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@ -4,7 +4,7 @@ |
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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) 2009-2010 Math.NET
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// Copyright (c) 2009-2013 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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@ -28,8 +28,8 @@ |
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// OTHER DEALINGS IN THE SOFTWARE.
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// </copyright>
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using MathNet.Numerics.Properties; |
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using System; |
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using MathNet.Numerics.Properties; |
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namespace MathNet.Numerics.LinearAlgebra.Single.Factorization |
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{ |
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@ -54,7 +54,7 @@ namespace MathNet.Numerics.LinearAlgebra.Single.Factorization |
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/// conditioned, or even singular, so the validity of the equation
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/// A = V*D*Inverse(V) depends upon V.Condition().
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/// </remarks>
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public class UserEvd : Evd |
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public sealed class UserEvd : Evd |
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{ |
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/// <summary>
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/// Initializes a new instance of the <see cref="UserEvd"/> class. This object will compute the
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@ -63,13 +63,8 @@ namespace MathNet.Numerics.LinearAlgebra.Single.Factorization |
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/// <param name="matrix">The matrix to factor.</param>
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/// <exception cref="ArgumentNullException">If <paramref name="matrix"/> is <c>null</c>.</exception>
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/// <exception cref="ArgumentException">If EVD algorithm failed to converge with matrix <paramref name="matrix"/>.</exception>
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public UserEvd(Matrix<float> matrix) |
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public static UserEvd Create(Matrix<float> matrix) |
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{ |
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if (matrix == null) |
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{ |
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throw new ArgumentNullException("matrix"); |
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} |
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if (matrix.RowCount != matrix.ColumnCount) |
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{ |
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throw new ArgumentException(Resources.ArgumentMatrixSquare); |
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@ -78,57 +73,64 @@ namespace MathNet.Numerics.LinearAlgebra.Single.Factorization |
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var order = matrix.RowCount; |
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// Initialize matricies for eigenvalues and eigenvectors
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EigenVectors = matrix.CreateMatrix(order, order); |
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D = matrix.CreateMatrix(order, order); |
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EigenValues = new LinearAlgebra.Complex.DenseVector(order); |
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IsSymmetric = true; |
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var eigenVectors = matrix.CreateMatrix(order, order); |
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var blockDiagonal = matrix.CreateMatrix(order, order); |
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var eigenValues = new LinearAlgebra.Complex.DenseVector(order); |
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var isSymmetric = true; |
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for (var i = 0; IsSymmetric && i < order; i++) |
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for (var i = 0; isSymmetric && i < order; i++) |
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{ |
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for (var j = 0; IsSymmetric && j < order; j++) |
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for (var j = 0; isSymmetric && j < order; j++) |
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{ |
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IsSymmetric &= matrix.At(i, j) == matrix.At(j, i); |
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isSymmetric &= matrix.At(i, j) == matrix.At(j, i); |
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} |
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} |
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var d = new float[order]; |
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var e = new float[order]; |
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if (IsSymmetric) |
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if (isSymmetric) |
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{ |
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matrix.CopyTo(EigenVectors); |
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d = EigenVectors.Row(order - 1).ToArray(); |
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matrix.CopyTo(eigenVectors); |
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d = eigenVectors.Row(order - 1).ToArray(); |
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SymmetricTridiagonalize(d, e, order); |
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SymmetricDiagonalize(d, e, order); |
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SymmetricTridiagonalize(eigenVectors, d, e, order); |
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SymmetricDiagonalize(eigenVectors, d, e, order); |
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} |
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else |
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{ |
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var matrixH = matrix.ToArray(); |
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NonsymmetricReduceToHessenberg(matrixH, order); |
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NonsymmetricReduceHessenberToRealSchur(matrixH, d, e, order); |
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NonsymmetricReduceToHessenberg(eigenVectors, matrixH, order); |
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NonsymmetricReduceHessenberToRealSchur(eigenVectors, matrixH, d, e, order); |
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} |
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for (var i = 0; i < order; i++) |
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{ |
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D.At(i, i, d[i]); |
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blockDiagonal.At(i, i, d[i]); |
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if (e[i] > 0) |
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{ |
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D.At(i, i + 1, e[i]); |
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blockDiagonal.At(i, i + 1, e[i]); |
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} |
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else if (e[i] < 0) |
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{ |
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D.At(i, i - 1, e[i]); |
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blockDiagonal.At(i, i - 1, e[i]); |
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} |
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} |
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for (var i = 0; i < order; i++) |
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{ |
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EigenValues[i] = new Complex(d[i], e[i]); |
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eigenValues[i] = new Complex(d[i], e[i]); |
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} |
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return new UserEvd(eigenVectors, eigenValues, blockDiagonal, isSymmetric); |
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} |
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UserEvd(Matrix<float> eigenVectors, Vector<Complex> eigenValues, Matrix<float> blockDiagonal, bool isSymmetric) |
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: base(eigenVectors, eigenValues, blockDiagonal, isSymmetric) |
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{ |
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} |
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/// <summary>
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@ -141,7 +143,7 @@ namespace MathNet.Numerics.LinearAlgebra.Single.Factorization |
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/// Bowdler, Martin, Reinsch, and Wilkinson, Handbook for
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/// Auto. Comp., Vol.ii-Linear Algebra, and the corresponding
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/// Fortran subroutine in EISPACK.</remarks>
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private void SymmetricTridiagonalize(float[] d, float[] e, int order) |
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static void SymmetricTridiagonalize(Matrix<float> eigenVectors, float[] d, float[] e, int order) |
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{ |
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// Householder reduction to tridiagonal form.
