forked from tsai/mathnet-numerics
Christoph Ruegg
13 years ago
8 changed files with 253 additions and 59 deletions
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// <copyright file="NewtonRaphson.cs" company="Math.NET">
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// Math.NET Numerics, part of the Math.NET Project
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// http://numerics.mathdotnet.com
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// http://github.com/mathnet/mathnet-numerics
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// http://mathnetnumerics.codeplex.com
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//
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// Copyright (c) 2009-2013 Math.NET
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//
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// Permission is hereby granted, free of charge, to any person
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// obtaining a copy of this software and associated documentation
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// files (the "Software"), to deal in the Software without
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// restriction, including without limitation the rights to use,
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// copy, modify, merge, publish, distribute, sublicense, and/or sell
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// copies of the Software, and to permit persons to whom the
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// Software is furnished to do so, subject to the following
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// conditions:
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//
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// The above copyright notice and this permission notice shall be
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// included in all copies or substantial portions of the Software.
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//
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// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
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// EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES
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// OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
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// NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT
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// HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY,
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// WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
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// FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
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// OTHER DEALINGS IN THE SOFTWARE.
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// </copyright>
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using System; |
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namespace MathNet.Numerics.RootFinding |
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{ |
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/// <summary>
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/// Pure Newton-Raphson root-finding algorithm without any recovery measures in cases it behaves badly.
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/// The algorithm aborts immediately if the root leaves the bound interval.
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/// </summary>
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/// <seealso cref="RobustNewtonRaphson"/>
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public static class NewtonRaphson |
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{ |
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/// <summary>Find a solution of the equation f(x)=0.</summary>
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/// <param name="f">The function to find roots from.</param>
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/// <param name="df">The first derivative of the function to find roots from.</param>
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/// <param name="initialGuess">Initial guess of the root.</param>
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/// <param name="lowerBound">The low value of the range where the root is supposed to be. Aborts if it leaves the interval.</param>
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/// <param name="upperBound">The high value of the range where the root is supposed to be. Aborts if it leaves the interval.</param>
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/// <param name="accuracy">Desired accuracy. The root will be refined until the accuracy or the maximum number of iterations is reached. Example: 1e-14.</param>
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/// <param name="maxIterations">Maximum number of iterations. Example: 100.</param>
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/// <returns>Returns the root with the specified accuracy.</returns>
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/// <exception cref="NonConvergenceException"></exception>
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public static double FindRoot(Func<double, double> f, Func<double, double> df, double initialGuess, double lowerBound, double upperBound, double accuracy, int maxIterations) |
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{ |
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double root; |
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if (TryFindRoot(f, df, initialGuess, lowerBound, upperBound, accuracy, maxIterations, out root)) |
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{ |
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return root; |
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} |
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throw new NonConvergenceException("The algorithm failed or has exceeded the number of iterations allowed. Consider to use RobustNewtonRaphson instead."); |
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} |
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/// <summary>Find a solution of the equation f(x)=0.</summary>
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/// <param name="f">The function to find roots from.</param>
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/// <param name="df">The first derivative of the function to find roots from.</param>
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/// <param name="initialGuess">Initial guess of the root.</param>
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/// <param name="lowerBound">The low value of the range where the root is supposed to be. Aborts if it leaves the interval.</param>
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/// <param name="upperBound">The high value of the range where the root is supposed to be. Aborts if it leaves the interval.</param>
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/// <param name="accuracy">Desired accuracy. The root will be refined until the accuracy or the maximum number of iterations is reached. Example: 1e-14.</param>
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/// <param name="maxIterations">Maximum number of iterations. Example: 100.</param>
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/// <param name="root">The root that was found, if any. Undefined if the function returns false.</param>
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/// <returns>True if a root with the specified accuracy was found, else false.</returns>
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public static bool TryFindRoot(Func<double, double> f, Func<double, double> df, double initialGuess, double lowerBound, double upperBound, double accuracy, int maxIterations, out double root) |
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{ |
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root = initialGuess; |
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for (int i = 0; i < maxIterations && root >= lowerBound && root <= upperBound; i++) |
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{ |
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// Evaluation
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double fx = f(root); |
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double dfx = df(root); |
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// Netwon-Raphson step
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double step = fx/dfx; |
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root -= step; |
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if (Math.Abs(step) < accuracy && Math.Abs(fx) < accuracy) |
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{ |
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return true; |
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} |
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} |
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return false; |
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} |
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} |
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} |
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@ -0,0 +1,127 @@ |
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// <copyright file="BrentTest.cs" company="Math.NET">
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// Math.NET Numerics, part of the Math.NET Project
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// http://numerics.mathdotnet.com
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// http://github.com/mathnet/mathnet-numerics
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// http://mathnetnumerics.codeplex.com
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//
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// Copyright (c) 2009-2013 Math.NET
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//
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// Permission is hereby granted, free of charge, to any person
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// obtaining a copy of this software and associated documentation
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// files (the "Software"), to deal in the Software without
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// restriction, including without limitation the rights to use,
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// copy, modify, merge, publish, distribute, sublicense, and/or sell
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// copies of the Software, and to permit persons to whom the
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// Software is furnished to do so, subject to the following
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// conditions:
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//
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// The above copyright notice and this permission notice shall be
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// included in all copies or substantial portions of the Software.
