forked from tsai/mathnet-numerics
Christoph Ruegg
10 years ago
committed by
GitHub
GitHub
15 changed files with 775 additions and 670 deletions
@ -0,0 +1,158 @@ |
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// <copyright file="BfgsTest.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) 2009-2016 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 MathNet.Numerics.LinearAlgebra; |
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using MathNet.Numerics.LinearAlgebra.Double; |
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using MathNet.Numerics.Optimization.LineSearch; |
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using System; |
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namespace MathNet.Numerics.Optimization |
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{ |
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public abstract class BfgsMinimizerBase |
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{ |
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public double GradientTolerance { get; set; } |
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public double ParameterTolerance { get; set; } |
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public double FunctionProgressTolerance { get; set; } |
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public int MaximumIterations { get; set; } |
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protected const double VerySmall = 1e-15; |
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/// <summary>
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/// Creates a base class for BFGS minimization
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/// </summary>
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/// <param name="gradientTolerance">The gradient tolerance</param>
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/// <param name="parameterTolerance">The parameter tolerance</param>
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/// <param name="functionProgressTolerance">The funciton progress tolerance</param>
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/// <param name="maximumIterations">The maximum number of iterations</param>
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public BfgsMinimizerBase(double gradientTolerance, double parameterTolerance, double functionProgressTolerance, int maximumIterations) |
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{ |
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GradientTolerance = gradientTolerance; |
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ParameterTolerance = parameterTolerance; |
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FunctionProgressTolerance = functionProgressTolerance; |
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MaximumIterations = maximumIterations; |
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} |
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protected MinimizationResult.ExitCondition ExitCriteriaSatisfied(IObjectiveFunction candidatePoint, IObjectiveFunction lastPoint, int iterations) |
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{ |
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Vector<double> relGrad = new DenseVector(candidatePoint.Point.Count); |
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double relativeGradient = 0.0; |
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double normalizer = Math.Max(Math.Abs(candidatePoint.Value), 1.0); |
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for (int ii = 0; ii < relGrad.Count; ++ii) |
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{ |
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double projectedGradient = GetProjectedGradient(candidatePoint, ii); |
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double tmp = projectedGradient * |
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Math.Max(Math.Abs(candidatePoint.Point[ii]), 1.0) / normalizer; |
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relativeGradient = Math.Max(relativeGradient, Math.Abs(tmp)); |
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} |
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if (relativeGradient < GradientTolerance) |
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{ |
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return MinimizationResult.ExitCondition.RelativeGradient; |
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} |
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if (lastPoint != null) |
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{ |
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double mostProgress = 0.0; |
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for (int ii = 0; ii < candidatePoint.Point.Count; ++ii) |
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{ |
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var tmp = Math.Abs(candidatePoint.Point[ii] - lastPoint.Point[ii]) / |
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Math.Max(Math.Abs(lastPoint.Point[ii]), 1.0); |
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mostProgress = Math.Max(mostProgress, tmp); |
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} |
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if (mostProgress < ParameterTolerance) |
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{ |
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return MinimizationResult.ExitCondition.LackOfProgress; |
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} |
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double functionChange = candidatePoint.Value - lastPoint.Value; |
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if (iterations > 500 && functionChange < 0 && Math.Abs(functionChange) < FunctionProgressTolerance) |
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return MinimizationResult.ExitCondition.LackOfProgress; |
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} |
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return MinimizationResult.ExitCondition.None; |
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} |
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protected virtual double GetProjectedGradient(IObjectiveFunction candidatePoint, int ii) |
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{ |
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return candidatePoint.Gradient[ii]; |
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} |
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protected void ValidateGradientAndObjective(IObjectiveFunction eval) |
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{ |
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foreach (var x in eval.Gradient) |
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{ |
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if (Double.IsNaN(x) || Double.IsInfinity(x)) |
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throw new EvaluationException("Non-finite gradient returned.", eval); |
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} |
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if (Double.IsNaN(eval.Value) || Double.IsInfinity(eval.Value)) |
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throw new EvaluationException("Non-finite objective function returned.", eval); |
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} |
