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1222 lines
44 KiB
1222 lines
44 KiB
// <copyright file="UserEvd.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) 2009-2010 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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namespace MathNet.Numerics.LinearAlgebra.Double.Factorization
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{
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using System;
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using System.Numerics;
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using Generic;
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using Properties;
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/// <summary>
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/// Eigenvalues and eigenvectors of a real matrix.
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/// </summary>
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/// <remarks>
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/// If A is symmetric, then A = V*D*V' where the eigenvalue matrix D is
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/// diagonal and the eigenvector matrix V is orthogonal.
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/// I.e. A = V*D*V' and V*VT=I.
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/// If A is not symmetric, then the eigenvalue matrix D is block diagonal
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/// with the real eigenvalues in 1-by-1 blocks and any complex eigenvalues,
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/// lambda + i*mu, in 2-by-2 blocks, [lambda, mu; -mu, lambda]. The
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/// columns of V represent the eigenvectors in the sense that A*V = V*D,
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/// i.e. A.Multiply(V) equals V.Multiply(D). The matrix V may be badly
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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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{
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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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/// the eigenvalue decomposition when the constructor is called and cache it's decomposition.
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/// </summary>
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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<double> 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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}
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var order = matrix.RowCount;
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// Initialize matricies for eigenvalues and eigenvectors
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MatrixEv = matrix.CreateMatrix(order, order);
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MatrixD = matrix.CreateMatrix(order, order);
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VectorEv = new LinearAlgebra.Complex.DenseVector(order);
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IsSymmetric = true;
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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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{
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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 double[order];
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var e = new double[order];
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if (IsSymmetric)
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{
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matrix.CopyTo(MatrixEv);
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d = MatrixEv.Row(order - 1).ToArray();
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SymmetricTridiagonalize(d, e, order);
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SymmetricDiagonalize(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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}
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for (var i = 0; i < order; i++)
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{
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MatrixD.At(i, i, d[i]);
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if (e[i] > 0)
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{
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MatrixD.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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MatrixD.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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VectorEv[i] = new Complex(d[i], e[i]);
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}
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}
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/// <summary>
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/// Symmetric Householder reduction to tridiagonal form.
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/// </summary>
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/// <param name="d">Arrays for internal storage of real parts of eigenvalues</param>
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/// <param name="e">Arrays for internal storage of imaginary parts of eigenvalues</param>
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/// <param name="order">Order of initial matrix</param>
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/// <remarks>This is derived from the Algol procedures tred2 by
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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(double[] d, double[] 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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{
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// Scale to avoid under/overflow.
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var scale = 0.0;
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var h = 0.0;
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for (var k = 0; k < i; k++)
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{
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scale = scale + Math.Abs(d[k]);
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}
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if (scale == 0.0)
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{
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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] = MatrixEv.At(i - 1, j);
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MatrixEv.At(i, j, 0.0);
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MatrixEv.At(j, i, 0.0);
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}
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}
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else
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{
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// Generate Householder vector.
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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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}
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var f = d[i - 1];
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var g = 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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d[i - 1] = f - g;
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for (var j = 0; j < i; j++)
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{
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e[j] = 0.0;
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}
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// Apply similarity transformation to remaining columns.