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for (var i = order - 1; i > 0; i--) |
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@ -160,9 +162,9 @@ namespace MathNet.Numerics.LinearAlgebra.Single.Factorization |
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e[i] = d[i - 1]; |
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for (var j = 0; j < i; j++) |
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{ |
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d[j] = EigenVectors.At(i - 1, j); |
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EigenVectors.At(i, j, 0.0f); |
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EigenVectors.At(j, i, 0.0f); |
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d[j] = eigenVectors.At(i - 1, j); |
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eigenVectors.At(i, j, 0.0f); |
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eigenVectors.At(j, i, 0.0f); |
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} |
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} |
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else |
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@ -171,18 +173,18 @@ namespace MathNet.Numerics.LinearAlgebra.Single.Factorization |
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for (var k = 0; k < i; k++) |
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{ |
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d[k] /= scale; |
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h += d[k] * d[k]; |
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h += d[k]*d[k]; |
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} |
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var f = d[i - 1]; |
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var g = (float)Math.Sqrt(h); |
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var g = (float) Math.Sqrt(h); |
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if (f > 0) |
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{ |
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g = -g; |
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} |
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e[i] = scale * g; |
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h = h - (f * g); |
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e[i] = scale*g; |
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h = h - (f*g); |
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d[i - 1] = f - g; |
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for (var j = 0; j < i; j++) |
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@ -194,13 +196,13 @@ namespace MathNet.Numerics.LinearAlgebra.Single.Factorization |
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for (var j = 0; j < i; j++) |
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{ |
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f = d[j]; |
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EigenVectors.At(j, i, f); |
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g = e[j] + (EigenVectors.At(j, j) * f); |
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eigenVectors.At(j, i, f); |
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g = e[j] + (eigenVectors.At(j, j)*f); |
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for (var k = j + 1; k <= i - 1; k++) |
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{ |
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g += EigenVectors.At(k, j) * d[k]; |
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e[k] += EigenVectors.At(k, j) * f; |
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g += eigenVectors.At(k, j)*d[k]; |
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e[k] += eigenVectors.At(k, j)*f; |
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} |
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e[j] = g; |
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@ -211,14 +213,14 @@ namespace MathNet.Numerics.LinearAlgebra.Single.Factorization |
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for (var j = 0; j < i; j++) |
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{ |
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e[j] /= h; |
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f += e[j] * d[j]; |
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f += e[j]*d[j]; |
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} |
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var hh = f / (h + h); |
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var hh = f/(h + h); |
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for (var j = 0; j < i; j++) |
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{ |
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e[j] -= hh * d[j]; |
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e[j] -= hh*d[j]; |
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} |
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for (var j = 0; j < i; j++) |
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@ -228,11 +230,11 @@ namespace MathNet.Numerics.LinearAlgebra.Single.Factorization |
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for (var k = j; k <= i - 1; k++) |
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{ |
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EigenVectors.At(k, j, EigenVectors.At(k, j) - (f * e[k]) - (g * d[k])); |
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eigenVectors.At(k, j, eigenVectors.At(k, j) - (f*e[k]) - (g*d[k])); |
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} |
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d[j] = EigenVectors.At(i - 1, j); |
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EigenVectors.At(i, j, 0.0f); |
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d[j] = eigenVectors.At(i - 1, j); |
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eigenVectors.At(i, j, 0.0f); |
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} |
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} |
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@ -242,14 +244,14 @@ namespace MathNet.Numerics.LinearAlgebra.Single.Factorization |
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// Accumulate transformations.
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for (var i = 0; i < order - 1; i++) |
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{ |
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EigenVectors.At(order - 1, i, EigenVectors.At(i, i)); |
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EigenVectors.At(i, i, 1.0f); |
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eigenVectors.At(order - 1, i, eigenVectors.At(i, i)); |
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eigenVectors.At(i, i, 1.0f); |
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var h = d[i + 1]; |
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if (h != 0.0f) |
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{ |
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for (var k = 0; k <= i; k++) |
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{ |
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d[k] = EigenVectors.At(k, i + 1) / h; |
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d[k] = eigenVectors.At(k, i + 1)/h; |
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} |
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for (var j = 0; j <= i; j++) |
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@ -257,29 +259,29 @@ namespace MathNet.Numerics.LinearAlgebra.Single.Factorization |
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var g = 0.0f; |
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for (var k = 0; k <= i; k++) |
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{ |
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g += EigenVectors.At(k, i + 1) * EigenVectors.At(k, j); |
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g += eigenVectors.At(k, i + 1)*eigenVectors.At(k, j); |
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} |
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for (var k = 0; k <= i; k++) |
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{ |
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EigenVectors.At(k, j, EigenVectors.At(k, j) - g * d[k]); |
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eigenVectors.At(k, j, eigenVectors.At(k, j) - g*d[k]); |
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} |
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} |
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} |
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for (var k = 0; k <= i; k++) |
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{ |
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EigenVectors.At(k, i + 1, 0.0f); |
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eigenVectors.At(k, i + 1, 0.0f); |
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} |
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} |
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for (var j = 0; j < order; j++) |
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{ |
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d[j] = EigenVectors.At(order - 1, j); |
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EigenVectors.At(order - 1, j, 0.0f); |
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d[j] = eigenVectors.At(order - 1, j); |
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eigenVectors.At(order - 1, j, 0.0f); |
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} |
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EigenVectors.At(order - 1, order - 1, 1.0f); |
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eigenVectors.At(order - 1, order - 1, 1.0f); |
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e[0] = 0.0f; |
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} |
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@ -294,7 +296,7 @@ namespace MathNet.Numerics.LinearAlgebra.Single.Factorization |
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/// Auto. Comp., Vol.ii-Linear Algebra, and the corresponding
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/// Fortran subroutine in EISPACK.</remarks>
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/// <exception cref="NonConvergenceException"></exception>
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private void SymmetricDiagonalize(float[] d, float[] e, int order) |
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static void SymmetricDiagonalize(Matrix<float> eigenVectors, float[] d, float[] e, int order) |
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{ |
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const int maxiter = 1000; |
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@ -315,7 +317,7 @@ namespace MathNet.Numerics.LinearAlgebra.Single.Factorization |
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var m = l; |
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while (m < order) |
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{ |
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if (Math.Abs(e[m]) <= eps * tst1) |
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if (Math.Abs(e[m]) <= eps*tst1) |
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{ |
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break; |
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} |
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@ -334,15 +336,15 @@ namespace MathNet.Numerics.LinearAlgebra.Single.Factorization |
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// Compute implicit shift
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var g = d[l]; |
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var p = (d[l + 1] - g) / (2.0f * e[l]); |
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var p = (d[l + 1] - g)/(2.0f*e[l]); |
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var r = SpecialFunctions.Hypotenuse(p, 1.0f); |
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if (p < 0) |
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{ |
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r = -r; |
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} |
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d[l] = e[l] / (p + r); |
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d[l + 1] = e[l] * (p + r); |
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d[l] = e[l]/(p + r); |
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d[l + 1] = e[l]*(p + r); |
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var dl1 = d[l + 1]; |
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var h = g - d[l]; |
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@ -366,27 +368,27 @@ namespace MathNet.Numerics.LinearAlgebra.Single.Factorization |
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c3 = c2; |
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c2 = c; |
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s2 = s; |
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g = c * e[i]; |
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h = c * p; |
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g = c*e[i]; |
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h = c*p; |
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r = SpecialFunctions.Hypotenuse(p, e[i]); |
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e[i + 1] = s * r; |
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s = e[i] / r; |
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c = p / r; |
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p = (c * d[i]) - (s * g); |
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d[i + 1] = h + (s * ((c * g) + (s * d[i]))); |
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e[i + 1] = s*r; |
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s = e[i]/r; |
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c = p/r; |
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p = (c*d[i]) - (s*g); |
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d[i + 1] = h + (s*((c*g) + (s*d[i]))); |
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// Accumulate transformation.
