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RootFinding: new implementation for complex cubic roots, tests

pull/197/head
Christoph Ruegg 13 years ago
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035c5e2915
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b0ed692ab2
3 changed files with 99 additions and 27 deletions
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  1. 4
      src/Numerics/FindRoots.cs
  2. 81
      src/Numerics/RootFinding/Cubic.cs
  3. 41
      src/UnitTests/RootFindingTests/CubicTest.cs

4
src/Numerics/FindRoots.cs
View File

@ -4,7 +4,7 @@
// http://github.com/mathnet/mathnet-numerics
// http://mathnetnumerics.codeplex.com
//
// Copyright (c) 2009-2013 Math.NET
// Copyright (c) 2009-2014 Math.NET
//
// Permission is hereby granted, free of charge, to any person
// obtaining a copy of this software and associated documentation
@ -112,7 +112,7 @@ namespace MathNet.Numerics
/// </summary>
public static Tuple<Complex, Complex, Complex> Cubic(double d, double c, double b, double a)
{
return RootFinding.Cubic.Roots(b/a, c/a, d/a);
return RootFinding.Cubic.Roots(d, c, b, a);
}
/// <summary>

81
src/Numerics/RootFinding/Cubic.cs
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@ -1,4 +1,34 @@
using System;
// <copyright file="Cubic.cs" company="Math.NET">
// Math.NET Numerics, part of the Math.NET Project
// http://numerics.mathdotnet.com
// http://github.com/mathnet/mathnet-numerics
// http://mathnetnumerics.codeplex.com
//
// Copyright (c) 2009-2014 Math.NET
//
// Permission is hereby granted, free of charge, to any person
// obtaining a copy of this software and associated documentation
// files (the "Software"), to deal in the Software without
// restriction, including without limitation the rights to use,
// copy, modify, merge, publish, distribute, sublicense, and/or sell
// copies of the Software, and to permit persons to whom the
// Software is furnished to do so, subject to the following
// conditions:
//
// The above copyright notice and this permission notice shall be
// included in all copies or substantial portions of the Software.
//
// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
// EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES
// OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
// NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT
// HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY,
// WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
// FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
// OTHER DEALINGS IN THE SOFTWARE.
// </copyright>
using System;
#if !NOSYSNUMERICS
using Complex = System.Numerics.Complex;
@ -34,7 +64,11 @@ namespace MathNet.Numerics.RootFinding
return Math.Pow(Math.Abs(n), 1d / 3d) * Math.Sign(n);
}
public static Tuple<double, double, double> RealRoots(double a2, double a1, double a0)
/// <summary>
/// Find all real-valued roots of the cubic equation a0 + a1*x + a2*x^2 + x^3 = 0.
/// Note the special coefficient order ascending by exponent (consistent with polynomials).
/// </summary>
public static Tuple<double, double, double> RealRoots(double a0, double a1, double a2)
{
double Q, R;
QR(a2, a1, a0, out Q, out R);
@ -68,27 +102,38 @@ namespace MathNet.Numerics.RootFinding
return new Tuple<double, double, double>(x1, x2, x3);
}
public static Tuple<Complex, Complex, Complex> Roots(double a2, double a1, double a0)
/// <summary>
/// Find all three complex roots of the cubic equation d + c*x + b*x^2 + a*x^3 = 0.
/// Note the special coefficient order ascending by exponent (consistent with polynomials).
/// </summary>
public static Tuple<Complex, Complex, Complex> Roots(double d, double c, double b, double a)
{
// use eqn (54)-(56)
double Q, R;
QR(a2, a1, a0, out Q, out R);
double A = b*b - 3*a*c;
double B = 2*b*b*b - 9*a*b*c + 27*a*a*d;
double s = -1/(3*a);
var D = Q * Q * Q + R * R;
double D = (B*B - 4*A*A*A)/(-27*a*a);
if (D == 0d)
{
if (A == 0d)
{
var u = new Complex(s*b, 0d);
return new Tuple<Complex, Complex, Complex>(u, u, u);
}
var rootD = Complex.Sqrt(D);
var S = Complex.Pow(R + rootD, 1d / 3d);
var T = Complex.Pow(R - rootD, 1d / 3d);
var shift = -a2 / 3d;
var sharedI = 0.5 * Complex.ImaginaryOne * Constants.Sqrt3 * (S - T);
var v = new Complex((9*a*d - b*c)/(2*A), 0d);
var w = new Complex((4*a*b*c - 9*a*a*d - b*b*b)/(a*A), 0d);
return new Tuple<Complex, Complex, Complex>(v, v, w);
}
var x1 = shift + (S + T);
var x2 = shift - 0.5 * (S + T);
var x3 = x2;
x2 += sharedI;
x3 -= sharedI;
var C = (A == 0)
? new Complex(B, 0d).CubicRoots()
: ((B + Complex.Sqrt(B*B - 4*A*A*A))/2).CubicRoots();
return new Tuple<Complex, Complex, Complex>(x1, x2, x3);
return new Tuple<Complex, Complex, Complex>(
s*(b + C.Item1 + A/C.Item1),
s*(b + C.Item2 + A/C.Item2),
s*(b + C.Item3 + A/C.Item3));
}
}
}