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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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MatrixEv.At(j, i, f);
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g = e[j] + (MatrixEv.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 += MatrixEv.At(k, j) * d[k];
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e[k] += MatrixEv.At(k, j) * f;
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}
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e[j] = g;
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}
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f = 0.0;
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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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}
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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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}
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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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g = e[j];
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for (var k = j; k <= i - 1; k++)
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{
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MatrixEv.At(k, j, MatrixEv.At(k, j) - (f * e[k]) - (g * d[k]));
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}
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d[j] = MatrixEv.At(i - 1, j);
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MatrixEv.At(i, j, 0.0);
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}
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}
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d[i] = h;
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}
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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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MatrixEv.At(order - 1, i, MatrixEv.At(i, i));
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MatrixEv.At(i, i, 1.0);
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var h = d[i + 1];
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if (h != 0.0)
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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] = MatrixEv.At(k, i + 1) / h;
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}
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for (var j = 0; j <= i; j++)
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{
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var g = 0.0;
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for (var k = 0; k <= i; k++)
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{
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g += MatrixEv.At(k, i + 1) * MatrixEv.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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MatrixEv.At(k, j, MatrixEv.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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MatrixEv.At(k, i + 1, 0.0);
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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] = MatrixEv.At(order - 1, j);
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MatrixEv.At(order - 1, j, 0.0);
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}
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MatrixEv.At(order - 1, order - 1, 1.0);
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e[0] = 0.0;
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}
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/// <summary>
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/// Symmetric tridiagonal QL algorithm.
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/// </summary>
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/// <param name="d">Arrays for internal storage of real parts of eigenvalues</param>
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/// <param name="e">Arrays for internal storage of imaginary parts of eigenvalues</param>
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/// <param name="order">Order of initial matrix</param>
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/// <remarks>This is derived from the Algol procedures tql2, by
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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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/// <exception cref="NonConvergenceException"></exception>
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private void SymmetricDiagonalize(double[] d, double[] e, int order)
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{
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const int Maxiter = 1000;
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for (var i = 1; i < order; i++)
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{
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e[i - 1] = e[i];
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}
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e[order - 1] = 0.0;
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var f = 0.0;
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var tst1 = 0.0;
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var eps = Precision.DoubleMachinePrecision;
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for (var l = 0; l < order; l++)
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{
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// Find small subdiagonal element
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tst1 = Math.Max(tst1, Math.Abs(d[l]) + Math.Abs(e[l]));
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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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{
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break;
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}
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m++;
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}
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// If m == l, d[l] is an eigenvalue,
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// otherwise, iterate.
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if (m > l)
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{
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var iter = 0;
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do
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{
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iter = iter + 1; // (Could check iteration count here.)
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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) / (2.0 * e[l]);
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var r = SpecialFunctions.Hypotenuse(p, 1.0);
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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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var dl1 = d[l + 1];
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var h = g - d[l];
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for (var i = l + 2; i < order; i++)
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{
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d[i] -= h;
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}
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f = f + h;
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// Implicit QL transformation.
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p = d[m];
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var c = 1.0;
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var c2 = c;
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var c3 = c;
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var el1 = e[l + 1];
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var s = 0.0;
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var s2 = 0.0;
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for (var i = m - 1; i >= l; i--)
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{
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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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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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// Accumulate transformation.
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for (var k = 0; k < order; k++)
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{
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h = MatrixEv.At(k, i + 1);
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MatrixEv.At(k, i + 1, (s * MatrixEv.At(k, i)) + (c * h));
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MatrixEv.At(k, i, (c * MatrixEv.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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// Check for convergence. If too many iterations have been performed,
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// throw exception that Convergence Failed
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if (iter >= Maxiter)
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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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}
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d[l] = d[l] + f;
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e[l] = 0.0;
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}
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// Sort eigenvalues and corresponding vectors.
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for (var i = 0; i < order - 1; i++)
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{
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var k = i;
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var p = d[i];
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for (var j = i + 1; j < order; j++)
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{
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if (d[j] < p)
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{
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k = j;
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p = d[j];
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}
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}
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if (k != i)
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{
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d[k] = d[i];
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d[i] = p;
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for (var j = 0; j < order; j++)
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{
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p = MatrixEv.At(j, i);
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MatrixEv.At(j, i, MatrixEv.At(j, k));
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MatrixEv.At(j, k, p);
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}
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}
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}
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}
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/// <summary>
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/// Nonsymmetric reduction to Hessenberg form.
