
*/
/*
* Cephes Math Library Release 2.8: June, 2000
* Copyright 1984, 1987, 1988, 2000 by Stephen L. Moshier
*/
/*
* Algorithm for Kn.
* n-1
* -n - (n-k-1)! 2 k
* K (x) = 0.5 (x/2) > -------- (-x /4)
* n - k!
* k=0
*
* inf. 2 k
* n n - (x /4)
* + (-1) 0.5(x/2) > {p(k+1) + p(n+k+1) - 2log(x/2)} ---------
* - k! (n+k)!
* k=0
*
* where p(m) is the psi function: p(1) = -EUL and
*
* m-1
* -
* p(m) = -EUL + > 1/k
* -
* k=1
*
* For large x,
* 2 2 2
* u-1 (u-1 )(u-3 )
* K (z) = sqrt(pi/2z) exp(-z) { 1 + ------- + ------------ + ...}
* v 1 2
* 1! (8z) 2! (8z)
* asymptotically, where
*
* 2
* u = 4 v .
*
*/
#![allow(clippy::excessive_precision)]
const EUL: f64 = 5.772156649015328606065e-1;
const MAXFAC: isize = 31;
use crate::cephes64::consts::{M_PI, MACHEP, MAXLOG};
pub fn kn(nn: isize, x: f64) -> f64 {
//! Modified Bessel function, third kind, integer order
//!
//! ## DESCRIPTION:
//!
//! Returns modified Bessel function of the third kind
//! of order n of the argument.
//!
//! The range is partitioned into the two intervals [0,9.55] and
//! (9.55, infinity). An ascending power series is used in the
//! low range, and an asymptotic expansion in the high range.
//!
//! ## ACCURACY:
//!
//! Relative error:
//!
//!<table>
//! <tr>
//! <th>Arithmetic</th>
//! <th>Domain</th>
//! <th># Trials</th>
//! <th>Peak</th>
//! <th>RMS</th>
//! </tr>
//! <tr>
//! <td>IEEE</td>
//! <td>0, 30</td>
//! <td>90000</td>
//! <td>1.8e-8</td>
//! <td>3.0e-10</td>
//! </tr>
//!</table>
//!
//! Error is high only near the crossover point x = 9.55
//! between the two expansions used.
let n = nn.abs();
if n > MAXFAC {
//overf:
//sf_error("kn", SF_ERROR_OVERFLOW, NULL);
return f64::INFINITY;
} else if x <= 0.0 {
if x < 0.0 {
//sf_error("kn", SF_ERROR_DOMAIN, NULL);
return f64::NAN;
}
else {
//sf_error("kn", SF_ERROR_SINGULAR, NULL);
return f64::INFINITY;
}
}
if x > 9.55 {
/* Asymptotic expansion for Kn(x) */
/* Converges to 1.4e-17 for x > 18.4 */
if x > MAXLOG {
//sf_error("kn", SF_ERROR_UNDERFLOW, NULL);
return 0.0;
}
let k = n;
let pn = 4.0 * (k as f64 * k as f64);
let mut pk = 1.0;
let z0 = 8.0 * x;
let mut fn_ = 1.0;
let mut t: f64 = 1.0;
let mut s = t;
let mut nkf = f64::INFINITY;
let mut i = 0;
loop {
let z = pn - pk * pk;
t = t * z / (fn_ * z0);
let nk1f = (t).abs();
if (i >= n) && (nk1f > nkf) {
break;
}
nkf = nk1f;
s += t;
fn_ += 1.0;
pk += 2.0;
i += 1;
if !((t / s).abs() > MACHEP) {
break;
}
}
let ans = (-x).exp() * (M_PI / (2.0 * x)).sqrt() * s;
return ans;
}
let mut ans = 0.0;
let z0 = 0.25 * x * x;
let mut fn_ = 1.0;
let mut pn = 0.0;
let mut zmn = 1.0;
let tox = 2.0 / x;
if n > 0 {
/* compute factorial of n and psi(n) */
pn = -EUL;
let mut k = 1.0;
for _ in 1..n {
pn += 1.0 / k;
k += 1.0;
fn_ *= k;
}
zmn = tox;
if n == 1 {
ans = 1.0 / x;
} else {
let mut nk1f = fn_ / (n as f64);
let mut kf = 1.0;
let mut s = nk1f;
let z = -z0;
let mut zn = 1.0;
let mut t;
for i in 1..n {
nk1f /= (n - i) as f64;
kf *= i as f64;
zn *= z;
t = nk1f * zn / kf;
s += t;
if (f64::MAX - t.abs()) < s.abs() {
return f64::INFINITY;
}
if (tox > 1.0) && ((f64::MAX / tox) < zmn) {
return f64::INFINITY;
}
zmn *= tox;
}
s *= 0.5;
t = s.abs();
if (zmn > 1.0) && ((f64::MAX / zmn) < t){
return f64::INFINITY;
}
if (t > 1.0) && ((f64::MAX / t) < zmn) {
return f64::INFINITY;
}
ans = s * zmn;
}
}
let tlg = 2.0 * (0.5 * x).ln();
let mut pk = -EUL;
let mut t = if n == 0 {
pn = pk;