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//
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// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
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// EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES
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// OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
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// NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT
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// HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY,
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// WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
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// FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
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// OTHER DEALINGS IN THE SOFTWARE.
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// </copyright>
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using System; |
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using MathNet.Numerics.RootFinding; |
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using NUnit.Framework; |
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namespace MathNet.Numerics.UnitTests.RootFindingTests |
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{ |
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[TestFixture] |
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public class RobustNewtonRaphsonTest |
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{ |
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[Test] |
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public void MultipleRoots() |
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{ |
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// Roots at -2, 2
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Func<double, double> f1 = x => x * x - 4; |
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Func<double, double> df1 = x => 2 * x; |
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Assert.AreEqual(0, f1(RobustNewtonRaphson.FindRoot(f1, df1, -5, 5, 1e-14, 100, 20))); |
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Assert.AreEqual(-2, RobustNewtonRaphson.FindRoot(f1, df1, -5, -1, 1e-14, 100, 20)); |
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Assert.AreEqual(2, RobustNewtonRaphson.FindRoot(f1, df1, 1, 4, 1e-14, 100, 20)); |
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Assert.AreEqual(0, f1(RobustNewtonRaphson.FindRoot(x => -f1(x), x => -df1(x), -5, 5, 1e-14, 100, 20))); |
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Assert.AreEqual(-2, RobustNewtonRaphson.FindRoot(x => -f1(x), x => -df1(x), -5, -1, 1e-14, 100, 20)); |
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Assert.AreEqual(2, RobustNewtonRaphson.FindRoot(x => -f1(x), x => -df1(x), 1, 4, 1e-14, 100, 20)); |
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// Roots at 3, 4
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Func<double, double> f2 = x => (x - 3) * (x - 4); |
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Func<double, double> df2 = x => 2 * x - 7; |
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Assert.AreEqual(0, f2(RobustNewtonRaphson.FindRoot(f2, df2, -5, 5, 1e-14, 100, 20))); |
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Assert.AreEqual(3, RobustNewtonRaphson.FindRoot(f2, df2, -5, 3.5, 1e-14, 100, 20)); |
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Assert.AreEqual(4, RobustNewtonRaphson.FindRoot(f2, df2, 3.2, 5, 1e-14, 100, 20)); |
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Assert.AreEqual(3, RobustNewtonRaphson.FindRoot(f2, df2, 2.1, 3.9, 0.001, 50, 20), 0.001); |
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Assert.AreEqual(3, RobustNewtonRaphson.FindRoot(f2, df2, 2.1, 3.4, 0.001, 50, 20), 0.001); |
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} |
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[Test] |
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public void LocalMinima() |
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{ |
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Func<double, double> f1 = x => x * x * x - 2 * x + 2; |
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Func<double, double> df1 = x => 3 * x * x - 2; |
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Assert.AreEqual(0, f1(RobustNewtonRaphson.FindRoot(f1, df1, -5, 5, 1e-14, 100, 20))); |
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Assert.AreEqual(0, f1(RobustNewtonRaphson.FindRoot(f1, df1, -2, 4, 1e-14, 100, 20))); |
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} |
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[Test] |
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public void Pole() |
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{ |
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Func<double, double> f1 = x => 1/(x - 2) + 2; |
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Func<double, double> df1 = x => -1/(x*x - 4*x + 4); |
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Assert.AreEqual(1.5, RobustNewtonRaphson.FindRoot(f1, df1, 1, 2, 1e-14, 100, 20)); |