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for (var k = 0; k < order; k++) |
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{ |
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h = EigenVectors.At(k, i + 1); |
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EigenVectors.At(k, i + 1, (s * EigenVectors.At(k, i)) + (c * h)); |
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EigenVectors.At(k, i, (c * EigenVectors.At(k, i)) - (s * h)); |
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h = eigenVectors.At(k, i + 1); |
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eigenVectors.At(k, i + 1, (s*eigenVectors.At(k, i)) + (c*h)); |
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eigenVectors.At(k, i, (c*eigenVectors.At(k, i)) - (s*h)); |
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} |
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} |
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p = (-s) * s2 * c3 * el1 * e[l] / dl1; |
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e[l] = s * p; |
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d[l] = c * p; |
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p = (-s)*s2*c3*el1*e[l]/dl1; |
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e[l] = s*p; |
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d[l] = c*p; |
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// Check for convergence. If too many iterations have been performed,
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// throw exception that Convergence Failed
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@ -394,8 +396,7 @@ namespace MathNet.Numerics.LinearAlgebra.Single.Factorization |
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{ |
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throw new NonConvergenceException(); |
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} |
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} |
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while (Math.Abs(e[l]) > eps * tst1); |
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} while (Math.Abs(e[l]) > eps*tst1); |
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} |
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|
d[l] = d[l] + f; |
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|
@ -422,9 +423,9 @@ namespace MathNet.Numerics.LinearAlgebra.Single.Factorization |
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|
d[i] = p; |
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|
for (var j = 0; j < order; j++) |
|
|
|
{ |
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|
|
p = EigenVectors.At(j, i); |
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|
|
EigenVectors.At(j, i, EigenVectors.At(j, k)); |
|
|
|
EigenVectors.At(j, k, p); |
|
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|
p = eigenVectors.At(j, i); |
|
|
|
eigenVectors.At(j, i, eigenVectors.At(j, k)); |
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|
eigenVectors.At(j, k, p); |
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|
} |
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|
} |
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|
} |
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|
@ -439,7 +440,7 @@ namespace MathNet.Numerics.LinearAlgebra.Single.Factorization |
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|
/// by Martin and Wilkinson, Handbook for Auto. Comp.,
|
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|
/// Vol.ii-Linear Algebra, and the corresponding
|
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|
|
/// Fortran subroutines in EISPACK.</remarks>
|
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|
private void NonsymmetricReduceToHessenberg(float[,] matrixH, int order) |
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|
static void NonsymmetricReduceToHessenberg(Matrix<float> eigenVectors, float[,] matrixH, int order) |
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|
{ |
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|
var ort = new float[order]; |
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|
@ -458,17 +459,17 @@ namespace MathNet.Numerics.LinearAlgebra.Single.Factorization |
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var h = 0.0f; |
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for (var i = order - 1; i >= m; i--) |
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{ |
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|
ort[i] = matrixH[i, m - 1] / scale; |
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|
h += ort[i] * ort[i]; |
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|
ort[i] = matrixH[i, m - 1]/scale; |
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|
h += ort[i]*ort[i]; |
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|
} |
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|
var g = (float)Math.Sqrt(h); |
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|
var g = (float) Math.Sqrt(h); |
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|
if (ort[m] > 0) |
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|
{ |
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|
g = -g; |
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|
} |
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|
h = h - (ort[m] * g); |
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|
h = h - (ort[m]*g); |
