// 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 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. // 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 }