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@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 // Cancelation 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 // Ensure that imag(r) has the same sign as imag(x) for imag(x) == signed zero. 61 if real(x) == 0 { 62 return complex(0, imag(x)) 63 } 64 if real(x) < 0 { 65 return complex(0, math.Copysign(math.Sqrt(-real(x)), imag(x))) 66 } 67 return complex(math.Sqrt(real(x)), imag(x)) 68 } 69 if real(x) == 0 { 70 if imag(x) < 0 { 71 r := math.Sqrt(-0.5 * imag(x)) 72 return complex(r, -r) 73 } 74 r := math.Sqrt(0.5 * imag(x)) 75 return complex(r, r) 76 } 77 a := real(x) 78 b := imag(x) 79 var scale float64 80 // Rescale to avoid internal overflow or underflow. 81 if math.Abs(a) > 4 || math.Abs(b) > 4 { 82 a *= 0.25 83 b *= 0.25 84 scale = 2 85 } else { 86 a *= 1.8014398509481984e16 // 2**54 87 b *= 1.8014398509481984e16 88 scale = 7.450580596923828125e-9 // 2**-27 89 } 90 r := math.Hypot(a, b) 91 var t float64 92 if a > 0 { 93 t = math.Sqrt(0.5*r + 0.5*a) 94 r = scale * math.Abs((0.5*b)/t) 95 t *= scale 96 } else { 97 r = math.Sqrt(0.5*r - 0.5*a) 98 t = scale * math.Abs((0.5*b)/r) 99 r *= scale 100 } 101 if b < 0 { 102 return complex(t, -r) 103 } 104 return complex(t, r) 105 } 106