// <copyright file="Broyden.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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//
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// Copyright (c) 2009-2020 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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// 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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// OTHER DEALINGS IN THE SOFTWARE.
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// </copyright>
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using System;
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using IStation.Numerics.LinearAlgebra;
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using IStation.Numerics.LinearAlgebra.Double;
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namespace IStation.Numerics.RootFinding
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{
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/// <summary>
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/// Algorithm by Broyden.
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/// Implementation inspired by Press, Teukolsky, Vetterling, and Flannery, "Numerical Recipes in C", 2nd edition, Cambridge University Press
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/// </summary>
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public static class Broyden
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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="initialGuess">Initial guess of the root.</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. Default 1e-8. Must be greater than 0.</param>
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/// <param name="maxIterations">Maximum number of iterations. Default 100.</param>
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/// <param name="jacobianStepSize">Relative step size for calculating the Jacobian matrix at first step. Default 1.0e-4</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, double[] initialGuess, double accuracy = 1e-8, int maxIterations = 100, double jacobianStepSize = 1.0e-4)
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{
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double[] root;
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if (TryFindRootWithJacobianStep(f, initialGuess, accuracy, maxIterations, jacobianStepSize, 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 has failed, exceeded the number of iterations allowed or there is no root within the provided bounds.");
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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="initialGuess">Initial guess of the root.</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. Must be greater than 0.</param>
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/// <param name="maxIterations">Maximum number of iterations. Usually 100.</param>
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/// <param name="jacobianStepSize">Relative step size for calculating the Jacobian matrix at first step.</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 TryFindRootWithJacobianStep(Func<double[], double[]> f, double[] initialGuess, double accuracy, int maxIterations, double jacobianStepSize, out double[] root)
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{
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if (accuracy <= 0)
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{
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throw new ArgumentOutOfRangeException(nameof(accuracy), "Must be greater than zero.");
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}
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var x = new DenseVector(initialGuess);
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double[] y0 = f(initialGuess);
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var y = new DenseVector(y0);
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double g = y.L2Norm();
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Matrix<double> B = CalculateApproximateJacobian(f, initialGuess, y0, jacobianStepSize);
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try
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{
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for (int i = 0; i <= maxIterations; i++)
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{
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var dx = (DenseVector) (-B.LU().Solve(y));
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var xnew = x + dx;
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var ynew = new DenseVector(f(xnew.Values));
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double gnew = ynew.L2Norm();
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if (gnew > g)
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{
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double g2 = g*g;
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double scale = g2/(g2 + gnew*gnew);
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if (scale == 0.0)
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{
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scale = 1.0e-4;
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}
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dx = scale*dx;
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xnew = x + dx;
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ynew = new DenseVector(f(xnew.Values));
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gnew = ynew.L2Norm();
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}
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if (gnew < accuracy)
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{
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root = xnew.Values;
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return true;
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}
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// update Jacobian B
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DenseVector dF = ynew - y;
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Matrix<double> dB = (dF - B.Multiply(dx)).ToColumnMatrix() * dx.Multiply(1.0 / Math.Pow(dx.L2Norm(), 2)).ToRowMatrix();
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B = B + dB;
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x = xnew;
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y = ynew;
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g = gnew;
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}
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}
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catch (InvalidParameterException)
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{
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root = null;
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return false;
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}
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root = null;
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return false;
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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="initialGuess">Initial guess of the root.</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. Must be greater than 0.</param>
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/// <param name="maxIterations">Maximum number of iterations. Usually 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, double[] initialGuess, double accuracy, int maxIterations, out double[] root)
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{
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return TryFindRootWithJacobianStep(f, initialGuess, accuracy, maxIterations, 1.0e-4, out root);
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}
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/// <summary>
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/// Helper method to calculate an approximation of the Jacobian.
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/// </summary>
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/// <param name="f">The function.</param>
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/// <param name="x0">The argument (initial guess).</param>
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/// <param name="y0">The result (of initial guess).</param>
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/// <param name="jacobianStepSize">Relative step size for calculating the Jacobian.</param>
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static Matrix<double> CalculateApproximateJacobian(Func<double[], double[]> f, double[] x0, double[] y0, double jacobianStepSize)
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{
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int dim = x0.Length;
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var B = new DenseMatrix(dim);
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var x = new double[dim];
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Array.Copy(x0, 0, x, 0, dim);
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for (int j = 0; j < dim; j++)
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{
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double h = (1.0+Math.Abs(x0[j]))*jacobianStepSize;
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var xj = x[j];
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x[j] = xj + h;
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double[] y = f(x);
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x[j] = xj;
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for (int i = 0; i < dim; i++)
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{
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B.At(i, j, (y[i] - y0[i])/h);
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}
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}
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return B;
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}
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}
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}
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