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Source file src/math/jn.go

Documentation: math

  // 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 n & 3 {
  			case 0:
  				temp = Cos(x) + Sin(x)
  			case 1:
  				temp = -Cos(x) + Sin(x)
  			case 2:
  				temp = -Cos(x) - Sin(x)
  			case 3:
  				temp = Cos(x) - Sin(x)
  			}
  			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 n & 3 {
  		case 0:
  			temp = Sin(x) - Cos(x)
  		case 1:
  			temp = -Sin(x) - Cos(x)
  		case 2:
  			temp = -Sin(x) + Cos(x)
  		case 3:
  			temp = Sin(x) + Cos(x)
  		}
  		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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