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|
ort[m] = ort[m] - g; |
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|
// Apply Householder similarity transformation
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@ -478,13 +479,13 @@ namespace MathNet.Numerics.LinearAlgebra.Single.Factorization |
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|
var f = 0.0f; |
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|
for (var i = order - 1; i >= m; i--) |
|
|
|
{ |
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|
|
f += ort[i] * matrixH[i, j]; |
|
|
|
f += ort[i]*matrixH[i, j]; |
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|
|
} |
|
|
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|
|
f = f / h; |
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|
|
f = f/h; |
|
|
|
for (var i = m; i < order; i++) |
|
|
|
{ |
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|
|
matrixH[i, j] -= f * ort[i]; |
|
|
|
matrixH[i, j] -= f*ort[i]; |
|
|
|
} |
|
|
|
} |
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|
@ -493,18 +494,18 @@ namespace MathNet.Numerics.LinearAlgebra.Single.Factorization |
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|
|
var f = 0.0f; |
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|
|
for (var j = order - 1; j >= m; j--) |
|
|
|
{ |
|
|
|
f += ort[j] * matrixH[i, j]; |
|
|
|
f += ort[j]*matrixH[i, j]; |
|
|
|
} |
|
|
|
|
|
|
|
f = f / h; |
|
|
|
f = f/h; |
|
|
|
for (var j = m; j < order; j++) |
|
|
|
{ |
|
|
|
matrixH[i, j] -= f * ort[j]; |
|
|
|
matrixH[i, j] -= f*ort[j]; |
|
|
|
} |
|
|
|
} |
|
|
|
|
|
|
|
ort[m] = scale * ort[m]; |
|
|
|
matrixH[m, m - 1] = scale * g; |
|
|
|
ort[m] = scale*ort[m]; |
|
|
|
matrixH[m, m - 1] = scale*g; |
|
|
|
} |
|
|
|
} |
|
|
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|
|
@ -513,7 +514,7 @@ namespace MathNet.Numerics.LinearAlgebra.Single.Factorization |
|
|
|
{ |
|
|
|
for (var j = 0; j < order; j++) |
|
|
|
{ |
|
|
|
EigenVectors.At(i, j, i == j ? 1.0f : 0.0f); |
|
|
|
eigenVectors.At(i, j, i == j ? 1.0f : 0.0f); |
|
|
|
} |
|
|
|
} |
|
|
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|
|
@ -531,14 +532,14 @@ namespace MathNet.Numerics.LinearAlgebra.Single.Factorization |
|
|
|
var g = 0.0f; |
|
|
|
for (var i = m; i < order; i++) |
|
|
|
{ |
|
|
|
g += ort[i] * EigenVectors.At(i, j); |
|
|
|
g += ort[i]*eigenVectors.At(i, j); |
|
|
|
} |
|
|
|
|
|
|
|
// Double division avoids possible underflow
|
|
|
|
g = (g / ort[m]) / matrixH[m, m - 1]; |
|
|
|
g = (g/ort[m])/matrixH[m, m - 1]; |
|
|
|
for (var i = m; i < order; i++) |
|
|
|
{ |
|
|
|
EigenVectors.At(i, j, EigenVectors.At(i, j) + g * ort[i]); |
|
|
|
eigenVectors.At(i, j, eigenVectors.At(i, j) + g*ort[i]); |
|
|
|
} |
|
|
|
} |
|
|
|
} |
|
|
|
@ -556,11 +557,11 @@ namespace MathNet.Numerics.LinearAlgebra.Single.Factorization |
|
|
|
/// by Martin and Wilkinson, Handbook for Auto. Comp.,
|
|
|
|
/// Vol.ii-Linear Algebra, and the corresponding
|
|
|
|
/// Fortran subroutine in EISPACK.</remarks>
|
|
|
|
private void NonsymmetricReduceHessenberToRealSchur(float[,] matrixH, float[] d, float[] e, int order) |
|
|
|
static void NonsymmetricReduceHessenberToRealSchur(Matrix<float> eigenVectors, float[,] matrixH, float[] d, float[] e, int order) |
|
|
|
{ |
|
|
|
// Initialize
|
|
|
|
var n = order - 1; |
|
|
|
var eps = (float)Precision.SingleMachinePrecision; |
|
|
|
var eps = (float) Precision.SingleMachinePrecision; |
|
|
|
var exshift = 0.0f; |
|
|
|
float p = 0, q = 0, r = 0, s = 0, z = 0, w, x, y; |
|
|
|
|
|
|
|
@ -589,7 +590,7 @@ namespace MathNet.Numerics.LinearAlgebra.Single.Factorization |
|
|
|
s = norm; |
|
|
|
} |
|
|
|
|
|
|
|
if (Math.Abs(matrixH[l, l - 1]) < eps * s) |
|
|
|
if (Math.Abs(matrixH[l, l - 1]) < eps*s) |
|
|
|
{ |
|
|
|
break; |
|
|
|
} |
|
|
|
@ -611,10 +612,10 @@ namespace MathNet.Numerics.LinearAlgebra.Single.Factorization |
|
|
|
} |
|
|
|
else if (l == n - 1) |
|
|
|
{ |
|
|
|
w = matrixH[n, n - 1] * matrixH[n - 1, n]; |
|
|
|
p = (matrixH[n - 1, n - 1] - matrixH[n, n]) / 2.0f; |
|
|
|
q = (p * p) + w; |
|
|
|
z = (float)Math.Sqrt(Math.Abs(q)); |
|
|
|
w = matrixH[n, n - 1]*matrixH[n - 1, n]; |
|
|
|
p = (matrixH[n - 1, n - 1] - matrixH[n, n])/2.0f; |
|
|
|
q = (p*p) + w; |
|
|
|
z = (float) Math.Sqrt(Math.Abs(q)); |
|
|
|
matrixH[n, n] = matrixH[n, n] + exshift; |
|
|
|
matrixH[n - 1, n - 1] = matrixH[n - 1, n - 1] + exshift; |
|
|
|
x = matrixH[n, n]; |
|
|
|
@ -636,41 +637,41 @@ namespace MathNet.Numerics.LinearAlgebra.Single.Factorization |
|
|
|
d[n] = d[n - 1]; |
|
|
|
if (z != 0.0f) |
|
|
|
{ |
|
|
|
d[n] = x - (w / z); |
|
|
|
d[n] = x - (w/z); |
|
|
|
} |
|
|
|
|
|
|
|
e[n - 1] = 0.0f; |
|
|
|
e[n] = 0.0f; |
|
|
|
x = matrixH[n, n - 1]; |
|
|
|
s = Math.Abs(x) + Math.Abs(z); |
|
|
|
p = x / s; |
|
|
|
q = z / s; |
|
|
|
r = (float)Math.Sqrt((p * p) + (q * q)); |
|
|
|
p = p / r; |
|
|
|
q = q / r; |
|
|
|
p = x/s; |
|
|
|
q = z/s; |
|
|
|
r = (float) Math.Sqrt((p*p) + (q*q)); |
|
|
|
p = p/r; |
|
|
|
q = q/r; |
|
|
|
|
|
|
|
// Row modification
|
|
|
|
for (var j = n - 1; j < order; j++) |
|
|
|
{ |
|
|
|
z = matrixH[n - 1, j]; |
|
|
|
matrixH[n - 1, j] = (q * z) + (p * matrixH[n, j]); |
|
|
|
matrixH[n, j] = (q * matrixH[n, j]) - (p * z); |
|
|
|
matrixH[n - 1, j] = (q*z) + (p*matrixH[n, j]); |
|
|
|
matrixH[n, j] = (q*matrixH[n, j]) - (p*z); |
|
|
|
} |
|
|
|
|
|
|
|
// Column modification
|
|
|
|
for (var i = 0; i <= n; i++) |
|
|
|
{ |
|
|