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/// </summary>
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/// <param name="matrixH">Array for internal storage of nonsymmetric Hessenberg form.</param>
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/// <param name="order">Order of initial matrix</param>
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/// <remarks>This is derived from the Algol procedures orthes and ortran,
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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(double[,] matrixH, int order)
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{
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var ort = new double[order];
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for (var m = 1; m < order - 1; m++)
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{
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// Scale column.
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var scale = 0.0;
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for (var i = m; i < order; i++)
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{
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scale = scale + Math.Abs(matrixH[i, m - 1]);
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}
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if (scale != 0.0)
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{
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// Compute Householder transformation.
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var h = 0.0;
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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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}
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var g = 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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ort[m] = ort[m] - g;
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// Apply Householder similarity transformation
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// H = (I-u*u'/h)*H*(I-u*u')/h)
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for (var j = m; j < order; j++)
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{
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var f = 0.0;
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for (var i = order - 1; i >= m; i--)
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{
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f += ort[i] * matrixH[i, j];
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}
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f = f / h;
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for (var i = m; i < order; i++)
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{
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matrixH[i, j] -= f * ort[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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var f = 0.0;
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for (var j = order - 1; j >= m; j--)
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{
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f += ort[j] * matrixH[i, j];
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}
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f = f / h;
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for (var j = m; j < order; j++)
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{
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matrixH[i, j] -= f * ort[j];
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}
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}
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ort[m] = scale * ort[m];
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matrixH[m, m - 1] = scale * g;
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}
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}
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// Accumulate transformations (Algol's ortran).
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for (var i = 0; i < order; i++)
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{
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for (var j = 0; j < order; j++)
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{
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MatrixEv.At(i, j, i == j ? 1.0 : 0.0);
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}
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}
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for (var m = order - 2; m >= 1; m--)
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{
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if (matrixH[m, m - 1] != 0.0)
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{