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protected int DoBfgsUpdate(ref MinimizationResult.ExitCondition currentExitCondition, WolfeLineSearch lineSearcher, ref Matrix<double> inversePseudoHessian, ref Vector<double> lineSearchDirection, ref IObjectiveFunction previousPoint, ref LineSearchResult lineSearchResult, ref IObjectiveFunction candidate, ref Vector<double> step, ref int totalLineSearchSteps, ref int iterationsWithNontrivialLineSearch) |
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{ |
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int iterations; |
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for (iterations = 1; iterations < MaximumIterations; ++iterations) |
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{ |
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double startingStepSize; |
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double maxLineSearchStep; |
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lineSearchDirection = CalculateSearchDirection(ref inversePseudoHessian, out maxLineSearchStep, out startingStepSize, previousPoint, candidate, step); |
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try |
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{ |
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lineSearchResult = lineSearcher.FindConformingStep(candidate, lineSearchDirection, startingStepSize, maxLineSearchStep); |
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} |
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catch (Exception e) |
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{ |
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throw new InnerOptimizationException("Line search failed.", e); |
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} |
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iterationsWithNontrivialLineSearch += lineSearchResult.Iterations > 0 ? 1 : 0; |
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totalLineSearchSteps += lineSearchResult.Iterations; |
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step = lineSearchResult.FunctionInfoAtMinimum.Point - candidate.Point; |
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previousPoint = candidate; |
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candidate = lineSearchResult.FunctionInfoAtMinimum; |
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currentExitCondition = ExitCriteriaSatisfied(candidate, previousPoint, iterations); |
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if (currentExitCondition != MinimizationResult.ExitCondition.None) |
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break; |
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} |
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return iterations; |
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} |
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protected abstract Vector<double> CalculateSearchDirection(ref Matrix<double> inversePseudoHessian, |
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out double maxLineSearchStep, |
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out double startingStepSize, |
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IObjectiveFunction previousPoint, |
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IObjectiveFunction candidate, |
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Vector<double> step); |
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} |
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} |
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@ -1,111 +1,50 @@ |
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using System; |
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using MathNet.Numerics.LinearAlgebra; |
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// <copyright file="BfgsTest.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) 2009-2016 Math.NET
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//
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// Permission is hereby granted, free of charge, to any person
|
|||
// obtaining a copy of this software and associated documentation
|
|||
// files (the "Software"), to deal in the Software without
|
|||
// restriction, including without limitation the rights to use,
|
|||
// copy, modify, merge, publish, distribute, sublicense, and/or sell
|
|||
// copies of the Software, and to permit persons to whom the
|
|||
// Software is furnished to do so, subject to the following
|
|||
// conditions:
|
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//
|
|||
// The above copyright notice and this permission notice shall be
|
|||
// included in all copies or substantial portions of the Software.
|
|||
//
|
|||
// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
|
|||
// EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES
|
|||
// OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
|
|||
// NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT
|
|||
// HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY,
|
|||
// WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
|
|||
// 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.Optimization.LineSearch |
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{ |
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public class StrongWolfeLineSearch |
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public class StrongWolfeLineSearch : WolfeLineSearch |
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{ |
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public double C1 { get; set; } |
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public double C2 { get; set; } |
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public double ParameterTolerance { get; set; } |
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public int MaximumIterations { get; set; } |
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public StrongWolfeLineSearch(double c1, double c2, double parameterTolerance, int maxIterations = 10) |
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: base(c1, c2, parameterTolerance, maxIterations) |
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{ |
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C1 = c1; |
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C2 = c2; |
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ParameterTolerance = parameterTolerance; |
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MaximumIterations = maxIterations; |
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// Argument validation in base class
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} |
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// Implemented following http://www.math.washington.edu/~burke/crs/408/lectures/L9-weak-Wolfe.pdf
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public LineSearchResult FindConformingStep(IObjectiveFunctionEvaluation objective, Vector<double> searchDirection, double initialStep, double upperBound = Double.PositiveInfinity) |
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{ |
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double lowerBound = 0.0; |
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double step = initialStep; |
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double initialValue = objective.Value; |
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Vector<double> initialGradient = objective.Gradient; |
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double initialDd = searchDirection*initialGradient; |
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int ii; |
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IObjectiveFunction candidateEval = objective.CreateNew(); |