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Assert.AreEqual(1.5, RobustNewtonRaphson.FindRoot(f1, df1, 1, 6, 1e-14, 100, 20)); |
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Assert.AreEqual(1.5, FindRoots.OfFunctionAndDerivative(f1, df1, 1, 6)); |
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Func<double, double> f2 = x => -1/(x - 2) + 2; |
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Func<double, double> df2 = x => 1/(x*x - 4*x + 4); |
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Assert.AreEqual(2.5, RobustNewtonRaphson.FindRoot(f2, df2, 2, 3, 1e-14, 100, 20)); |
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Assert.AreEqual(2.5, RobustNewtonRaphson.FindRoot(f2, df2, -2, 3, 1e-14, 100, 20)); |
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Assert.AreEqual(2.5, FindRoots.OfFunctionAndDerivative(f2, df2, -2, 3)); |
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Func<double, double> f3 = x => 1/(x - 2) + x + 2; |
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Func<double, double> df3 = x => -1/(x*x - 4*x + 4) + 1; |
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Assert.AreEqual(-Math.Sqrt(3), RobustNewtonRaphson.FindRoot(f3, df3, -2, -1, 1e-14, 100, 20), 1e-14); |
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Assert.AreEqual(Math.Sqrt(3), RobustNewtonRaphson.FindRoot(f3, df3, 1, 1.99, 1e-14, 100, 20)); |
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Assert.AreEqual(Math.Sqrt(3), RobustNewtonRaphson.FindRoot(f3, df3, -1.5, 1.99, 1e-14, 100, 20)); |
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Assert.AreEqual(Math.Sqrt(3), RobustNewtonRaphson.FindRoot(f3, df3, 1, 6, 1e-14, 100, 20)); |
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Func<double, double> f4 = x => 1/(2 - x) - x + 6; |
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Func<double, double> df4 = x => 1/(x*x - 4*x + 4) - 1; |
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Assert.AreEqual(4 + Math.Sqrt(3), RobustNewtonRaphson.FindRoot(f4, df4, 5, 6, 1e-14, 100, 20), 1e-14); |
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Assert.AreEqual(4 - Math.Sqrt(3), RobustNewtonRaphson.FindRoot(f4, df4, 2.01, 3, 1e-14, 100, 20)); |
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Assert.AreEqual(4 - Math.Sqrt(3), RobustNewtonRaphson.FindRoot(f4, df4, 2.01, 5, 1e-14, 100, 20)); |
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Assert.AreEqual(4 - Math.Sqrt(3), RobustNewtonRaphson.FindRoot(f4, df4, -2, 4, 1e-14, 100, 20)); |
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} |
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[Test] |
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public void Cubic() |
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{ |
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// with complex roots (looking for the real root only): 3x^3 + 4x^2 + 5x + 6, derivative 9x^2 + 8x + 5
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Func<double, double> f1 = x => Evaluate.Polynomial(x, 6, 5, 4, 3); |
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Func<double, double> df1 = x => Evaluate.Polynomial(x, 5, 8, 9); |
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Assert.AreEqual(-1.265328088928, RobustNewtonRaphson.FindRoot(f1, df1, -2, -1, 1e-10, 100, 20), 1e-6); |
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Assert.AreEqual(-1.265328088928, RobustNewtonRaphson.FindRoot(f1, df1, -5, 5, 1e-10, 100, 20), 1e-6); |
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// real roots only: 2x^3 + 4x^2 - 50x + 6, derivative 6x^2 + 8x - 50
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Func<double, double> f2 = x => Evaluate.Polynomial(x, 6, -50, 4, 2); |
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Func<double, double> df2 = x => Evaluate.Polynomial(x, -50, 8, 6); |
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Assert.AreEqual(-6.1466562197069, RobustNewtonRaphson.FindRoot(f2, df2, -8, -5, 1e-10, 100, 20), 1e-6); |
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Assert.AreEqual(0.12124737195841, RobustNewtonRaphson.FindRoot(f2, df2, -1, 1, 1e-10, 100, 20), 1e-6); |
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Assert.AreEqual(4.0254088477485, RobustNewtonRaphson.FindRoot(f2, df2, 3, 5, 1e-10, 100, 20), 1e-6); |
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} |
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[Test] |
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public void NoRoot() |
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{ |
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Func<double, double> f1 = x => x * x + 4; |
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Func<double, double> df1 = x => 2 * x; |
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Assert.Throws<NonConvergenceException>(() => RobustNewtonRaphson.FindRoot(f1, df1, -5, 5, 1e-14, 50, 20)); |
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} |
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} |
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} |
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