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for (var i = m + 1; i < order; i++)
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{
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ort[i] = matrixH[i, m - 1];
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}
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for (var j = m; j < order; j++)
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{
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var g = 0.0;
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for (var i = m; i < order; i++)
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{
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g += ort[i] * MatrixEv.At(i, j);
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}
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// Double division avoids possible underflow
|
|
g = (g / ort[m]) / matrixH[m, m - 1];
|
|
for (var i = m; i < order; i++)
|
|
{
|
|
MatrixEv.At(i, j, MatrixEv.At(i, j) + g * ort[i]);
|
|
}
|
|
}
|
|
}
|
|
}
|
|
}
|
|
|
|
/// <summary>
|
|
/// Nonsymmetric reduction from Hessenberg to real Schur form.
|
|
/// </summary>
|
|
/// <param name="matrixH">Array for internal storage of nonsymmetric Hessenberg form.</param>
|
|
/// <param name="d">Arrays for internal storage of real parts of eigenvalues</param>
|
|
/// <param name="e">Arrays for internal storage of imaginary parts of eigenvalues</param>
|
|
/// <param name="order">Order of initial matrix</param>
|
|
/// <remarks>This is derived from the Algol procedure hqr2,
|
|
/// by Martin and Wilkinson, Handbook for Auto. Comp.,
|
|
/// Vol.ii-Linear Algebra, and the corresponding
|
|
/// Fortran subroutine in EISPACK.</remarks>
|
|
private void NonsymmetricReduceHessenberToRealSchur(double[,] matrixH, double[] d, double[] e, int order)
|
|
{
|
|
// Initialize
|
|
var n = order - 1;
|
|
var eps = Precision.DoubleMachinePrecision;
|
|
var exshift = 0.0;
|
|
double p = 0, q = 0, r = 0, s = 0, z = 0, w, x, y;
|
|
|
|
// Store roots isolated by balanc and compute matrix norm
|
|
var norm = 0.0;
|
|
for (var i = 0; i < order; i++)
|
|
{
|
|
for (var j = Math.Max(i - 1, 0); j < order; j++)
|
|
{
|
|
norm = norm + Math.Abs(matrixH[i, j]);
|
|
}
|
|
}
|
|
|
|
// Outer loop over eigenvalue index
|
|
var iter = 0;
|
|
while (n >= 0)
|
|
{
|
|
// Look for single small sub-diagonal element
|
|
var l = n;
|
|
while (l > 0)
|
|
{
|
|
s = Math.Abs(matrixH[l - 1, l - 1]) + Math.Abs(matrixH[l, l]);
|
|
|
|
if (s == 0.0)
|
|
{
|
|
s = norm;
|
|
}
|
|
|
|
if (Math.Abs(matrixH[l, l - 1]) < eps * s)
|
|
{
|
|
break;
|
|
}
|
|
|
|
l--;
|
|
}
|
|
|
|
// Check for convergence
|
|
// One root found
|
|
if (l == n)
|
|
{
|
|
matrixH[n, n] = matrixH[n, n] + exshift;
|
|
d[n] = matrixH[n, n];
|
|
e[n] = 0.0;
|
|
n--;
|
|
iter = 0;
|
|
|
|
// Two roots found
|
|
}
|
|
else if (l == n - 1)
|
|
{
|
|
w = matrixH[n, n - 1] * matrixH[n - 1, n];
|
|
p = (matrixH[n - 1, n - 1] - matrixH[n, n]) / 2.0;
|
|
q = (p * p) + w;
|
|
z = 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];
|
|
|
|
// Real pair
|
|
if (q >= 0)
|
|
{
|
|
if (p >= 0)
|
|
{
|
|
z = p + z;
|
|
}
|
|
else
|
|
{
|
|
z = p - z;
|
|
}
|
|
|
|
d[n - 1] = x + z;
|
|
|
|
d[n] = d[n - 1];
|
|
if (z != 0.0)
|
|
{
|
|
d[n] = x - (w / z);
|
|
}
|
|
|
|
e[n - 1] = 0.0;
|
|
e[n] = 0.0;
|
|
x = matrixH[n, n - 1];
|
|
s = Math.Abs(x) + Math.Abs(z);
|
|
p = x / s;
|
|
q = z / s;
|
|
r = 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);
|
|
}
|
|
|
|
// 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);
|
|
}
|
|
|
|
// Accumulate transformations
|
|
for (var i = 0; i < order; i++)
|
|
{
|
|
z = MatrixEv.At(i, n - 1);
|
|
MatrixEv.At(i, n - 1, (q * z) + (p * MatrixEv.At(i, n)));
|
|
MatrixEv.At(i, n, (q * MatrixEv.At(i, n)) - (p * z));
|
|
}
|
|
|
|
// Complex pair
|
|
}
|
|
else
|
|
{
|
|
d[n - 1] = x + p;
|
|
d[n] = x + p;
|
|
e[n - 1] = z;
|
|
e[n] = -z;
|
|
}
|
|
|
|
n = n - 2;
|
|
iter = 0;
|
|
|
|
// No convergence yet
|
|
}
|
|
else
|
|
{
|
|
// Form shift
|
|
x = matrixH[n, n];
|
|
y = 0.0;
|
|
w = 0.0;
|
|
if (l < n)
|
|
{
|
|
y = matrixH[n - 1, n - 1];
|
|
w = matrixH[n, n - 1] * matrixH[n - 1, n];
|
|
}
|
|
|
|
// Wilkinson's original ad hoc shift
|
|
if (iter == 10)
|
|
{
|
|
exshift += x;
|
|
for (var i = 0; i <= n; i++)
|
|
{
|
|
matrixH[i, i] -= x;
|
|
}
|
|
|
|
s = Math.Abs(matrixH[n, n - 1]) + Math.Abs(matrixH[n - 1, n - 2]);
|
|
x = y = 0.75 * s;
|
|
w = (-0.4375) * s * s;
|
|
}
|
|
|
|
// MATLAB's new ad hoc shift
|
|
if (iter == 30)
|
|
{
|
|
s = (y - x) / 2.0;
|
|
s = (s * s) + w;
|
|
if (s > 0)
|
|
{
|
|
s = Math.Sqrt(s);
|
|
if (y < x)
|
|
{
|
|
s = -s;
|
|
}
|
|
|
|
s = x - (w / (((y - x) / 2.0) + s));
|
|
for (var i = 0; i <= n; i++)
|
|
{
|
|
matrixH[i, i] -= s;
|
|
}
|
|
|
|
exshift += s;
|
|
x = y = w = 0.964;
|
|
}
|
|
}
|
|
|
|
iter = iter + 1; // (Could check iteration count here.)
