// <copyright file="Brent.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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// 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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/// Algorithm by Brent, Van Wijngaarden, Dekker et al.
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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 Brent
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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="guessLowerBound">Guess for the low value of the range where the root is supposed to be. Will be expanded if needed.</param>
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/// <param name="guessUpperBound">Guess for the high value of the range where the root is supposed to be. Will be expanded if needed.</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="expandFactor">Factor at which to expand the bounds, if needed. Default 1.6.</param>
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/// <param name="maxExpandIteratons">Maximum number of expand iterations. Default 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 FindRootExpand(Func<double, double> f, double guessLowerBound, double guessUpperBound, double accuracy = 1e-8, int maxIterations = 100, double expandFactor = 1.6, int maxExpandIteratons = 100)
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{
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ZeroCrossingBracketing.ExpandReduce(f, ref guessLowerBound, ref guessUpperBound, expandFactor, maxExpandIteratons, maxExpandIteratons*10);
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return FindRoot(f, guessLowerBound, guessUpperBound, accuracy, maxIterations);
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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="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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/// <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 lowerBound, double upperBound, double accuracy = 1e-8, int maxIterations = 100)
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{
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double root;
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if (TryFindRoot(f, 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 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="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. 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 lowerBound, double upperBound, double accuracy, int maxIterations, 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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double froot = fmax;
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double d = 0.0, e = 0.0;
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root = upperBound;
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double xMid = double.NaN;
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// Root must be bracketed.
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if (Math.Sign(fmin) == Math.Sign(fmax))
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{
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return false;
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}
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for (int i = 0; i <= maxIterations; i++)
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{
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// adjust bounds
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if (Math.Sign(froot) == Math.Sign(fmax))
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{
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upperBound = lowerBound;
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fmax = fmin;
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e = d = root - lowerBound;
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}
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if (Math.Abs(fmax) < Math.Abs(froot))
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{
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lowerBound = root;
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root = upperBound;
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upperBound = lowerBound;
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fmin = froot;
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froot = fmax;
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fmax = fmin;
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}
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// convergence check
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double xAcc1 = Precision.PositiveDoublePrecision*Math.Abs(root) + 0.5*accuracy;
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double xMidOld = xMid;
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xMid = (upperBound - root)/2.0;
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if (Math.Abs(xMid) <= xAcc1 || froot.AlmostEqualNormRelative(0, froot, accuracy))
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{
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return true;
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}
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if (xMid == xMidOld)
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{
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// accuracy not sufficient, but cannot be improved further
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return false;
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}
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if (Math.Abs(e) >= xAcc1 && Math.Abs(fmin) > Math.Abs(froot))
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{
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// Attempt inverse quadratic interpolation
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double s = froot/fmin;
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double p;
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double q;
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if (lowerBound.AlmostEqualRelative(upperBound))
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{
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p = 2.0*xMid*s;
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q = 1.0 - s;
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}
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else
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{
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q = fmin/fmax;
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double r = froot/fmax;
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p = s*(2.0*xMid*q*(q - r) - (root - lowerBound)*(r - 1.0));
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q = (q - 1.0)*(r - 1.0)*(s - 1.0);
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}
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if (p > 0.0)
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{
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// Check whether in bounds
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q = -q;
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}
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p = Math.Abs(p);
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if (2.0*p < Math.Min(3.0*xMid*q - Math.Abs(xAcc1*q), Math.Abs(e*q)))
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{
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// Accept interpolation
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e = d;
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d = p/q;
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}
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else
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{
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// Interpolation failed, use bisection
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d = xMid;
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e = d;
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}
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}
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else
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{
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// Bounds decreasing too slowly, use bisection
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d = xMid;
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e = d;
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}
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lowerBound = root;
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fmin = froot;
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if (Math.Abs(d) > xAcc1)
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{
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root += d;
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}
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else
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{
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root += Sign(xAcc1, xMid);
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}
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froot = f(root);
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}
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return false;
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}
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/// <summary>Helper method useful for preventing rounding errors.</summary>
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/// <returns>a*sign(b)</returns>
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static double Sign(double a, double b)
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{
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return b >= 0 ? (a >= 0 ? a : -a) : (a >= 0 ? -a : a);
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}
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}
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}
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