// <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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//
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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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// files (the "Software"), to deal in the Software without
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// restriction, including without limitation the rights to use,
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// copy, modify, merge, publish, distribute, sublicense, and/or sell
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// copies of the Software, and to permit persons to whom the
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// Software is furnished to do so, subject to the following
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// conditions:
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//
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// The above copyright notice and this permission notice shall be
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// included in all copies or substantial portions of the Software.
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//
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// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
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// EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES
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// OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
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// NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT
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// HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY,
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// WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
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// FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
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// OTHER DEALINGS IN THE SOFTWARE.
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// </copyright>
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using System;
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namespace IStation.Numerics.LinearAlgebra.Complex.Factorization
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{
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using Complex = System.Numerics.Complex;
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/// <summary>
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/// Eigenvalues and eigenvectors of a complex matrix.
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/// </summary>
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/// <remarks>
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/// If A is Hermitian, then A = V*D*V' where the eigenvalue matrix D is
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/// diagonal and the eigenvector matrix V is Hermitian.
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/// I.e. A = V*D*V' and V*VH=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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internal 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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/// 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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/// <param name="symmetricity">If it is known whether the matrix is symmetric or not the routine can skip checking it itself.</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 static UserEvd Create(Matrix<Complex> matrix, Symmetricity symmetricity)
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{
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if (matrix.RowCount != matrix.ColumnCount)
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{
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throw new ArgumentException("Matrix must be square.");
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}
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var order = matrix.RowCount;
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// Initialize matrices for eigenvalues and eigenvectors
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var eigenVectors = DenseMatrix.CreateIdentity(order);
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var blockDiagonal = Matrix<Complex>.Build.SameAs(matrix, order, order);
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var eigenValues = new DenseVector(order);
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bool isSymmetric;
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switch (symmetricity)
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{
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case Symmetricity.Hermitian:
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isSymmetric = true;
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break;
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case Symmetricity.Asymmetric:
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isSymmetric = false;
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break;
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default:
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isSymmetric = matrix.IsHermitian();
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break;
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}
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if (isSymmetric)
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{
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var matrixCopy = matrix.ToArray();
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var tau = new Complex[order];
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var d = new double[order];
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var e = new double[order];
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SymmetricTridiagonalize(matrixCopy, d, e, tau, order);
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SymmetricDiagonalize(eigenVectors, d, e, order);
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SymmetricUntridiagonalize(eigenVectors, matrixCopy, tau, order);
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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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}
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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(eigenVectors, matrixH, order);
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NonsymmetricReduceHessenberToRealSchur(eigenVectors, eigenValues, matrixH, order);
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}
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blockDiagonal.SetDiagonal(eigenValues);
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return new UserEvd(eigenVectors, eigenValues, blockDiagonal, isSymmetric);
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}
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UserEvd(Matrix<Complex> eigenVectors, Vector<Complex> eigenValues, Matrix<Complex> 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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/// Reduces a complex Hermitian matrix to a real symmetric tridiagonal matrix using unitary similarity transformations.
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/// </summary>
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/// <param name="matrixA">Source matrix to reduce</param>
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/// <param name="d">Output: Arrays for internal storage of real parts of eigenvalues</param>
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/// <param name="e">Output: Arrays for internal storage of imaginary parts of eigenvalues</param>
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/// <param name="tau">Output: Arrays that contains further information about the transformations.</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 HTRIDI by
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/// Smith, Boyle, Dongarra, Garbow, Ikebe, Klema, Moler, 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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static void SymmetricTridiagonalize(Complex[,] matrixA, double[] d, double[] e, Complex[] tau, int order)
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{
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double hh;
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tau[order - 1] = Complex.One;
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for (var i = 0; i < order; i++)
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{
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d[i] = matrixA[i, i].Real;
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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(matrixA[i, k].Real) + Math.Abs(matrixA[i, k].Imaginary);
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}
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if (scale == 0.0)
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{
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tau[i - 1] = Complex.One;
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e[i] = 0.0;
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}
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else
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{
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for (var k = 0; k < i; k++)
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{
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matrixA[i, k] /= scale;
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h += matrixA[i, k].MagnitudeSquared();
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}
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Complex g = Math.Sqrt(h);
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e[i] = scale*g.Real;
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Complex temp;
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var f = matrixA[i, i - 1];
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if (f.Magnitude != 0)
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{
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temp = -(matrixA[i, i - 1].Conjugate()*tau[i].Conjugate())/f.Magnitude;
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h += f.Magnitude*g.Real;
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g = 1.0 + (g/f.Magnitude);
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matrixA[i, i - 1] *= g;
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}
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else
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{
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temp = -tau[i].Conjugate();
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matrixA[i, i - 1] = g;
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}
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if ((f.Magnitude == 0) || (i != 1))
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{
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f = Complex.Zero;
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for (var j = 0; j < i; j++)
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{
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var tmp = Complex.Zero;
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// Form element of A*U.
