// <copyright file="RobustNewtonRaphson.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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// 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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// OTHER DEALINGS IN THE SOFTWARE.
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// </copyright>
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using System;
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namespace IStation.Numerics.RootFinding
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{
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/// <summary>
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/// Robust Newton-Raphson root-finding algorithm that falls back to bisection when overshooting or converging too slow, or to subdivision on lacking bracketing.
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/// </summary>
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/// <seealso cref="NewtonRaphson"/>
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public static class RobustNewtonRaphson
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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="lowerBound">The low value of the range where the root is supposed to be.</param>
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/// <param name="upperBound">The high value of the range where the root is supposed to be.</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="subdivision">How many parts an interval should be split into for zero crossing scanning in case of lacking bracketing. Default 20.</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 lowerBound, double upperBound, double accuracy = 1e-8, int maxIterations = 100, int subdivision = 20)
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{
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double root;
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if (TryFindRoot(f, df, lowerBound, upperBound, accuracy, maxIterations, subdivision, 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="df">The first derivative of the function to find roots from.</param>
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/// <param name="lowerBound">The low value of the range where the root is supposed to be.</param>
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/// <param name="upperBound">The high value of the range where the root is supposed to be.</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. Must be greater than 0.</param>
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/// <param name="maxIterations">Maximum number of iterations. Example: 100.</param>
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/// <param name="subdivision">How many parts an interval should be split into for zero crossing scanning in case of lacking bracketing. Example: 20.</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 lowerBound, double upperBound, double accuracy, int maxIterations, int subdivision, 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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double fmin = f(lowerBound);
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double fmax = f(upperBound);
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if (Math.Abs(fmin) < accuracy)
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{
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root = lowerBound;
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return true;
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}
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if (Math.Abs(fmax) < accuracy)
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{
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root = upperBound;
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return true;
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}
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root = 0.5*(lowerBound + upperBound);
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double fx = f(root);
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double lastStep = Math.Abs(upperBound - lowerBound);
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for (int i = 0; i < maxIterations; i++)
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{
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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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bool overshoot = root > upperBound, undershoot = root < lowerBound;
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if (overshoot || undershoot || Math.Abs(2*fx) > Math.Abs(lastStep*dfx))
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{
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// Newton-Raphson step failed
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// If same signs, try subdivision to scan for zero crossing intervals
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if (Math.Sign(fmin) == Math.Sign(fmax) && TryScanForCrossingsWithRoots(f, df, lowerBound, upperBound, accuracy, maxIterations - i - 1, subdivision, out root))
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{
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return true;
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}
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// Bisection
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root = 0.5*(upperBound + lowerBound);
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fx = f(root);
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lastStep = 0.5*Math.Abs(upperBound - lowerBound);
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if (Math.Sign(fx) == Math.Sign(fmin))
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{
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lowerBound = root;
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fmin = fx;
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if (overshoot)
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{
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root = upperBound;
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fx = fmax;
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}
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}
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else
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{
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upperBound = root;
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fmax = fx;
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if (undershoot)
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{
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root = lowerBound;
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fx = fmin;
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}
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}
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continue;
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}
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// Evaluation
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fx = f(root);
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lastStep = step;
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// Update bounds
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if (Math.Sign(fx) != Math.Sign(fmin))
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{
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upperBound = root;
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fmax = fx;
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}
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else if (Math.Sign(fx) != Math.Sign(fmax))
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{
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lowerBound = root;
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fmin = fx;
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}
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else if (Math.Sign(fmin) != Math.Sign(fmax) && 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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static bool TryScanForCrossingsWithRoots(Func<double, double> f, Func<double, double> df, double lowerBound, double upperBound, double accuracy, int maxIterations, int subdivision, out double root)
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{
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var zeroCrossings = ZeroCrossingBracketing.FindIntervalsWithin(f, lowerBound, upperBound, subdivision);
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foreach (Tuple<double, double> bounds in zeroCrossings)
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{
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if (TryFindRoot(f, df, bounds.Item1, bounds.Item2, accuracy, maxIterations, subdivision, out root))
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{
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return true;
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}
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}
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root = double.NaN;
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return false;
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}
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}
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}
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