1 // Copyright 2010 The Go Authors. All rights reserved. 2 // Use of this source code is governed by a BSD-style 3 // license that can be found in the LICENSE file. 4 5 package cmplx 6 7 import "math" 8 9 // The original C code, the long comment, and the constants 10 // below are from http://netlib.sandia.gov/cephes/c9x-complex/clog.c. 11 // The go code is a simplified version of the original C. 12 // 13 // Cephes Math Library Release 2.8: June, 2000 14 // Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier 15 // 16 // The readme file at http://netlib.sandia.gov/cephes/ says: 17 // Some software in this archive may be from the book _Methods and 18 // Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster 19 // International, 1989) or from the Cephes Mathematical Library, a 20 // commercial product. In either event, it is copyrighted by the author. 21 // What you see here may be used freely but it comes with no support or 22 // guarantee. 23 // 24 // The two known misprints in the book are repaired here in the 25 // source listings for the gamma function and the incomplete beta 26 // integral. 27 // 28 // Stephen L. Moshier 29 // moshier (a] na-net.ornl.gov 30 31 // Complex square root 32 // 33 // DESCRIPTION: 34 // 35 // If z = x + iy, r = |z|, then 36 // 37 // 1/2 38 // Re w = [ (r + x)/2 ] , 39 // 40 // 1/2 41 // Im w = [ (r - x)/2 ] . 42 // 43 // Cancellation error in r-x or r+x is avoided by using the 44 // identity 2 Re w Im w = y. 45 // 46 // Note that -w is also a square root of z. The root chosen 47 // is always in the right half plane and Im w has the same sign as y. 48 // 49 // ACCURACY: 50 // 51 // Relative error: 52 // arithmetic domain # trials peak rms 53 // DEC -10,+10 25000 3.2e-17 9.6e-18 54 // IEEE -10,+10 1,000,000 2.9e-16 6.1e-17 55 56 // Sqrt returns the square root of x. 57 // The result r is chosen so that real(r) 0 and imag(r) has the same sign as imag(x). 58 func Sqrt(x complex128) complex128 { 59 if imag(x) == 0 { 60 if real(x) == 0 { 61 return complex(0, 0) 62 } 63 if real(x) < 0 { 64 return complex(0, math.Sqrt(-real(x))) 65 } 66 return complex(math.Sqrt(real(x)), 0) 67 } 68 if real(x) == 0 { 69 if imag(x) < 0 { 70 r := math.Sqrt(-0.5 * imag(x)) 71 return complex(r, -r) 72 } 73 r := math.Sqrt(0.5 * imag(x)) 74 return complex(r, r) 75 } 76 a := real(x) 77 b := imag(x) 78 var scale float64 79 // Rescale to avoid internal overflow or underflow. 80 if math.Abs(a) > 4 || math.Abs(b) > 4 { 81 a *= 0.25 82 b *= 0.25 83 scale = 2 84 } else { 85 a *= 1.8014398509481984e16 // 2**54 86 b *= 1.8014398509481984e16 87 scale = 7.450580596923828125e-9 // 2**-27 88 } 89 r := math.Hypot(a, b) 90 var t float64 91 if a > 0 { 92 t = math.Sqrt(0.5*r + 0.5*a) 93 r = scale * math.Abs((0.5*b)/t) 94 t *= scale 95 } else { 96 r = math.Sqrt(0.5*r - 0.5*a) 97 t = scale * math.Abs((0.5*b)/r) 98 r *= scale 99 } 100 if b < 0 { 101 return complex(t, -r) 102 } 103 return complex(t, r) 104 } 105
