| // 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 cmath |
| |
| import "math" |
| |
| // The original C code, the long comment, and the constants |
| // below are from http://netlib.sandia.gov/cephes/c9x-complex/clog.c. |
| // The go code is a simplified version of the original C. |
| // |
| // Cephes Math Library Release 2.8: June, 2000 |
| // Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier |
| // |
| // The readme file at http://netlib.sandia.gov/cephes/ says: |
| // Some software in this archive may be from the book _Methods and |
| // Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster |
| // International, 1989) or from the Cephes Mathematical Library, a |
| // commercial product. In either event, it is copyrighted by the author. |
| // What you see here may be used freely but it comes with no support or |
| // guarantee. |
| // |
| // The two known misprints in the book are repaired here in the |
| // source listings for the gamma function and the incomplete beta |
| // integral. |
| // |
| // Stephen L. Moshier |
| // moshier@na-net.ornl.gov |
| |
| // Complex circular arc sine |
| // |
| // DESCRIPTION: |
| // |
| // Inverse complex sine: |
| // 2 |
| // w = -i clog( iz + csqrt( 1 - z ) ). |
| // |
| // casin(z) = -i casinh(iz) |
| // |
| // ACCURACY: |
| // |
| // Relative error: |
| // arithmetic domain # trials peak rms |
| // DEC -10,+10 10100 2.1e-15 3.4e-16 |
| // IEEE -10,+10 30000 2.2e-14 2.7e-15 |
| // Larger relative error can be observed for z near zero. |
| // Also tested by csin(casin(z)) = z. |
| |
| // Asin returns the inverse sine of x. |
| func Asin(x complex128) complex128 { |
| if imag(x) == 0 { |
| if math.Fabs(real(x)) > 1 { |
| return cmplx(math.Pi/2, 0) // DOMAIN error |
| } |
| return cmplx(math.Asin(real(x)), 0) |
| } |
| ct := cmplx(-imag(x), real(x)) // i * x |
| xx := x * x |
| x1 := cmplx(1-real(xx), -imag(xx)) // 1 - x*x |
| x2 := Sqrt(x1) // x2 = sqrt(1 - x*x) |
| w := Log(ct + x2) |
| return cmplx(imag(w), -real(w)) // -i * w |
| } |
| |
| // Asinh returns the inverse hyperbolic sine of x. |
| func Asinh(x complex128) complex128 { |
| // TODO check range |
| if imag(x) == 0 { |
| if math.Fabs(real(x)) > 1 { |
| return cmplx(math.Pi/2, 0) // DOMAIN error |
| } |
| return cmplx(math.Asinh(real(x)), 0) |
| } |
| xx := x * x |
| x1 := cmplx(1+real(xx), imag(xx)) // 1 + x*x |
| return Log(x + Sqrt(x1)) // log(x + sqrt(1 + x*x)) |
| } |
| |
| // Complex circular arc cosine |
| // |
| // DESCRIPTION: |
| // |
| // w = arccos z = PI/2 - arcsin z. |
| // |
| // ACCURACY: |
| // |
| // Relative error: |
| // arithmetic domain # trials peak rms |
| // DEC -10,+10 5200 1.6e-15 2.8e-16 |
| // IEEE -10,+10 30000 1.8e-14 2.2e-15 |
| |
| // Acos returns the inverse cosine of x. |
| func Acos(x complex128) complex128 { |
| w := Asin(x) |
| return cmplx(math.Pi/2-real(w), -imag(w)) |
| } |
| |
| // Acosh returns the inverse hyperbolic cosine of x. |
| func Acosh(x complex128) complex128 { |
| w := Acos(x) |
| if imag(w) <= 0 { |
| return cmplx(-imag(w), real(w)) // i * w |
| } |
| return cmplx(imag(w), -real(w)) // -i * w |
| } |
| |
| // Complex circular arc tangent |
| // |
| // DESCRIPTION: |
| // |
| // If |
| // z = x + iy, |
| // |
| // then |
| // 1 ( 2x ) |
| // Re w = - arctan(-----------) + k PI |
| // 2 ( 2 2) |
| // (1 - x - y ) |
| // |
| // ( 2 2) |
| // 1 (x + (y+1) ) |
| // Im w = - log(------------) |
| // 4 ( 2 2) |
| // (x + (y-1) ) |
| // |
| // Where k is an arbitrary integer. |
| // |
| // catan(z) = -i catanh(iz). |
| // |
| // ACCURACY: |
| // |
| // Relative error: |
| // arithmetic domain # trials peak rms |
| // DEC -10,+10 5900 1.3e-16 7.8e-18 |
| // IEEE -10,+10 30000 2.3e-15 8.5e-17 |
| // The check catan( ctan(z) ) = z, with |x| and |y| < PI/2, |
| // had peak relative error 1.5e-16, rms relative error |
| // 2.9e-17. See also clog(). |
| |
| // Atan returns the inverse tangent of x. |
| func Atan(x complex128) complex128 { |
| if real(x) == 0 && imag(x) > 1 { |
| return NaN() |
| } |
| |
| x2 := real(x) * real(x) |
| a := 1 - x2 - imag(x)*imag(x) |
| if a == 0 { |
| return NaN() |
| } |
| t := 0.5 * math.Atan2(2*real(x), a) |
| w := reducePi(t) |
| |
| t = imag(x) - 1 |
| b := x2 + t*t |
| if b == 0 { |
| return NaN() |
| } |
| t = imag(x) + 1 |
| c := (x2 + t*t) / b |
| return cmplx(w, 0.25*math.Log(c)) |
| } |
| |
| // Atanh returns the inverse hyperbolic tangent of x. |
| func Atanh(x complex128) complex128 { |
| z := cmplx(-imag(x), real(x)) // z = i * x |
| z = Atan(z) |
| return cmplx(imag(z), -real(z)) // z = -i * z |
| } |