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MinimizationResult.ExitCondition reasonForExit = MinimizationResult.ExitCondition.None; |
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for (ii = 0; ii < this.MaximumIterations; ++ii) |
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{ |
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candidateEval.EvaluateAt(objective.Point + searchDirection*step); |
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double stepDd = searchDirection*candidateEval.Gradient; |
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if (candidateEval.Value > initialValue + C1*step*initialDd) |
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{ |
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upperBound = step; |
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step = 0.5*(lowerBound + upperBound); |
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} |
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else if (Math.Abs(stepDd) > C2*Math.Abs(initialDd)) |
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{ |
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lowerBound = step; |
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step = Double.IsPositiveInfinity(upperBound) ? 2*lowerBound : 0.5*(lowerBound + upperBound); |
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} |
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else |
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{ |
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reasonForExit = MinimizationResult.ExitCondition.StrongWolfeCriteria; |
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break; |
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} |
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if (!Double.IsInfinity(upperBound)) |
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{ |
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double maxRelChange = 0.0; |
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for (int jj = 0; jj < candidateEval.Point.Count; ++jj) |
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{ |
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double tmp = Math.Abs(searchDirection[jj]*(upperBound - lowerBound))/Math.Max(Math.Abs(candidateEval.Point[jj]), 1.0); |
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maxRelChange = Math.Max(maxRelChange, tmp); |
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} |
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if (maxRelChange < ParameterTolerance) |
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{ |
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reasonForExit = MinimizationResult.ExitCondition.LackOfProgress; |
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break; |
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} |
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} |
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} |
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if (ii == MaximumIterations && Double.IsPositiveInfinity(upperBound)) |
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throw new MaximumIterationsException(String.Format("Maximum iterations ({0}) reached. Function appears to be unbounded in search direction.", MaximumIterations)); |
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if (ii == MaximumIterations) |
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throw new MaximumIterationsException(String.Format("Maximum iterations ({0}) reached.", MaximumIterations)); |
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return new LineSearchResult(candidateEval, ii, step, reasonForExit); |
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} |
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bool Conforms(IObjectiveFunction startingPoint, Vector<double> searchDirection, double step, IObjectiveFunction endingPoint) |
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{ |
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bool sufficientDecrease = endingPoint.Value <= startingPoint.Value + C1*step*(startingPoint.Gradient*searchDirection); |
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bool notTooSteep = endingPoint.Gradient*searchDirection >= C2*startingPoint.Gradient*searchDirection; |
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return step > 0 && sufficientDecrease && notTooSteep; |
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} |
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void ValidateValue(IObjectiveFunction eval) |
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{ |
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if (!IsFinite(eval.Value)) |
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throw new EvaluationException(String.Format("Non-finite value returned by objective function: {0}", eval.Value), eval); |
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} |
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void ValidateGradient(IObjectiveFunction eval) |
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{ |
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foreach (double x in eval.Gradient) |
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{ |
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if (!IsFinite(x)) |
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{ |
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throw new EvaluationException(String.Format("Non-finite value returned by gradient: {0}", x), eval); |
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} |
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} |
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} |
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protected override MinimizationResult.ExitCondition WolfeExitCondition { get { return MinimizationResult.ExitCondition.StrongWolfeCriteria; } } |
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bool IsFinite(double x) |
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protected override bool WolfeCondition(double stepDd, double initialDd) |
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{ |
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return !(Double.IsNaN(x) || Double.IsInfinity(x)); |
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return Math.Abs(stepDd) > C2 * Math.Abs(initialDd); |
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} |
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} |
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} |
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@ -0,0 +1,158 @@ |
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// <copyright file="BfgsTest.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) 2009-2016 Math.NET
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//
|
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// Permission is hereby granted, free of charge, to any person
|
|||
// obtaining a copy of this software and associated documentation
|
|||
// files (the "Software"), to deal in the Software without
|
|||
// restriction, including without limitation the rights to use,
|
|||
// copy, modify, merge, publish, distribute, sublicense, and/or sell
|
|||
// copies of the Software, and to permit persons to whom the
|
|||
// Software is furnished to do so, subject to the following
|
|||
// conditions:
|
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//
|
|||
// The above copyright notice and this permission notice shall be
|
|||
// included in all copies or substantial portions of the Software.