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for (var k = 0; k <= j; k++)
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{
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tmp += matrixA[j, k]*matrixA[i, k].Conjugate();
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}
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for (var k = j + 1; k <= i - 1; k++)
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{
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tmp += matrixA[k, j].Conjugate()*matrixA[i, k].Conjugate();
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}
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// Form element of P
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tau[j] = tmp/h;
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f += (tmp/h)*matrixA[i, j];
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}
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hh = f.Real/(h + h);
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// Form the reduced A.
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for (var j = 0; j < i; j++)
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{
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f = matrixA[i, j].Conjugate();
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g = tau[j] - (hh*f);
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tau[j] = g.Conjugate();
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for (var k = 0; k <= j; k++)
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{
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matrixA[j, k] -= (f*tau[k]) + (g*matrixA[i, 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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matrixA[i, k] *= scale;
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}
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tau[i - 1] = temp.Conjugate();
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}
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hh = d[i];
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d[i] = matrixA[i, i].Real;
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matrixA[i, i] = new Complex(hh, scale*Math.Sqrt(h));
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}
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hh = d[0];
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d[0] = matrixA[0, 0].Real;
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matrixA[0, 0] = hh;
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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="eigenVectors">The eigen vectors to work on.</param>
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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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static void SymmetricDiagonalize(Matrix<Complex> eigenVectors, 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.DoublePrecision;
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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 = eigenVectors.At(k, i + 1).Real;
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eigenVectors.At(k, i + 1, (s*eigenVectors.At(k, i).Real) + (c*h));
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eigenVectors.At(k, i, (c*eigenVectors.At(k, i).Real) - (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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} 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 = eigenVectors.At(j, i).Real;
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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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}
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/// <summary>
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/// Determines eigenvectors by undoing the symmetric tridiagonalize transformation
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/// </summary>
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/// <param name="eigenVectors">The eigen vectors to work on.</param>
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/// <param name="matrixA">Previously tridiagonalized matrix by <see cref="SymmetricTridiagonalize"/>.</param>
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/// <param name="tau">Contains further information about the transformations</param>
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/// <param name="order">Input matrix order</param>
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/// <remarks>This is derived from the Algol procedures HTRIBK, by
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/// by Smith, Boyle, Dongarra, Garbow, Ikebe, Klema, Moler, 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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static void SymmetricUntridiagonalize(Matrix<Complex> eigenVectors, Complex[,] matrixA, Complex[] tau, int order)
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{
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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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eigenVectors.At(i, j, eigenVectors.At(i, j).Real*tau[i].Conjugate());
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}
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}
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// Recover and apply the Householder matrices.
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for (var i = 1; i < order; i++)
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{
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var h = matrixA[i, i].Imaginary;
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if (h != 0)
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{
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for (var j = 0; j < order; j++)
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{
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var s = Complex.Zero;
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for (var k = 0; k < i; k++)
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{
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s += eigenVectors.At(k, j)*matrixA[i, k];
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}
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s = (s/h)/h;
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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) - s*matrixA[i, k].Conjugate());
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}
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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="eigenVectors">The eigen vectors to work on.</param>
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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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static void NonsymmetricReduceToHessenberg(Matrix<Complex> eigenVectors, Complex[,] matrixH, int order)
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{
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var ort = new Complex[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 += Math.Abs(matrixH[i, m - 1].Real) + Math.Abs(matrixH[i, m - 1].Imaginary);
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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].MagnitudeSquared();
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}
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var g = Math.Sqrt(h);
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if (ort[m].Magnitude != 0)
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{
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h = h + (ort[m].Magnitude*g);
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g /= ort[m].Magnitude;
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ort[m] = (1.0 + g)*ort[m];
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}
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else
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{
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ort[m] = g;
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matrixH[m, m - 1] = scale;
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}
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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 = Complex.Zero;
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for (var i = order - 1; i >= m; i--)
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{
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f += ort[i].Conjugate()*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 = Complex.Zero;
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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].Conjugate();
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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] *= -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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eigenVectors.At(i, j, i == j ? Complex.One : Complex.Zero);
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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] != Complex.Zero && ort[m] != Complex.Zero)
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{
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var norm = (matrixH[m, m - 1].Real*ort[m].Real) + (matrixH[m, m - 1].Imaginary*ort[m].Imaginary);
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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 = Complex.Zero;
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for (var i = m; i < order; i++)
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{
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g += ort[i].Conjugate()*eigenVectors.At(i, j);
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}
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// Double division avoids possible underflow
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g /= norm;
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for (var i = m; i < order; i++)
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{
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eigenVectors.At(i, j, eigenVectors.At(i, j) + g*ort[i]);
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}
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}
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}
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}
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// Create real subdiagonal elements.
