// https://github.com/JuliaLang/openlibm/blob/master/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. * ==================================================== */ /* * __ieee754_jn(n, x), __ieee754_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 0 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 nx, 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::{ bessel_j0_y0::{j0, y0}, bessel_j1_y1::{j1, y1}, bessel_util::{split_words, FRAC_2_SQRT_PI}, }; // Special cases are: // // $ J_n(n, ±\Infinity) = 0$ // $ J_n(n, NaN} = NaN $ // $ J_n(n, 0) = 0 $ pub(crate) fn jn(n: i32, x: f64) -> f64 { let (lx, mut hx) = split_words(x); let ix = 0x7fffffff & hx; // if J(n,NaN) is NaN if x.is_nan() { return x; } // if (ix | (/*(u_int32_t)*/(lx | -lx)) >> 31) > 0x7ff00000 { // return x + x; // } let (n, x) = if n < 0 { // hx ^= 0x80000000; hx = -hx; (-n, -x) } else { (n, x) }; if n == 0 { return j0(x); } if n == 1 { return j1(x); } let sign = (n & 1) & (hx >> 31); /* even n -- 0, odd n -- sign(x) */ // let sign = if x < 0.0 { -1 } else { 1 }; let x = x.abs(); let b = if (ix | lx) == 0 || ix >= 0x7ff00000 { // if x is 0 or inf 0.0 } else if n 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=x.sin(), c=x.cos(), * 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 */ let temp = match n & 3 { 0 => x.cos() + x.sin(), 1 => -x.cos() + x.sin(), 2 => -x.cos() - x.sin(), 3 => x.cos() - x.sin(), _ => { // Impossible: FIXME! // panic!("") 0.0 } }; FRAC_2_SQRT_PI * temp / x.sqrt() } else { let mut a = j0(x); let mut b = j1(x); for i in 1..n { let temp = b; b = b * (((i + i) as f64) / x) - a; /* avoid underflow */ a = temp; } b } } else { // x < 2^(-29) if ix < 0x3e100000 { // x is tiny, return the first Taylor expansion of J(n,x) // J(n,x) = 1/n!*(x/2)^n - ... if n > 33 { // underflow 0.0 } else { let temp = x * 0.5; let mut b = temp; let mut a = 1; for i in 2..=n { a *= i; /* a = n! */ b *= temp; /* b = (x/2)^n */ } b / (a as f64) } } 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 */ let w = ((n + n) as f64) / x; let h = 2.0 / x; let mut q0 = w; let mut z = w + h; let mut q1 = w * z - 1.0; let mut k = 1; while q1 < 1.0e9 { k += 1; z += h; let tmp = z * q1 - q0; q0 = q1; q1 = tmp; } let m = n + n; let mut t = 0.0; for i in (m..2 * (n + k)).step_by(2).rev() { t = 1.0 / ((i as f64) / x - t); } // for (t=0, i = 2*(n+k); i>=m; i -= 2) t = 1/(i/x-t); let mut a = t; let mut 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 0 */ let mut tmp = n as f64; let v = 2.0 / x; tmp = tmp * f64::ln(f64::abs(v * tmp)); if tmp < 7.097_827_128_933_84e2 { // for(i=n-1, di=(i+i); i>0; i--){ let mut di = 2.0 * ((n - 1) as f64); for _ in (1..=n - 1).rev() { let temp = b; b *= di; b = b / x - a; a = temp; di -= 2.0; } } else { // for(i=n-1, di=(i+i); i>0; i--) { let mut di = 2.0 * ((n - 1) as f64); for _ in (1..=n - 1).rev() { let temp = b; b *= di; b = b / x - a; a = temp; di -= 2.0; /* scale b to avoid spurious overflow */ if b > 1e100 { a /= b; t /= b; b = 1.0; } } } let z = j0(x); let w = j1(x); if z.abs() >= w.abs() { t * z / b } else { t * w / a } } }; if sign == 1 { -b } else { b } } // Yn returns the order-n Bessel function of the second kind. // // Special cases are: // // Y(n, +Inf) = 0 // Y(n ≥ 0, 0) = -Inf // Y(n < 0, 0) = +Inf if n is odd, -Inf if n is even // Y(n, x < 0) = NaN // Y(n, NaN) = NaN pub(crate) fn yn(n: i32, x: f64) -> f64 { let (lx, hx) = split_words(x); let ix = 0x7fffffff & hx; // if Y(n, NaN) is NaN if x.is_nan() { return x; } // if (ix | (/*(u_int32_t)*/(lx | -lx)) >> 31) > 0x7ff00000 { // return x + x; // } if (ix | lx) == 0 { return f64::NEG_INFINITY; } if hx < 0 { return f64::NAN; } let (n, sign) = if n < 0 { (-n, 1 - ((n & 1) << 1)) } else { (n, 1) }; if n == 0 { return y0(x); } if n == 1 { return (sign as f64) * y1(x); } if ix == 0x7ff00000 { return 0.0; } let b = 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=x.sin(), c=x.cos(), * 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 */ let temp = match n & 3 { 0 => x.sin() - x.cos(), 1 => -x.sin() - x.cos(), 2 => -x.sin() + x.cos(), 3 => x.sin() + x.cos(), _ => { // unreachable 0.0 } }; FRAC_2_SQRT_PI * temp / x.sqrt() } else { let mut a = y0(x); let mut b = y1(x); for i in 1..n { if b.is_infinite() { break; } // if high_word(b) != 0xfff00000 { // break; // } (a, b) = (b, ((2.0 * i as f64) / x) * b - a); } b }; if sign > 0 { b } else { -b } }