|
|||
//
|
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// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
|
|||
// EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES
|
|||
// OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
|
|||
// NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT
|
|||
// HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY,
|
|||
// WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
|
|||
// 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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|
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using MathNet.Numerics.LinearAlgebra; |
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using System; |
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namespace MathNet.Numerics.Optimization.LineSearch |
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{ |
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public abstract class WolfeLineSearch |
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{ |
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protected double C1 { get; } |
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protected double C2 { get; } |
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protected double ParameterTolerance { get; } |
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protected int MaximumIterations { get; } |
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public WolfeLineSearch(double c1, double c2, double parameterTolerance, int maxIterations = 10) |
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{ |
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if (c1 <= 0) |
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throw new ArgumentException(string.Format("c1 {0} should be greater than 0", c1)); |
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if (c2 <= c1) |
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throw new ArgumentException(string.Format("c1 {0} should be less than c2 {1}", c1, c2)); |
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if (c2 >= 1) |
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throw new ArgumentException(string.Format("c2 {0} should be less than 1", c2)); |
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C1 = c1; |
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C2 = c2; |
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ParameterTolerance = parameterTolerance; |
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MaximumIterations = maxIterations; |
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} |
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/// <summary>Implemented following http://www.math.washington.edu/~burke/crs/408/lectures/L9-weak-Wolfe.pdf</summary>
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/// <param name="startingPoint">The objective function being optimized, evaluated at the starting point of the search</param>
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/// <param name="searchDirection">Search direction</param>
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/// <param name="initialStep">Initial size of the step in the search direction</param>
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public LineSearchResult FindConformingStep(IObjectiveFunctionEvaluation startingPoint, Vector<double> searchDirection, double initialStep) |
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{ |
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return FindConformingStep(startingPoint, searchDirection, initialStep, double.PositiveInfinity); |
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} |
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/// <summary></summary>
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/// <param name="startingPoint">The objective function being optimized, evaluated at the starting point of the search</param>
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/// <param name="searchDirection">Search direction</param>
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/// <param name="initialStep">Initial size of the step in the search direction</param>
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/// <param name="upperBound">The upper bound</param>
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public LineSearchResult FindConformingStep(IObjectiveFunctionEvaluation startingPoint, Vector<double> searchDirection, double initialStep, double upperBound) |
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{ |
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ValidateInputArguments(startingPoint, searchDirection, initialStep, upperBound); |
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double lowerBound = 0.0; |
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double step = initialStep; |
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double initialValue = startingPoint.Value; |
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Vector<double> initialGradient = startingPoint.Gradient; |
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double initialDd = searchDirection * initialGradient; |
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IObjectiveFunction objective = startingPoint.CreateNew(); |
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int ii; |
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MinimizationResult.ExitCondition reasonForExit = MinimizationResult.ExitCondition.None; |
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for (ii = 0; ii < MaximumIterations; ++ii) |
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{ |
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objective.EvaluateAt(startingPoint.Point + searchDirection * step); |
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ValidateGradient(objective); |
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ValidateValue(objective); |
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double stepDd = searchDirection * objective.Gradient; |
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if (objective.Value > initialValue + C1 * step * initialDd) |
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{ |
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upperBound = step; |
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step = 0.5 * (lowerBound + upperBound); |
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} |
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else if (WolfeCondition(stepDd,initialDd)) |
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{ |
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lowerBound = step; |
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step = double.IsPositiveInfinity(upperBound) ? 2 * lowerBound : 0.5 * (lowerBound + upperBound); |
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} |
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else |
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{ |
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reasonForExit = WolfeExitCondition; |
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break; |
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} |
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if (!double.IsInfinity(upperBound)) |
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{ |
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double maxRelChange = 0.0; |
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for (int jj = 0; jj < objective.Point.Count; ++jj) |
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{ |
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double tmp = Math.Abs(searchDirection[jj] * (upperBound - lowerBound)) / Math.Max(Math.Abs(objective.Point[jj]), 1.0); |
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maxRelChange = Math.Max(maxRelChange, tmp); |
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} |
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if (maxRelChange < ParameterTolerance) |
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{ |
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reasonForExit = MinimizationResult.ExitCondition.LackOfProgress; |
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break; |
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} |
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} |
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} |
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if (ii == MaximumIterations && Double.IsPositiveInfinity(upperBound)) |
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{ |
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throw new MaximumIterationsException(String.Format("Maximum iterations ({0}) reached. Function appears to be unbounded in search direction.", MaximumIterations)); |
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} |
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|
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if (ii == MaximumIterations) |
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{ |
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throw new MaximumIterationsException(String.Format("Maximum iterations ({0}) reached.", MaximumIterations)); |
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} |
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return new LineSearchResult(objective, ii, step, reasonForExit); |
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} |
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protected abstract MinimizationResult.ExitCondition WolfeExitCondition { get; } |
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protected abstract bool WolfeCondition(double stepDd, double initialDd); |
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protected virtual void ValidateGradient(IObjectiveFunction objective) |
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{ |
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} |
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protected virtual void ValidateValue(IObjectiveFunction objective) |
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{ |
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} |
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protected virtual void ValidateInputArguments(IObjectiveFunctionEvaluation startingPoint, Vector<double> searchDirection, double initialStep, double upperBound) |
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{ |
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|
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} |
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} |
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} |
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@ -1,111 +0,0 @@ |
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// <copyright file="WolfeRule.cs" company="Math.NET">
|
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// Math.NET Numerics, part of the Math.NET Project
|
|||
// http://numerics.mathdotnet.com
|
|||
// http://github.com/mathnet/mathnet-numerics
|
|||
// http://mathnetnumerics.codeplex.com
|
|||
//
|
|||
// Copyright (c) 2009-2015 Math.NET
|
|||
//
|
|||
// Permission is hereby granted, free of charge, to any person
|
|||
// obtaining a copy of this software and associated documentation
|
|||
// files (the "Software"), to deal in the Software without
|
|||
// restriction, including without limitation the rights to use,
|
|||
// copy, modify, merge, publish, distribute, sublicense, and/or sell
|
|||
// copies of the Software, and to permit persons to whom the
|
|||
// Software is furnished to do so, subject to the following
|
|||
// conditions:
|
|||
//
|
|||
// The above copyright notice and this permission notice shall be
|
|||
// included in all copies or substantial portions of the Software.