|
z = matrixH[i, n - 1]; |
|
|
|
matrixH[i, n - 1] = (q * z) + (p * matrixH[i, n]); |
|
|
|
matrixH[i, n] = (q * matrixH[i, n]) - (p * z); |
|
|
|
matrixH[i, n - 1] = (q*z) + (p*matrixH[i, n]); |
|
|
|
matrixH[i, n] = (q*matrixH[i, n]) - (p*z); |
|
|
|
} |
|
|
|
|
|
|
|
// Accumulate transformations
|
|
|
|
for (var i = 0; i < order; i++) |
|
|
|
{ |
|
|
|
z = EigenVectors.At(i, n - 1); |
|
|
|
EigenVectors.At(i, n - 1, (q * z) + (p * EigenVectors.At(i, n))); |
|
|
|
EigenVectors.At(i, n, (q * EigenVectors.At(i, n)) - (p * z)); |
|
|
|
z = eigenVectors.At(i, n - 1); |
|
|
|
eigenVectors.At(i, n - 1, (q*z) + (p*eigenVectors.At(i, n))); |
|
|
|
eigenVectors.At(i, n, (q*eigenVectors.At(i, n)) - (p*z)); |
|
|
|
} |
|
|
|
|
|
|
|
// Complex pair
|
|
|
|
@ -697,7 +698,7 @@ namespace MathNet.Numerics.LinearAlgebra.Single.Factorization |
|
|
|
if (l < n) |
|
|
|
{ |
|
|
|
y = matrixH[n - 1, n - 1]; |
|
|
|
w = matrixH[n, n - 1] * matrixH[n - 1, n]; |
|
|
|
w = matrixH[n, n - 1]*matrixH[n - 1, n]; |
|
|
|
} |
|
|
|
|
|
|
|
// Wilkinson's original ad hoc shift
|
|
|
|
@ -710,24 +711,24 @@ namespace MathNet.Numerics.LinearAlgebra.Single.Factorization |
|
|
|
} |
|
|
|
|
|
|
|
s = Math.Abs(matrixH[n, n - 1]) + Math.Abs(matrixH[n - 1, n - 2]); |
|
|
|
x = y = 0.75f * s; |
|
|
|
w = (-0.4375f) * s * s; |
|
|
|
x = y = 0.75f*s; |
|
|
|
w = (-0.4375f)*s*s; |
|
|
|
} |
|
|
|
|
|
|
|
// MATLAB's new ad hoc shift
|
|
|
|
if (iter == 30) |
|
|
|
{ |
|
|
|
s = (y - x) / 2.0f; |
|
|
|
s = (s * s) + w; |
|
|
|
s = (y - x)/2.0f; |
|
|
|
s = (s*s) + w; |
|
|
|
if (s > 0) |
|
|
|
{ |
|
|
|
s = (float)Math.Sqrt(s); |
|
|
|
s = (float) Math.Sqrt(s); |
|
|
|
if (y < x) |
|
|
|
{ |
|
|
|
s = -s; |
|
|
|
} |
|
|
|
|
|
|
|
s = x - (w / (((y - x) / 2.0f) + s)); |
|
|
|
s = x - (w/(((y - x)/2.0f) + s)); |
|
|
|
for (var i = 0; i <= n; i++) |
|
|
|
{ |
|
|
|
matrixH[i, i] -= s; |
|
|
|
@ -747,20 +748,20 @@ namespace MathNet.Numerics.LinearAlgebra.Single.Factorization |
|
|
|
z = matrixH[m, m]; |
|
|
|
r = x - z; |
|
|
|
s = y - z; |
|
|
|
p = (((r * s) - w) / matrixH[m + 1, m]) + matrixH[m, m + 1]; |
|
|
|
p = (((r*s) - w)/matrixH[m + 1, m]) + matrixH[m, m + 1]; |
|
|
|
q = matrixH[m + 1, m + 1] - z - r - s; |
|
|
|
r = matrixH[m + 2, m + 1]; |
|
|
|
s = Math.Abs(p) + Math.Abs(q) + Math.Abs(r); |
|
|
|
p = p / s; |
|
|
|
q = q / s; |
|
|
|
r = r / s; |
|
|
|
p = p/s; |
|
|
|
q = q/s; |
|
|
|
r = r/s; |
|
|
|
|
|
|
|
if (m == l) |
|
|
|
{ |
|
|
|
break; |
|
|
|
} |
|
|
|
|
|
|
|
if (Math.Abs(matrixH[m, m - 1]) * (Math.Abs(q) + Math.Abs(r)) < eps * (Math.Abs(p) * (Math.Abs(matrixH[m - 1, m - 1]) + Math.Abs(z) + Math.Abs(matrixH[m + 1, m + 1])))) |
|
|
|
if (Math.Abs(matrixH[m, m - 1])*(Math.Abs(q) + Math.Abs(r)) < eps*(Math.Abs(p)*(Math.Abs(matrixH[m - 1, m - 1]) + Math.Abs(z) + Math.Abs(matrixH[m + 1, m + 1])))) |
|
|
|
{ |
|
|
|
break; |
|
|
|
} |
|
|
|
@ -790,9 +791,9 @@ namespace MathNet.Numerics.LinearAlgebra.Single.Factorization |
|
|
|
x = Math.Abs(p) + Math.Abs(q) + Math.Abs(r); |
|
|
|
if (x != 0.0f) |
|
|
|
{ |
|
|
|
p = p / x; |
|
|
|
q = q / x; |
|
|
|
r = r / x; |
|
|
|
p = p/x; |
|
|
|
q = q/x; |
|
|
|
r = r/x; |
|
|
|
} |
|
|
|
} |
|
|
|
|
|
|
|
@ -801,7 +802,7 @@ namespace MathNet.Numerics.LinearAlgebra.Single.Factorization |
|
|
|
break; |
|
|
|
} |
|
|
|
|
|
|
|
s = (float)Math.Sqrt((p * p) + (q * q) + (r * r)); |
|
|
|
s = (float) Math.Sqrt((p*p) + (q*q) + (r*r)); |
|
|
|
if (p < 0) |
|
|
|
{ |
|
|
|
s = -s; |
|
|
|
@ -811,7 +812,7 @@ namespace MathNet.Numerics.LinearAlgebra.Single.Factorization |
|
|
|
{ |
|
|
|
if (k != m) |
|
|
|
{ |
|
|
|
matrixH[k, k - 1] = (-s) * x; |
|
|
|
matrixH[k, k - 1] = (-s)*x; |
|
|
|
} |
|
|
|
else if (l != m) |
|
|
|
{ |
|
|
|
@ -819,55 +820,55 @@ namespace MathNet.Numerics.LinearAlgebra.Single.Factorization |
|
|
|
} |
|
|
|
|
|
|
|
p = p + s; |
|
|
|
x = p / s; |
|
|
|
y = q / s; |
|
|
|
z = r / s; |
|
|
|
q = q / p; |
|
|
|
r = r / p; |
|
|
|
x = p/s; |
|
|
|
y = q/s; |
|
|
|
z = r/s; |
|
|
|
q = q/p; |
|
|
|
r = r/p; |
|
|
|
|
|
|
|
// Row modification
|
|
|
|
for (var j = k; j < order; j++) |
|
|
|
{ |
|
|
|
p = matrixH[k, j] + (q * matrixH[k + 1, j]); |
|
|
|
p = matrixH[k, j] + (q*matrixH[k + 1, j]); |
|
|
|
|
|
|
|
if (notlast) |
|
|
|
{ |
|
|
|
p = p + (r * matrixH[k + 2, j]); |
|
|
|
matrixH[k + 2, j] = matrixH[k + 2, j] - (p * z); |
|
|
|
p = p + (r*matrixH[k + 2, j]); |
|
|
|
matrixH[k + 2, j] = matrixH[k + 2, j] - (p*z); |
|
|
|
} |
|
|
|
|
|
|
|
matrixH[k, j] = matrixH[k, j] - (p * x); |
|
|
|
matrixH[k + 1, j] = matrixH[k + 1, j] - (p * y); |
|
|
|
matrixH[k, j] = matrixH[k, j] - (p*x); |
|
|
|
matrixH[k + 1, j] = matrixH[k + 1, j] - (p*y); |
|
|
|
} |
|
|
|
|
|
|
|
// Column modification
|
|
|
|
for (var i = 0; i <= Math.Min(n, k + 3); i++) |
|
|
|
{ |
|
|
|
p = (x * matrixH[i, k]) + (y * matrixH[i, k + 1]); |
|
|
|
p = (x*matrixH[i, k]) + (y*matrixH[i, k + 1]); |
|
|
|
|
|
|
|
if (notlast) |
|
|
|
{ |
|
|
|
p = p + (z * matrixH[i, k + 2]); |
|
|
|
matrixH[i, k + 2] = matrixH[i, k + 2] - (p * r); |
|
|
|
p = p + (z*matrixH[i, k + 2]); |
|
|
|
matrixH[i, k + 2] = matrixH[i, k + 2] - (p*r); |
|
|
|
} |
|
|
|
|
|
|
|
matrixH[i, k] = matrixH[i, k] - p; |
|
|
|
matrixH[i, k + 1] = matrixH[i, k + 1] - (p * q); |
|
|
|
matrixH[i, k + 1] = matrixH[i, k + 1] - (p*q); |
|
|
|
} |
|
|
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// Accumulate transformations
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for (var i = 0; i < order; i++) |