|
|
|
|
// Look for two consecutive small sub-diagonal elements
|
|
var m = n - 2;
|
|
while (m >= l)
|
|
{
|
|
z = matrixH[m, m];
|
|
r = x - z;
|
|
s = y - z;
|
|
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;
|
|
|
|
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]))))
|
|
{
|
|
break;
|
|
}
|
|
|
|
m--;
|
|
}
|
|
|
|
for (var i = m + 2; i <= n; i++)
|
|
{
|
|
matrixH[i, i - 2] = 0.0;
|
|
if (i > m + 2)
|
|
{
|
|
matrixH[i, i - 3] = 0.0;
|
|
}
|
|
}
|
|
|
|
// Double QR step involving rows l:n and columns m:n
|
|
for (var k = m; k <= n - 1; k++)
|
|
{
|
|
bool notlast = k != n - 1;
|
|
|
|
if (k != m)
|
|
{
|
|
p = matrixH[k, k - 1];
|
|
q = matrixH[k + 1, k - 1];
|
|
r = notlast ? matrixH[k + 2, k - 1] : 0.0;
|
|
x = Math.Abs(p) + Math.Abs(q) + Math.Abs(r);
|
|
if (x != 0.0)
|
|
{
|
|
p = p / x;
|
|
q = q / x;
|
|
r = r / x;
|
|
}
|
|
}
|
|
|
|
if (x == 0.0)
|
|
{
|
|
break;
|
|
}
|
|
|
|
s = Math.Sqrt((p * p) + (q * q) + (r * r));
|
|
if (p < 0)
|
|
{
|
|
s = -s;
|
|
}
|
|
|
|
if (s != 0.0)
|
|
{
|
|
if (k != m)
|
|
{
|
|
matrixH[k, k - 1] = (-s) * x;
|
|
}
|
|
else if (l != m)
|
|
{
|
|
matrixH[k, k - 1] = -matrixH[k, k - 1];
|
|
}
|
|
|
|
p = p + s;
|
|
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]);
|
|
|
|
if (notlast)
|
|
{
|
|
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);
|
|
}
|
|
|
|
// Column modification
|
|
for (var i = 0; i <= Math.Min(n, k + 3); i++)
|
|
{
|
|
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);
|
|
}
|
|
|
|
matrixH[i, k] = matrixH[i, k] - p;
|
|
matrixH[i, k + 1] = matrixH[i, k + 1] - (p * q);
|
|
}
|
|
|
|
// Accumulate transformations
|
|
for (var i = 0; i < order; i++)
|
|
{
|
|
p = (x * MatrixEv.At(i, k)) + (y * MatrixEv.At(i, k + 1));
|
|
|
|
if (notlast)
|
|
{
|
|
p = p + (z * MatrixEv.At(i, k + 2));
|
|
MatrixEv.At(i, k + 2, MatrixEv.At(i, k + 2) - (p * r));
|
|
}
|
|
|
|
MatrixEv.At(i, k, MatrixEv.At(i, k) - p);
|
|
MatrixEv.At(i, k + 1, MatrixEv.At(i, k + 1) - (p * q));
|
|
}
|
|
} // (s != 0)
|
|
} // k loop
|
|
} // check convergence
|
|
} // while (n >= low)
|
|
|
|
// Backsubstitute to find vectors of upper triangular form
|
|
if (norm == 0.0)
|
|
{
|
|
return;
|
|
}
|
|
|
|
for (n = order - 1; n >= 0; n--)
|
|
{
|
|
double t;
|
|
|
|
p = d[n];
|
|
q = e[n];
|
|
|
|
// Real vector
|
|
if (q == 0.0)
|
|
{
|
|
var l = n;
|
|
matrixH[n, n] = 1.0;
|
|
for (var i = n - 1; i >= 0; i--)
|
|
{
|
|
w = matrixH[i, i] - p;