41
src/UnitTests/RootFindingTests/CubicTest.cs
View File

@ -1,5 +1,4 @@
using System;
using MathNet.Numerics.RootFinding;
using MathNet.Numerics.RootFinding;
using NUnit.Framework;
namespace MathNet.Numerics.UnitTests.RootFindingTests
@ -13,8 +12,7 @@ namespace MathNet.Numerics.UnitTests.RootFindingTests
var a0 = 1d;
var a1 = 5d;
var a2 = 2d;
var roots = Cubic.RealRoots(a2, a1, a0);
var roots = Cubic.RealRoots(a0, a1, a2);
var root = roots.Item1;
var funcValue = root * (root * (root + a2) + a1) + a0;
Assert.That(funcValue, Is.EqualTo(0).Within(1e-14));
@ -28,7 +26,7 @@ namespace MathNet.Numerics.UnitTests.RootFindingTests
var a0 = -2d;
var a1 = 5d;
var a2 = -4d;
var roots = Cubic.RealRoots(a2, a1, a0);
var roots = Cubic.RealRoots(a0, a1, a2);
Assert.That(roots.Item1, Is.EqualTo(2).Within(1e-14));
Assert.That(roots.Item2, Is.EqualTo(1).Within(1e-14));
Assert.AreEqual(double.NaN, roots.Item3);
@ -40,7 +38,7 @@ namespace MathNet.Numerics.UnitTests.RootFindingTests
var a0 = -4d;
var a1 = 8d;
var a2 = -5d;
var roots = Cubic.RealRoots(a2, a1, a0);
var roots = Cubic.RealRoots(a0, a1, a2);
Assert.That(roots.Item1, Is.EqualTo(1).Within(1e-14));
Assert.That(roots.Item2, Is.EqualTo(2).Within(1e-14));
Assert.AreEqual(double.NaN, roots.Item3);
@ -52,10 +50,39 @@ namespace MathNet.Numerics.UnitTests.RootFindingTests
var a0 = 6d;
var a1 = -5d;
var a2 = -2d;
var roots = Cubic.RealRoots(a2, a1, a0);
var roots = Cubic.RealRoots(a0, a1, a2);
Assert.That(roots.Item1, Is.EqualTo(3).Within(1e-14));
Assert.That(roots.Item2, Is.EqualTo(-2).Within(1e-14));
Assert.That(roots.Item3, Is.EqualTo(1).Within(1e-14));
}
[TestCase(6.0, -5.0, -2.0, 1.0, -2.0, 3.0, 1.0)]
public void ComplexRoots_TripleReal(double d, double c, double b, double a, double x1, double x2, double x3)
{
var roots = FindRoots.Cubic(d, c, b, a);
Assert.That(roots.Item1.Real, Is.EqualTo(x1).Within(1e-14));
Assert.That(roots.Item1.Imaginary, Is.EqualTo(0).Within(1e-14));
Assert.That(roots.Item2.Real, Is.EqualTo(x2).Within(1e-14));
Assert.That(roots.Item2.Imaginary, Is.EqualTo(0).Within(1e-14));
Assert.That(roots.Item3.Real, Is.EqualTo(x3).Within(1e-14));
Assert.That(roots.Item3.Imaginary, Is.EqualTo(0).Within(1e-14));
}
[TestCase(-350.0, 162.0, -30.0, 2.0)]
[TestCase(6.0, -5.0, -2.0, 1.0)]
[TestCase(1.0, 5.0, 2.0, 1.0)]
[TestCase(1.0, 5.0, 0.0, 1.0)]
[TestCase(1.0, 0.0, 2.0, 1.0)]
[TestCase(0.0, 0.0, 0.0, 2.0)]
public void ComplexRootsAreRoots(double d, double c, double b, double a)
{
var roots = FindRoots.Cubic(d, c, b, a);
Assert.That(Evaluate.Polynomial(roots.Item1, d, c, b, a).Real, Is.EqualTo(0).Within(1e-12));
Assert.That(Evaluate.Polynomial(roots.Item1, d, c, b, a).Imaginary, Is.EqualTo(0).Within(1e-12));
Assert.That(Evaluate.Polynomial(roots.Item2, d, c, b, a).Real, Is.EqualTo(0).Within(1e-12));
Assert.That(Evaluate.Polynomial(roots.Item2, d, c, b, a).Imaginary, Is.EqualTo(0).Within(1e-12));
Assert.That(Evaluate.Polynomial(roots.Item3, d, c, b, a).Real, Is.EqualTo(0).Within(1e-12));
Assert.That(Evaluate.Polynomial(roots.Item3, d, c, b, a).Imaginary, Is.EqualTo(0).Within(1e-12));
}
}
}

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