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{ |
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p = (x * EigenVectors.At(i, k)) + (y * EigenVectors.At(i, k + 1)); |
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p = (x*eigenVectors.At(i, k)) + (y*eigenVectors.At(i, k + 1)); |
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if (notlast) |
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{ |
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p = p + (z * EigenVectors.At(i, k + 2)); |
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EigenVectors.At(i, k + 2, EigenVectors.At(i, k + 2) - (p * r)); |
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p = p + (z*eigenVectors.At(i, k + 2)); |
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eigenVectors.At(i, k + 2, eigenVectors.At(i, k + 2) - (p*r)); |
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} |
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EigenVectors.At(i, k, EigenVectors.At(i, k) - p); |
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EigenVectors.At(i, k + 1, EigenVectors.At(i, k + 1) - (p * q)); |
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eigenVectors.At(i, k, eigenVectors.At(i, k) - p); |
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eigenVectors.At(i, k + 1, eigenVectors.At(i, k + 1) - (p*q)); |
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} |
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} // (s != 0)
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} // k loop
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@ -898,7 +899,7 @@ namespace MathNet.Numerics.LinearAlgebra.Single.Factorization |
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r = 0.0f; |
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for (var j = l; j <= n; j++) |
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{ |
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r = r + (matrixH[i, j] * matrixH[j, n]); |
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r = r + (matrixH[i, j]*matrixH[j, n]); |
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} |
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if (e[i] < 0.0f) |
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@ -913,11 +914,11 @@ namespace MathNet.Numerics.LinearAlgebra.Single.Factorization |
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{ |
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if (w != 0.0f) |
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{ |
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matrixH[i, n] = (-r) / w; |
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matrixH[i, n] = (-r)/w; |
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} |
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else |
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{ |
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matrixH[i, n] = (-r) / (eps * norm); |
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matrixH[i, n] = (-r)/(eps*norm); |
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} |
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// Solve real equations
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@ -926,26 +927,26 @@ namespace MathNet.Numerics.LinearAlgebra.Single.Factorization |
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{ |
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x = matrixH[i, i + 1]; |
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y = matrixH[i + 1, i]; |
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q = ((d[i] - p) * (d[i] - p)) + (e[i] * e[i]); |
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t = ((x * s) - (z * r)) / q; |
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q = ((d[i] - p)*(d[i] - p)) + (e[i]*e[i]); |
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t = ((x*s) - (z*r))/q; |
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matrixH[i, n] = t; |
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if (Math.Abs(x) > Math.Abs(z)) |
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{ |
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matrixH[i + 1, n] = (-r - (w * t)) / x; |
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matrixH[i + 1, n] = (-r - (w*t))/x; |
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} |
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else |
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{ |
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matrixH[i + 1, n] = (-s - (y * t)) / z; |
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matrixH[i + 1, n] = (-s - (y*t))/z; |
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} |
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} |
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// Overflow control
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t = Math.Abs(matrixH[i, n]); |
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if ((eps * t) * t > 1) |
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if ((eps*t)*t > 1) |
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{ |
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for (var j = i; j <= n; j++) |
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{ |
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matrixH[j, n] = matrixH[j, n] / t; |
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matrixH[j, n] = matrixH[j, n]/t; |