|
|
r = 0.0;
|
|
for (var j = l; j <= n; j++)
|
|
{
|
|
r = r + (matrixH[i, j] * matrixH[j, n]);
|
|
}
|
|
|
|
if (e[i] < 0.0)
|
|
{
|
|
z = w;
|
|
s = r;
|
|
}
|
|
else
|
|
{
|
|
l = i;
|
|
if (e[i] == 0.0)
|
|
{
|
|
if (w != 0.0)
|
|
{
|
|
matrixH[i, n] = (-r) / w;
|
|
}
|
|
else
|
|
{
|
|
matrixH[i, n] = (-r) / (eps * norm);
|
|
}
|
|
|
|
// Solve real equations
|
|
}
|
|
else
|
|
{
|
|
x = matrixH[i, i + 1];
|
|
y = matrixH[i + 1, i];
|
|
q = ((d[i] - p) * (d[i] - p)) + (e[i] * e[i]);
|
|
t = ((x * s) - (z * r)) / q;
|
|
matrixH[i, n] = t;
|
|
if (Math.Abs(x) > Math.Abs(z))
|
|
{
|
|
matrixH[i + 1, n] = (-r - (w * t)) / x;
|
|
}
|
|
else
|
|
{
|
|
matrixH[i + 1, n] = (-s - (y * t)) / z;
|
|
}
|
|
}
|
|
|
|
// Overflow control
|
|
t = Math.Abs(matrixH[i, n]);
|
|
if ((eps * t) * t > 1)
|
|
{
|
|
for (var j = i; j <= n; j++)
|
|
{
|
|
matrixH[j, n] = matrixH[j, n] / t;
|
|
}
|
|
}
|
|
}
|
|
}
|
|
|
|
// Complex vector
|
|
}
|
|
else if (q < 0)
|
|
{
|
|
var l = n - 1;
|
|
|
|
// Last vector component imaginary so matrix is triangular
|
|
if (Math.Abs(matrixH[n, n - 1]) > Math.Abs(matrixH[n - 1, n]))
|
|
{
|
|
matrixH[n - 1, n - 1] = q / matrixH[n, n - 1];
|
|
matrixH[n - 1, n] = (-(matrixH[n, n] - p)) / matrixH[n, n - 1];
|
|
}
|
|
else
|
|
{
|
|
var res = Cdiv(0.0, -matrixH[n - 1, n], matrixH[n - 1, n - 1] - p, q);
|
|
matrixH[n - 1, n - 1] = res.Real;
|
|
matrixH[n - 1, n] = res.Imaginary;
|
|
}
|
|
|
|
matrixH[n, n - 1] = 0.0;
|
|
matrixH[n, n] = 1.0;
|
|
for (var i = n - 2; i >= 0; i--)
|
|
{
|
|
double ra = 0.0;
|
|
double sa = 0.0;
|
|
for (var j = l; j <= n; j++)
|
|
{
|
|
ra = ra + (matrixH[i, j] * matrixH[j, n - 1]);
|
|
sa = sa + (matrixH[i, j] * matrixH[j, n]);
|
|
}
|
|
|
|
w = matrixH[i, i] - p;
|
|
|
|
if (e[i] < 0.0)
|
|
{
|
|
z = w;
|
|
r = ra;
|
|
s = sa;
|
|
}
|
|
else
|
|
{
|
|
l = i;
|
|
if (e[i] == 0.0)
|
|
{
|
|
var res = Cdiv(-ra, -sa, w, q);
|
|
matrixH[i, n - 1] = res.Real;
|
|
matrixH[i, n] = res.Imaginary;
|
|
}
|
|
else
|
|
{
|
|
// Solve complex equations
|
|
x = matrixH[i, i + 1];
|
|
y = matrixH[i + 1, i];
|
|
|
|
double vr = ((d[i] - p) * (d[i] - p)) + (e[i] * e[i]) - (q * q);
|
|
double vi = (d[i] - p) * 2.0 * q;
|
|
if ((vr == 0.0) && (vi == 0.0))
|
|
{
|
|
vr = eps * norm * (Math.Abs(w) + Math.Abs(q) + Math.Abs(x) + Math.Abs(y) + Math.Abs(z));
|
|
}
|
|
|
|
var res = Cdiv((x * r) - (z * ra) + (q * sa), (x * s) - (z * sa) - (q * ra), vr, vi);
|
|
