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+// 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
+}