mathnet-numerics/src/Numerics/RootFinding/Broyden.cs

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using MathNet.Numerics.LinearAlgebra;
using MathNet.Numerics.LinearAlgebra.Double;
using MathNet.Numerics.Properties;
using System;

namespace MathNet.Numerics.RootFinding
{
    /// <summary>
    /// Algorithm by Broyden.
    /// Implementation inspired by Press, Teukolsky, Vetterling, and Flannery, "Numerical Recipes in C", 2nd edition, Cambridge University Press
    /// </summary>
    public static class Broyden
    {
        /// <summary>Find a solution of the equation f(x)=0.</summary>
        /// <param name="f">The function to find roots from.</param>
        /// <param name="initialGuess">Initial guess of the root.</param>
        /// <param name="accuracy">Desired accuracy. The root will be refined until the accuracy or the maximum number of iterations is reached. Default 1e-8.</param>
        /// <param name="maxIterations">Maximum number of iterations. Default 100.</param>
        /// <returns>Returns the root with the specified accuracy.</returns>
        /// <exception cref="NonConvergenceException"></exception>
        public static double[] FindRoot(Func<double[], double[]> f, double[] initialGuess, double accuracy = 1e-8, int maxIterations = 100)
        {
            double[] root;
            if (TryFindRoot(f, initialGuess, accuracy, maxIterations, out root))
            {
                return root;
            }

            throw new NonConvergenceException(Resources.RootFindingFailed);
        }

        /// <summary>Find a solution of the equation f(x)=0.</summary>
        /// <param name="f">The function to find roots from.</param>
        /// <param name="initialGuess">Initial guess of the root.</param>
        /// <param name="accuracy">Desired accuracy. The root will be refined until the accuracy or the maximum number of iterations is reached.</param>
        /// <param name="maxIterations">Maximum number of iterations. Usually 100.</param>
        /// <param name="root">The root that was found, if any. Undefined if the function returns false.</param>
        /// <returns>True if a root with the specified accuracy was found, else false.</returns>
        public static bool TryFindRoot(Func<double[], double[]> f, double[] initialGuess, double accuracy, int maxIterations, out double[] root)
        {
            var x = new DenseVector(initialGuess);

            double[] y0 = f(initialGuess);
            var y = new DenseVector(y0);
            double g = y.L2Norm();

            Matrix<double> B = CalculateApproximateJacobian(f, initialGuess, y0);

            for (int i = 0; i <= maxIterations; i++)
            {
                var dx = (DenseVector) (-B.LU().Solve(y));
                var xnew = x + dx;
                var ynew = new DenseVector(f(xnew.Values));
                double gnew = ynew.L2Norm();

                if (gnew > g)
                {
                    double g2 = g*g;
                    double scale = g2/(g2 + gnew*gnew);
                    if (scale == 0.0)
                    {
                        scale = 1.0e-4;
                    }

                    dx = scale*dx;
                    xnew = x + dx;
                    ynew = new DenseVector(f(xnew.Values));
                    gnew = ynew.L2Norm();
                }

                if (gnew < accuracy)
                {
                    root = xnew.Values;
                    return true;
                }

                // update Jacobian B
                DenseVector dF = ynew - y;
                Matrix<double> dB = (dF - B.Multiply(dx)).ToColumnMatrix() * dx.Multiply(1.0 / Math.Pow(dx.L2Norm(), 2)).ToRowMatrix();
                B = B + dB;

                x = xnew;
                y = ynew;
                g = gnew;
            }

            root = null;
            return false;
        }

        /// <summary>
        /// Helper method to calculate an approximation of the Jacobian.
        /// </summary>
        /// <param name="f">The function.</param>
        /// <param name="x0">The argument (initial guess).</param>
        /// <param name="y0">The result (of initial guess).</param>
        static Matrix<double> CalculateApproximateJacobian(Func<double[], double[]> f, double[] x0, double[] y0)
        {
            int dim = x0.Length;
            var B = new DenseMatrix(dim);

            var x = new double[dim];
            Array.Copy(x0, 0, x, 0, dim);

            for (int j = 0; j < dim; j++)
            {
                double h = Math.Abs(x0[j])*1.0e-4;
                if (h == 0.0)
                {
                    h = 1.0e-4;
                }

                var xj = x[j];
                x[j] = xj + h;
                double[] y = f(x);
                x[j] = xj;

                for (int i = 0; i < dim; i++)
                {
                    B.At(i, j, (y[i] - y0[i])/h);
                }
            }

            return B;
        }
    }
}