|
|||
//
|
|||
// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
|
|||
// EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES
|
|||
// OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
|
|||
// NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT
|
|||
// HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY,
|
|||
// WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
|
|||
// FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
|
|||
// OTHER DEALINGS IN THE SOFTWARE.
|
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// </copyright>
|
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|
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using System; |
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using MathNet.Numerics.LinearAlgebra; |
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|
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namespace MathNet.Numerics.Optimization.LineSearch |
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{ |
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/// <summary>
|
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/// Performs an inexact line search. This is used as a part of quasi-Newton optimization methods to figure
|
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/// out how far to move along a certain gradient.
|
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/// See http://en.wikipedia.org/wiki/Wolfe_conditions
|
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/// Inspired by implementation: https://github.com/PatWie/CppNumericalSolvers/blob/master/src/linesearch/WolfeRule.h
|
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/// </summary>
|
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internal static class WolfeRule |
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{ |
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/// <summary>
|
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/// Searches along a line to satisfy the Wolfe conditions (inexact search for minimum)
|
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/// </summary>
|
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/// <param name="x0">Starting point of search</param>
|
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/// <param name="z">Search direction</param>
|
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/// <param name="functionValue">Evaluates the function being minimized</param>
|
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/// <param name="functionGradient">Evaluates the gradient of the function</param>
|
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/// <param name="alphaInit">Initial value for the coefficient of z (distance to travel in z direction)</param>
|
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/// <returns></returns>
|
|||
public static double LineSearch( |
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Vector<double> x0, |
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Vector<double> z, |
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Func<Vector<double>, double> functionValue, |
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Func<Vector<double>, Vector<double>> functionGradient, |
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float alphaInit = 1) |
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{ |
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Vector<double> x = x0; |
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|
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// evaluate phi(0)
|
|||
double phi0 = functionValue(x0); |
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|
|||
// evaluate phi'(0)
|
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Vector<double> grad = functionGradient(x); |
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double phi0_dash = z * grad; |
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|
|||
double alpha = alphaInit; |
|||
|
|||
bool decrease_direction = true; |
|||
|
|||
// 200 guesses max
|
|||
for (int iter = 0; iter < 200; ++iter) { |
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|
|||
// new guess for phi(alpha)
|
|||
Vector<double> x_candidate = x + alpha * z; |
|||
double phi = functionValue(x_candidate); |
|||
|
|||
// decrease condition invalid --> shrink interval
|
|||
if (phi > phi0 + 0.0001 * alpha * phi0_dash) |
|||
{ |
|||
alpha *= 0.5; |
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decrease_direction = false; |
|||
} |
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else |
|||
{ |
|||
// valid decrease --> test strong wolfe condition
|
|||
Vector<double> grad2 = functionGradient(x_candidate); |
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double phi_dash = z * grad2; |
|||
|
|||
// curvature condition invalid ?
|
|||
if ((phi_dash < 0.9 * phi0_dash) || !decrease_direction) { |
|||
// increase interval
|
|||
alpha *= 4.0; |
|||
} |
|||
else { |
|||
// both condition are valid --> we are happy
|
|||
x = x_candidate; |
|||
grad = grad2; |
|||
phi0 = phi; |
|||
return alpha; |
|||
} |
|||
} |
|||
} |
|||
|
|||
|
|||
return alpha; |
|||
} |
|||
} |
|||
} |
|||
Loading…
Reference in new issue