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} |
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} |
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} |
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@ -960,8 +961,8 @@ namespace MathNet.Numerics.LinearAlgebra.Single.Factorization |
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// Last vector component imaginary so matrix is triangular
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if (Math.Abs(matrixH[n, n - 1]) > Math.Abs(matrixH[n - 1, n])) |
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{ |
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matrixH[n - 1, n - 1] = q / matrixH[n, n - 1]; |
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matrixH[n - 1, n] = (-(matrixH[n, n] - p)) / matrixH[n, n - 1]; |
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matrixH[n - 1, n - 1] = q/matrixH[n, n - 1]; |
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matrixH[n - 1, n] = (-(matrixH[n, n] - p))/matrixH[n, n - 1]; |
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} |
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else |
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{ |
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@ -978,8 +979,8 @@ namespace MathNet.Numerics.LinearAlgebra.Single.Factorization |
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float sa = 0.0f; |
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for (var j = l; j <= n; j++) |
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{ |
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ra = ra + (matrixH[i, j] * matrixH[j, n - 1]); |
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sa = sa + (matrixH[i, j] * matrixH[j, n]); |
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ra = ra + (matrixH[i, j]*matrixH[j, n - 1]); |
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sa = sa + (matrixH[i, j]*matrixH[j, n]); |
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} |
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w = matrixH[i, i] - p; |
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@ -1005,24 +1006,24 @@ namespace MathNet.Numerics.LinearAlgebra.Single.Factorization |
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x = matrixH[i, i + 1]; |
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y = matrixH[i + 1, i]; |
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float vr = ((d[i] - p) * (d[i] - p)) + (e[i] * e[i]) - (q * q); |
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float vi = (d[i] - p) * 2.0f * q; |
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float vr = ((d[i] - p)*(d[i] - p)) + (e[i]*e[i]) - (q*q); |
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|
float vi = (d[i] - p)*2.0f*q; |
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if ((vr == 0.0f) && (vi == 0.0f)) |
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{ |
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|
|
vr = eps * norm * (Math.Abs(w) + Math.Abs(q) + Math.Abs(x) + Math.Abs(y) + Math.Abs(z)); |
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|
vr = eps*norm*(Math.Abs(w) + Math.Abs(q) + Math.Abs(x) + Math.Abs(y) + Math.Abs(z)); |
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} |
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|
var res = Cdiv((x * r) - (z * ra) + (q * sa), (x * s) - (z * sa) - (q * ra), vr, vi); |
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var res = Cdiv((x*r) - (z*ra) + (q*sa), (x*s) - (z*sa) - (q*ra), vr, vi); |
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|
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matrixH[i, n - 1] = res.Real; |
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matrixH[i, n] = res.Imaginary; |
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|
|
if (Math.Abs(x) > (Math.Abs(z) + Math.Abs(q))) |
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|
{ |
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|
matrixH[i + 1, n - 1] = (-ra - (w * matrixH[i, n - 1]) + (q * matrixH[i, n])) / x; |
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matrixH[i + 1, n] = (-sa - (w * matrixH[i, n]) - (q * matrixH[i, n - 1])) / x; |
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|
matrixH[i + 1, n - 1] = (-ra - (w*matrixH[i, n - 1]) + (q*matrixH[i, n]))/x; |
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|
matrixH[i + 1, n] = (-sa - (w*matrixH[i, n]) - (q*matrixH[i, n - 1]))/x; |
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|
} |
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|
else |
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|
{ |
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|
|
res = Cdiv(-r - (y * matrixH[i, n - 1]), -s - (y * matrixH[i, n]), z, q); |
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|
res = Cdiv(-r - (y*matrixH[i, n - 1]), -s - (y*matrixH[i, n]), z, q); |
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|
|
matrixH[i + 1, n - 1] = res.Real; |
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|
matrixH[i + 1, n] = res.Imaginary; |
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|
} |
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|
@ -1030,12 +1031,12 @@ namespace MathNet.Numerics.LinearAlgebra.Single.Factorization |
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|
// Overflow control
|
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|