matrixH[i, n - 1] = res.Real;
|
|
matrixH[i, n] = res.Imaginary;
|
|
if (Math.Abs(x) > (Math.Abs(z) + Math.Abs(q)))
|
|
{
|
|
matrixH[i + 1, n - 1] = (-ra - (w * matrixH[i, n - 1]) + (q * matrixH[i, n])) / x;
|
|
matrixH[i + 1, n] = (-sa - (w * matrixH[i, n]) - (q * matrixH[i, n - 1])) / x;
|
|
}
|
|
else
|
|
{
|
|
res = Cdiv(-r - (y * matrixH[i, n - 1]), -s - (y * matrixH[i, n]), z, q);
|
|
matrixH[i + 1, n - 1] = res.Real;
|
|
matrixH[i + 1, n] = res.Imaginary;
|
|
}
|
|
}
|
|
|
|
// Overflow control
|
|
t = Math.Max(Math.Abs(matrixH[i, n - 1]), Math.Abs(matrixH[i, n]));
|
|
if ((eps * t) * t > 1)
|
|
{
|
|
for (var j = i; j <= n; j++)
|
|
{
|
|
matrixH[j, n - 1] = matrixH[j, n - 1] / t;
|
|
matrixH[j, n] = matrixH[j, n] / t;
|
|
}
|
|
}
|
|
}
|
|
}
|
|
}
|
|
}
|
|
|
|
// Back transformation to get eigenvectors of original matrix
|
|
for (var j = order - 1; j >= 0; j--)
|
|
{
|
|
for (var i = 0; i < order; i++)
|
|
{
|
|
z = 0.0;
|
|
for (var k = 0; k <= j; k++)
|
|
{
|
|
z = z + (MatrixEv.At(i, k) * matrixH[k, j]);
|
|
}
|
|
|
|
MatrixEv.At(i, j, z);
|
|
}
|
|
}
|
|
}
|
|
|
|
/// <summary>
|
|
/// Complex scalar division X/Y.
|
|
/// </summary>
|
|
/// <param name="xreal">Real part of X</param>
|
|
/// <param name="ximag">Imaginary part of X</param>
|
|
/// <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>
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private static Complex Cdiv(double xreal, double ximag, double yreal, double yimag)
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{
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if (Math.Abs(yimag) < Math.Abs(yreal))
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{
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return new Complex((xreal + (ximag * (yimag / yreal))) / (yreal + (yimag * (yimag / yreal))), (ximag - (xreal * (yimag / yreal))) / (yreal + (yimag * (yimag / yreal))));
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}
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return new Complex((ximag + (xreal * (yreal / yimag))) / (yimag + (yreal * (yreal / yimag))), (-xreal + (ximag * (yreal / yimag))) / (yimag + (yreal * (yreal / yimag))));
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}
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/// <summary>
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/// Solves a system of linear equations, <b>AX = B</b>, with A SVD factorized.