1.0
} else {
pn += 1.0 / (n as f64);
1.0 / fn_
};
let mut s = (pk + pn - tlg) * t;
let mut k = 1.0;
loop {
t *= z0 / (k * (k + n as f64));
pk += 1.0 / k;
pn += 1.0 / (k + n as f64);
s += (pk + pn - tlg) * t;
k += 1.0;
if !((t / s).abs() > MACHEP) {
break;
}
}
s = 0.5 * s / zmn;
if n & 1 != 0 {
s = -s;
}
ans += s;
ans
}
// TODO: This doesn't line up exactly with output from the C CEHPES library
#[cfg(test)]
mod kn_tests {
use super::*;
#[test]
fn yn_pos_tests() {
assert_eq!(kn(3, 1e-20), 8e+60);
assert_eq!(kn(3, 1.0), 7.1012628247379448);
assert_eq!(kn(3, 10.0), 0.000027252700261657538);
assert_eq!(kn(3, 1e4), 0.0);
assert_eq!(kn(10, 2e-20), 1.8143999999999999e+205);
assert_eq!(kn(10, 2.0), 162482.40397955917);
assert_eq!(kn(10, 20.0), 6.316214528321577e-09);
assert_eq!(kn(10, 2e4), 0.0);
assert_eq!(kn(30, 2e-5), 4.420880996854586e+180);
assert_eq!(kn(30, 2.0), 4.271125754887687e+30);
assert_eq!(kn(30, 20.0), 0.16883087719470813);
assert_eq!(kn(30, 2e4), 0.0);
}
#[test]
fn yn_neg_tests() {
assert_eq!(kn(-3, 1e-20), 8e+60);
assert_eq!(kn(-3, 1.0), 7.1012628247379448);
assert_eq!(kn(-3, 10.0), 0.000027252700261657538);
assert_eq!(kn(-3, 1e4), 0.0);
assert_eq!(kn(-10, 2e-20), 1.8143999999999999e+205);
assert_eq!(kn(-10, 2.0), 162482.40397955917);
assert_eq!(kn(-10, 20.0), 6.316214528321577e-09);
assert_eq!(kn(-10, 2e4), 0.0);
assert_eq!(kn(-30, 2e-5), 4.420880996854586e+180);
assert_eq!(kn(-30, 2.0), 4.271125754887687e+30);
assert_eq!(kn(-30, 20.0), 0.16883087719470813);
assert_eq!(kn(-30, 2e4), 0.0);
}
#[test]
fn kn_1_trivials() {
assert_eq!(kn(1, f64::INFINITY), 0.0);
assert_eq!(kn(1, -f64::INFINITY).is_nan(), true);
assert_eq!(kn(1, f64::NAN).is_nan(), true);
assert_eq!(kn(1, -1e-20).is_nan(), true);
assert_eq!(kn(1, -1.0).is_nan(), true);
assert_eq!(kn(1, 0.0), f64::INFINITY);
}
#[test]
fn kn_1_small_x() {
assert_eq!(kn(1, 1e-20), 1e+20);
assert_eq!(kn(1, 1e-10), 10000000000.0);
assert_eq!(kn(1, 0.1), 9.853844780870606);
assert_eq!(kn(1, 0.5), 1.6564411200033011);
assert_eq!(kn(1, 1.0), 0.6019072301972346);
assert_eq!(kn(1, 1.5), 0.27738780045684375);
assert_eq!(kn(1, 2.0), 0.13986588181652243);
}
#[test]
fn kn_1_large_x() {
assert_eq!(kn(1, 2.0 + 1e-16), 0.13986588181652243);
assert_eq!(kn(1, 3.0), 0.04015643112819417);
assert_eq!(kn(1, 10.0), 1.8648773457308253e-5);
assert_eq!(kn(1, 20.0), 5.883057969557036e-10);
assert_eq!(kn(1, 100.0), 4.679853735636909e-45);
assert_eq!(kn(1, 1000.0), 0.0);
}
#[test]
fn kn_0_trivials() {
assert_eq!(kn(0, f64::INFINITY), 0.0);
assert_eq!(kn(0, -f64::INFINITY).is_nan(), true);
assert_eq!(kn(0, f64::NAN).is_nan(), true);
assert_eq!(kn(0, -1e-20).is_nan(), true);
assert_eq!(kn(0, -1.0).is_nan(), true);
assert_eq!(kn(0, 0.0), f64::INFINITY);
}
#[test]
fn kn_0_small_x() {
assert_eq!(kn(0, 1e-20), 46.16763337553933);
assert_eq!(kn(0, 1e-10), 23.141782445598867);
assert_eq!(kn(0, 0.1), 2.427069024702017);
assert_eq!(kn(0, 0.5), 0.9244190712276656);
assert_eq!(kn(0, 1.0), 0.42102443824070834);
assert_eq!(kn(0, 1.5), 0.21380556264752568);
assert_eq!(kn(0, 2.0), 0.11389387274953341);
}
#[test]
fn kn_0_large_x() {
assert_eq!(kn(0, 2.0 + 1e-16), 0.11389387274953341);
assert_eq!(kn(0, 3.0), 0.03473950438627915);
assert_eq!(kn(0, 10.0), 1.7780062319409137e-5);
assert_eq!(kn(0, 20.0), 5.7412378153365248e-10);
assert_eq!(kn(0, 100.0), 4.656628229175901e-45);
assert_eq!(kn(0, 1000.0), 0.0);
}
}