t = Math.Max(Math.Abs(matrixH[i, n - 1]), Math.Abs(matrixH[i, n])); |
|
|
|
if ((eps * t) * t > 1) |
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|
|
if ((eps*t)*t > 1) |
|
|
|
{ |
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|
|
for (var j = i; j <= n; j++) |
|
|
|
{ |
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|
|
matrixH[j, n - 1] = matrixH[j, n - 1] / t; |
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|
|
matrixH[j, n] = matrixH[j, n] / t; |
|
|
|
matrixH[j, n - 1] = matrixH[j, n - 1]/t; |
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|
|
matrixH[j, n] = matrixH[j, n]/t; |
|
|
|
} |
|
|
|
} |
|
|
|
} |
|
|
|
@ -1051,10 +1052,10 @@ namespace MathNet.Numerics.LinearAlgebra.Single.Factorization |
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|
|
z = 0.0f; |
|
|
|
for (var k = 0; k <= j; k++) |
|
|
|
{ |
|
|
|
z = z + (EigenVectors.At(i, k) * matrixH[k, j]); |
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|
|
z = z + (eigenVectors.At(i, k)*matrixH[k, j]); |
|
|
|
} |
|
|
|
|
|
|
|
EigenVectors.At(i, j, z); |
|
|
|
eigenVectors.At(i, j, z); |
|
|
|
} |
|
|
|
} |
|
|
|
} |
|
|
|
@ -1067,14 +1068,14 @@ namespace MathNet.Numerics.LinearAlgebra.Single.Factorization |
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|
|
/// <param name="yreal">Real part of Y</param>
|
|
|
|
/// <param name="yimag">Imaginary part of Y</param>
|
|
|
|
/// <returns>Division result as a <see cref="Complex"/> number.</returns>
|
|
|
|
private static Complex32 Cdiv(float xreal, float ximag, float yreal, float yimag) |
|
|
|
static Complex32 Cdiv(float xreal, float ximag, float yreal, float yimag) |
|
|
|
{ |
|
|
|
if (Math.Abs(yimag) < Math.Abs(yreal)) |
|
|
|
{ |
|
|
|
return new Complex32((xreal + (ximag * (yimag / yreal))) / (yreal + (yimag * (yimag / yreal))), (ximag - (xreal * (yimag / yreal))) / (yreal + (yimag * (yimag / yreal)))); |
|
|
|
return new Complex32((xreal + (ximag*(yimag/yreal)))/(yreal + (yimag*(yimag/yreal))), (ximag - (xreal*(yimag/yreal)))/(yreal + (yimag*(yimag/yreal)))); |
|
|
|
} |
|
|
|
|
|
|
|
return new Complex32((ximag + (xreal * (yreal / yimag))) / (yimag + (yreal * (yreal / yimag))), (-xreal + (ximag * (yreal / yimag))) / (yimag + (yreal * (yreal / yimag)))); |
|
|
|
return new Complex32((ximag + (xreal*(yreal/yimag)))/(yimag + (yreal*(yreal/yimag))), (-xreal + (ximag*(yreal/yimag)))/(yimag + (yreal*(yreal/yimag)))); |
|
|
|
} |
|
|
|
|
|
|
|
/// <summary>
|
|
|
|
@ -1084,17 +1085,6 @@ namespace MathNet.Numerics.LinearAlgebra.Single.Factorization |
|
|
|
/// <param name="result">The left hand side <see cref="Matrix{T}"/>, <b>X</b>.</param>
|
|
|
|
public override void Solve(Matrix<float> input, Matrix<float> result) |
|
|
|
{ |
|
|
|
// Check for proper arguments.
|
|
|
|
if (input == null) |
|
|
|
{ |
|
|
|
throw new ArgumentNullException("input"); |
|
|
|
} |
|
|
|
|
|
|
|
if (result == null) |
|
|
|
{ |
|
|
|
throw new ArgumentNullException("result"); |
|
|
|
} |
|
|
|
|
|
|
|
// The solution X should have the same number of columns as B
|
|
|
|
if (input.ColumnCount != result.ColumnCount) |
|
|
|
{ |
|
|
|
@ -1127,10 +1117,10 @@ namespace MathNet.Numerics.LinearAlgebra.Single.Factorization |
|
|
|
{ |
|
|
|
for (var i = 0; i < order; i++) |
|
|
|
{ |
|
|
|
value += EigenVectors.At(i, j) * input.At(i, k); |
|
|
|
value += EigenVectors.At(i, j)*input.At(i, k); |
|
|
|
} |
|
|
|
|
|
|
|
value /= (float)EigenValues[j].Real; |
|
|
|
value /= (float) EigenValues[j].Real; |
|
|
|
} |
|
|
|
|
|
|
|
tmp[j] = value; |
|
|
|
@ -1141,7 +1131,7 @@ namespace MathNet.Numerics.LinearAlgebra.Single.Factorization |
|
|
|
float value = 0; |
|
|
|
for (var i = 0; i < order; i++) |
|
|
|
{ |
|
|
|
value += EigenVectors.At(j, i) * tmp[i]; |
|
|
|
value += EigenVectors.At(j, i)*tmp[i]; |
|
|
|
} |
|
|
|
|
|
|
|
result.At(j, k, value); |
|
|
|
@ -1161,16 +1151,6 @@ namespace MathNet.Numerics.LinearAlgebra.Single.Factorization |
|
|
|
/// <param name="result">The left hand side <see cref="Matrix{T}"/>, <b>x</b>.</param>
|
|
|
|
public override void Solve(Vector<float> input, Vector<float> result) |
|
|
|
{ |
|
|
|
if (input == null) |
|
|
|
{ |
|
|
|
throw new ArgumentNullException("input"); |
|
|
|
} |
|
|
|
|
|
|
|
if (result == null) |
|
|
|
{ |
|
|
|
throw new ArgumentNullException("result"); |
|
|
|
} |
|
|
|
|
|
|
|
// Ax=b where A is an m x m matrix
|
|
|
|
// Check that b is a column vector with m entries
|
|
|
|
if (EigenValues.Count != input.Count) |
|
|
|
@ -1198,10 +1178,10 @@ namespace MathNet.Numerics.LinearAlgebra.Single.Factorization |
|
|
|
{ |
|
|
|
for (var i = 0; i < order; i++) |
|
|
|
{ |
|
|
|
value += EigenVectors.At(i, j) * input[i]; |
|
|
|
value += EigenVectors.At(i, j)*input[i]; |
|
|
|
} |
|
|
|
|
|
|
|
value /= (float)EigenValues[j].Real; |
|
|
|
value /= (float) EigenValues[j].Real; |
|
|
|
} |
|
|
|
|
|
|
|
tmp[j] = value; |
|
|
|
@ -1212,7 +1192,7 @@ namespace MathNet.Numerics.LinearAlgebra.Single.Factorization |
|
|
|
value = 0; |
|
|
|
for (int i = 0; i < order; i++) |
|
|
|
{ |
|
|
|
value += EigenVectors.At(j, i) * tmp[i]; |
|
|
|
value += EigenVectors.At(j, i)*tmp[i]; |
|
|
|
} |
|
|
|
|
|
|
|
result[j] = value; |
|
|
|
@ -1224,4 +1204,4 @@ namespace MathNet.Numerics.LinearAlgebra.Single.Factorization |
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|
|
} |
|
|
|
} |
|
|
|
} |
|
|
|
} |
|
|
|
} |
|
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|
|
Christoph Ruegg
13 years ago