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/// </summary>
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/// <param name="input">The right hand side <see cref="Matrix{T}"/>, <b>B</b>.</param>
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/// <param name="result">The left hand side <see cref="Matrix{T}"/>, <b>X</b>.</param>
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public override void Solve(Matrix<double> input, Matrix<double> result)
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{
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// Check for proper arguments.
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if (input == null)
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{
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throw new ArgumentNullException("input");
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}
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if (result == null)
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{
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throw new ArgumentNullException("result");
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}
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// The solution X should have the same number of columns as B
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if (input.ColumnCount != result.ColumnCount)
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{
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throw new ArgumentException(Resources.ArgumentMatrixSameColumnDimension);
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}
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// The dimension compatibility conditions for X = A\B require the two matrices A and B to have the same number of rows
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if (VectorEv.Count != input.RowCount)
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{
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throw new ArgumentException(Resources.ArgumentMatrixSameRowDimension);
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}
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|
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// The solution X row dimension is equal to the column dimension of A
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if (VectorEv.Count != result.RowCount)
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{
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throw new ArgumentException(Resources.ArgumentMatrixSameColumnDimension);
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}
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if (IsSymmetric)
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{
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var order = VectorEv.Count;
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var tmp = new double[order];
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|
|
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for (var k = 0; k < order; k++)
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{
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for (var j = 0; j < order; j++)
|
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{
|
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double value = 0;
|
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if (j < order)
|
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{
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for (var i = 0; i < order; i++)
|
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{
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value += MatrixEv.At(i, j) * input.At(i, k);
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}
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|
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value /= VectorEv[j].Real;
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}
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|
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tmp[j] = value;
|
|
}
|
|
|
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for (var j = 0; j < order; j++)
|
|
{
|
|
double value = 0;
|
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for (var i = 0; i < order; i++)
|
|
{
|
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value += MatrixEv.At(j, i) * tmp[i];
|
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}
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|
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result.At(j, k, value);
|
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}
|
|
}
|
|
}
|
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else
|
|
{
|
|
throw new ArgumentException(Resources.ArgumentMatrixSymmetric);
|
|
}
|
|
}
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|
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/// <summary>
|
|
/// Solves a system of linear equations, <b>Ax = b</b>, with A EVD factorized.
|
|
/// </summary>
|
|
/// <param name="input">The right hand side vector, <b>b</b>.</param>
|
|
/// <param name="result">The left hand side <see cref="Matrix{T}"/>, <b>x</b>.</param>
|
|
public override void Solve(Vector<double> input, Vector<double> 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 (VectorEv.Count != input.Count)
|
|
{
|
|
throw new ArgumentException(Resources.ArgumentVectorsSameLength);
|
|
}
|
|
|
|
// Check that x is a column vector with n entries
|
|
if (VectorEv.Count != result.Count)
|
|
{
|
|
throw new ArgumentException(Resources.ArgumentMatrixDimensions);
|
|
}
|
|
|
|
if (IsSymmetric)
|
|
{
|
|
// Symmetric case -> x = V * inv(λ) * VT * b;
|
|
var order = VectorEv.Count;
|
|
var tmp = new double[order];
|
|
double value;
|
|
|
|
for (var j = 0; j < order; j++)
|
|
{
|
|
value = 0;
|
|
if (j < order)
|
|
{
|
|
for (var i = 0; i < order; i++)
|
|
{
|
|
value += MatrixEv.At(i, j) * input[i];
|
|
}
|
|
|
|
value /= VectorEv[j].Real;
|
|
}
|
|
|
|
tmp[j] = value;
|
|
}
|
|
|
|
for (var j = 0; j < order; j++)
|
|
{
|
|
value = 0;
|
|
for (int i = 0; i < order; i++)
|
|
{
|
|
value += MatrixEv.At(j, i) * tmp[i];
|
|
}
|
|
|
|
result[j] = value;
|
|
}
|
|
}
|
|
else
|
|
{
|
|
throw new ArgumentException(Resources.ArgumentMatrixSymmetric);
|
|
}
|
|
}
|
|
}
|
|
}
|