307 lines
7.2 KiB
Go
307 lines
7.2 KiB
Go
// Copyright 2010 The Go Authors. All rights reserved.
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// Use of this source code is governed by a BSD-style
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// license that can be found in the LICENSE file.
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package math
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/*
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Bessel function of the first and second kinds of order n.
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*/
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// The original C code and the long comment below are
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// from FreeBSD's /usr/src/lib/msun/src/e_jn.c and
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// came with this notice. The go code is a simplified
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// version of the original C.
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//
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// ====================================================
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// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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//
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// Developed at SunPro, a Sun Microsystems, Inc. business.
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// Permission to use, copy, modify, and distribute this
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// software is freely granted, provided that this notice
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// is preserved.
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// ====================================================
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//
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// __ieee754_jn(n, x), __ieee754_yn(n, x)
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// floating point Bessel's function of the 1st and 2nd kind
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// of order n
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//
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// Special cases:
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// y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal;
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// y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
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// Note 2. About jn(n,x), yn(n,x)
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// For n=0, j0(x) is called,
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// for n=1, j1(x) is called,
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// for n<x, forward recursion is used starting
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// from values of j0(x) and j1(x).
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// for n>x, a continued fraction approximation to
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// j(n,x)/j(n-1,x) is evaluated and then backward
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// recursion is used starting from a supposed value
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// for j(n,x). The resulting value of j(0,x) is
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// compared with the actual value to correct the
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// supposed value of j(n,x).
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//
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// yn(n,x) is similar in all respects, except
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// that forward recursion is used for all
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// values of n>1.
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// Jn returns the order-n Bessel function of the first kind.
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//
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// Special cases are:
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// Jn(n, ±Inf) = 0
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// Jn(n, NaN) = NaN
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func Jn(n int, x float64) float64 {
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const (
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TwoM29 = 1.0 / (1 << 29) // 2**-29 0x3e10000000000000
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Two302 = 1 << 302 // 2**302 0x52D0000000000000
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)
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// special cases
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switch {
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case IsNaN(x):
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return x
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case IsInf(x, 0):
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return 0
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}
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// J(-n, x) = (-1)**n * J(n, x), J(n, -x) = (-1)**n * J(n, x)
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// Thus, J(-n, x) = J(n, -x)
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if n == 0 {
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return J0(x)
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}
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if x == 0 {
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return 0
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}
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if n < 0 {
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n, x = -n, -x
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}
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if n == 1 {
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return J1(x)
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}
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sign := false
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if x < 0 {
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x = -x
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if n&1 == 1 {
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sign = true // odd n and negative x
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}
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}
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var b float64
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if float64(n) <= x {
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// Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x)
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if x >= Two302 { // x > 2**302
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// (x >> n**2)
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// Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
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// Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
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// Let s=sin(x), c=cos(x),
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// xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
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//
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// n sin(xn)*sqt2 cos(xn)*sqt2
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// ----------------------------------
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// 0 s-c c+s
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// 1 -s-c -c+s
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// 2 -s+c -c-s
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// 3 s+c c-s
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var temp float64
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switch n & 3 {
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case 0:
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temp = Cos(x) + Sin(x)
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case 1:
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temp = -Cos(x) + Sin(x)
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case 2:
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temp = -Cos(x) - Sin(x)
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case 3:
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temp = Cos(x) - Sin(x)
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}
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b = (1 / SqrtPi) * temp / Sqrt(x)
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} else {
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b = J1(x)
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for i, a := 1, J0(x); i < n; i++ {
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a, b = b, b*(float64(i+i)/x)-a // avoid underflow
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}
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}
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} else {
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if x < TwoM29 { // x < 2**-29
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// x is tiny, return the first Taylor expansion of J(n,x)
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// J(n,x) = 1/n!*(x/2)**n - ...
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if n > 33 { // underflow
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b = 0
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} else {
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temp := x * 0.5
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b = temp
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a := 1.0
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for i := 2; i <= n; i++ {
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a *= float64(i) // a = n!
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b *= temp // b = (x/2)**n
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}
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b /= a
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}
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} else {
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// use backward recurrence
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// x x**2 x**2
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// J(n,x)/J(n-1,x) = ---- ------ ------ .....
