// Copyright 2010 The Go Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.
package math
/*
Bessel function of the first and second kinds of order n.
*/
// The original C code and the long comment below are
// from FreeBSD's /usr/src/lib/msun/src/e_jn.c and
// came with this notice. The go code is a simplified
// version of the original C.
//
// ====================================================
// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
//
// Developed at SunPro, 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 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.
// Jn returns the order-n Bessel function of the first kind.
//
// Special cases are:
// Jn(n, ±Inf) = 0
// Jn(n, NaN) = NaN
func Jn(n int, x float64) float64 {
const (
TwoM29 = 1.0 / (1 << 29) // 2**-29 0x3e10000000000000
Two302 = 1 << 302 // 2**302 0x52D0000000000000
)
// special cases
switch {
case IsNaN(x):
return x
case IsInf(x, 0):
return 0
}
// J(-n, x) = (-1)**n * J(n, x), J(n, -x) = (-1)**n * J(n, x)
// Thus, J(-n, x) = J(n, -x)
if n == 0 {
return J0(x)
}
if x == 0 {
return 0
}
if n < 0 {
n, x = -n, -x
}
if n == 1 {
return J1(x)
}
sign := false
if x < 0 {
x = -x
if n&1 == 1 {
sign = true // odd n and negative x
}
}
var b float64
if float64(n) <= x {
// Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x)
if x >= Two302 { // 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
var temp float64
switch s, c := Sincos(x); n & 3 {
case 0:
temp = c + s
case 1:
temp = -c + s
case 2:
temp = -c - s
case 3:
temp = c - s
}
b = (1 / SqrtPi) * temp / Sqrt(x)
} else {
b = J1(x)
for i, a := 1, J0(x); i < n; i++ {
a, b = b, b*(float64(i+i)/x)-a // avoid underflow
}
}
} else {
if x < TwoM29 { // x < 2**-29
// x is tiny, return the first Taylor expansion of J(n,x)
// J(n,x) = 1/n!*(x/2)**n - ...
if n > 33 { // underflow
b = 0
} else {
temp := x * 0.5
b = temp
a := 1.0
for i := 2; i <= n; i++ {
a *= float64(i) // a = n!
b *= temp // b = (x/2)**n
}
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
w := float64(n+n) / x
h := 2 / x
q0 := w
z := w + h
q1 := w*z - 1
k := 1
for q1 < 1e9 {
k++
z += h
q0, q1 = q1, z*q1-q0
}
m := n + n
t := 0.0
for i := 2 * (n + k); i >= m; i -= 2 {
t = 1 / (float64(i)/x - t)
}
a := t
b = 1
// 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 := float64(n)
v := 2 / x
tmp = tmp * Log(Abs(v*tmp))
if tmp < 7.09782712893383973096e+02 {
for i := n - 1; i > 0; i-- {
di := float64(i + i)
a, b = b, b*di/x-a
}
} else {
for i := n - 1; i > 0; i-- {
di := float64(i + i)
a, b = b, b*di/x-a
// scale b to avoid spurious overflow
if b > 1e100 {
a /= b
t /= b
b = 1
}
}
}
b = t * J0(x) / b
}
}
if sign {
return -b
}
return b
}
// Yn returns the order-n Bessel function of the second kind.
//
// Special cases are:
// Yn(n, +Inf) = 0
// Yn(n ≥ 0, 0) = -Inf
// Yn(n < 0, 0) = +Inf if n is odd, -Inf if n is even
// Yn(n, x < 0) = NaN
// Yn(n, NaN) = NaN
func Yn(n int, x float64) float64 {
const Two302 = 1 << 302 // 2**302 0x52D0000000000000
// special cases
switch {
case x < 0 || IsNaN(x):
return NaN()
case IsInf(x, 1):
return 0
}
if n == 0 {
return Y0(x)
}
if x == 0 {
if n < 0 && n&1 == 1 {
return Inf(1)
}
return Inf(-1)
}
sign := false
if n < 0 {
n = -n
if n&1 == 1 {
sign = true // sign true if n < 0 && |n| odd
}
}
if n == 1 {
if sign {
return -Y1(x)
}
return Y1(x)
}
var b float64
if x >= Two302 { // 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
var temp float64
switch s, c := Sincos(x); n & 3 {
case 0:
temp = s - c
case 1:
temp = -s - c
case 2:
temp = -s + c
case 3:
temp = s + c
}
b = (1 / SqrtPi) * temp / Sqrt(x)
} else {
a := Y0(x)
b = Y1(x)
// quit if b is -inf
for i := 1; i < n && !IsInf(b, -1); i++ {
a, b = b, (float64(i+i)/x)*b-a
}
}
if sign {
return -b
}
return b
}
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