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// 2n - 2(n+1) - 2(n+2)
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//
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// 1 1 1
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// (for large x) = ---- ------ ------ .....
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// 2n 2(n+1) 2(n+2)
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// -- - ------ - ------ -
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// x x x
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//
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// Let w = 2n/x and h=2/x, then the above quotient
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// is equal to the continued fraction:
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// 1
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// = -----------------------
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// 1
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// w - -----------------
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// 1
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// w+h - ---------
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// w+2h - ...
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//
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// To determine how many terms needed, let
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// Q(0) = w, Q(1) = w(w+h) - 1,
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// Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
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// When Q(k) > 1e4 good for single
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// When Q(k) > 1e9 good for double
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// When Q(k) > 1e17 good for quadruple
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// determine k
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w := float64(n+n) / x
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h := 2 / x
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q0 := w
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z := w + h
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q1 := w*z - 1
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k := 1
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for q1 < 1e9 {
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k += 1
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z += h
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q0, q1 = q1, z*q1-q0
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}
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m := n + n
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t := 0.0
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for i := 2 * (n + k); i >= m; i -= 2 {
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t = 1 / (float64(i)/x - t)
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}
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a := t
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b = 1
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// estimate log((2/x)**n*n!) = n*log(2/x)+n*ln(n)
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// Hence, if n*(log(2n/x)) > ...
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// single 8.8722839355e+01
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// double 7.09782712893383973096e+02
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// long double 1.1356523406294143949491931077970765006170e+04
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// then recurrent value may overflow and the result is
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// likely underflow to zero
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tmp := float64(n)
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v := 2 / x
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tmp = tmp * Log(Abs(v*tmp))
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if tmp < 7.09782712893383973096e+02 {
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for i := n - 1; i > 0; i-- {
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di := float64(i + i)
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a, b = b, b*di/x-a
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di -= 2
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}
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} else {
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for i := n - 1; i > 0; i-- {
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di := float64(i + i)
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a, b = b, b*di/x-a
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di -= 2
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// scale b to avoid spurious overflow
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if b > 1e100 {
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a /= b
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t /= b
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b = 1
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}
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}
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}
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b = t * J0(x) / b
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}
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}
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if sign {
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return -b
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}
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return b
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}
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// Yn returns the order-n Bessel function of the second kind.
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//
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// Special cases are:
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// Yn(n, +Inf) = 0
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// Yn(n > 0, 0) = -Inf
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// Yn(n < 0, 0) = +Inf if n is odd, -Inf if n is even
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// Y1(n, x < 0) = NaN
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// Y1(n, NaN) = NaN
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func Yn(n int, x float64) float64 {
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const Two302 = 1 << 302 // 2**302 0x52D0000000000000
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// special cases
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switch {
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case x < 0 || IsNaN(x):
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return NaN()
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case IsInf(x, 1):
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return 0
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}
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if n == 0 {
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return Y0(x)
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}
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if x == 0 {
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if n < 0 && n&1 == 1 {
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return Inf(1)
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}
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return Inf(-1)
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}
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sign := false
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if n < 0 {
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n = -n
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if n&1 == 1 {
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sign = true // sign true if n < 0 && |n| odd
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}
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}
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if n == 1 {
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if sign {
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return -Y1(x)
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}
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return Y1(x)
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}
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var b float64
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if x >= Two302 { // x > 2**302
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// (x >> n**2)
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// Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
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// Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
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// Let s=sin(x), c=cos(x),
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// xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
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//
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// n sin(xn)*sqt2 cos(xn)*sqt2
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// ----------------------------------
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// 0 s-c c+s
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// 1 -s-c -c+s
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// 2 -s+c -c-s
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// 3 s+c c-s
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var temp float64
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switch n & 3 {
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case 0:
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temp = Sin(x) - Cos(x)
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case 1:
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temp = -Sin(x) - Cos(x)
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case 2:
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temp = -Sin(x) + Cos(x)
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case 3:
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temp = Sin(x) + Cos(x)
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}
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b = (1 / SqrtPi) * temp / Sqrt(x)
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} else {
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a := Y0(x)
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b = Y1(x)
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// quit if b is -inf
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for i := 1; i < n && !IsInf(b, -1); i++ {
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a, b = b, (float64(i+i)/x)*b-a
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}
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}
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if sign {
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return -b
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}
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return b
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}
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