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for (var i = 1; i < order; i++)
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{
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if (matrixH[i, i - 1].Imaginary != 0.0)
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{
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var y = matrixH[i, i - 1]/matrixH[i, i - 1].Magnitude;
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matrixH[i, i - 1] = matrixH[i, i - 1].Magnitude;
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for (var j = i; j < order; j++)
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{
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matrixH[i, j] *= y.Conjugate();
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}
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for (var j = 0; j <= Math.Min(i + 1, order - 1); j++)
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{
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matrixH[j, i] *= y;
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}
|
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for (var j = 0; j < order; j++)
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{
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eigenVectors.At(j, i, eigenVectors.At(j, i)*y);
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}
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}
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}
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}
|
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/// <summary>
|
/// Nonsymmetric reduction from Hessenberg to real Schur form.
|
/// </summary>
|
/// <param name="eigenVectors">The eigen vectors to work on.</param>
|
/// <param name="eigenValues">The eigen values to work on.</param>
|
/// <param name="matrixH">Array for internal storage of nonsymmetric Hessenberg form.</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>
|
static void NonsymmetricReduceHessenberToRealSchur(Matrix<Complex> eigenVectors, Vector<Complex> eigenValues, Complex[,] matrixH, int order)
|
{
|
// Initialize
|
var n = order - 1;
|
var eps = Precision.DoublePrecision;
|
|
double norm;
|
Complex x, y, z, exshift = Complex.Zero;
|
|
// Outer loop over eigenvalue index
|
var iter = 0;
|
while (n >= 0)
|
{
|
// Look for single small sub-diagonal element
|
var l = n;
|
while (l > 0)
|
{
|
var tst1 = Math.Abs(matrixH[l - 1, l - 1].Real) + Math.Abs(matrixH[l - 1, l - 1].Imaginary) + Math.Abs(matrixH[l, l].Real) + Math.Abs(matrixH[l, l].Imaginary);
|
if (Math.Abs(matrixH[l, l - 1].Real) < eps*tst1)
|
{
|
break;
|
}
|
|
l--;
|
}
|
|
// Check for convergence
|
// One root found
|
if (l == n)
|
{
|
matrixH[n, n] += exshift;
|
eigenValues[n] = matrixH[n, n];
|
n--;
|
iter = 0;
|
}
|
else
|
{
|
// Form shift
|
Complex s;
|
if (iter != 10 && iter != 20)
|
{
|
s = matrixH[n, n];
|
x = matrixH[n - 1, n]*matrixH[n, n - 1].Real;
|
|
if (x.Real != 0.0 || x.Imaginary != 0.0)
|
{
|
y = (matrixH[n - 1, n - 1] - s)/2.0;
|
z = ((y*y) + x).SquareRoot();
|
if ((y.Real*z.Real) + (y.Imaginary*z.Imaginary) < 0.0)
|
{
|
z *= -1.0;
|
}
|
|
x /= y + z;
|
s = s - x;
|
}
|
}
|
else
|
{
|
// Form exceptional shift
|
s = Math.Abs(matrixH[n, n - 1].Real) + Math.Abs(matrixH[n - 1, n - 2].Real);
|
}
|
|
for (var i = 0; i <= n; i++)
|
{
|
matrixH[i, i] -= s;
|
}
|
|
exshift += s;
|
iter++;
|
|
// Reduce to triangle (rows)
|
for (var i = l + 1; i <= n; i++)
|
{
|
s = matrixH[i, i - 1].Real;
|
norm = SpecialFunctions.Hypotenuse(matrixH[i - 1, i - 1].Magnitude, s.Real);
|
x = matrixH[i - 1, i - 1]/norm;
|
eigenValues[i - 1] = x;
|
matrixH[i - 1, i - 1] = norm;
|
matrixH[i, i - 1] = new Complex(0.0, s.Real/norm);
|
|
for (var j = i; j < order; j++)
|
{
|
y = matrixH[i - 1, j];
|
z = matrixH[i, j];
|
matrixH[i - 1, j] = (x.Conjugate()*y) + (matrixH[i, i - 1].Imaginary*z);
|
matrixH[i, j] = (x*z) - (matrixH[i, i - 1].Imaginary*y);
|
}
|
}
|
|
s = matrixH[n, n];
|
if (s.Imaginary != 0.0)
|
{
|
s /= matrixH[n, n].Magnitude;
|
matrixH[n, n] = matrixH[n, n].Magnitude;
|
|
for (var j = n + 1; j < order; j++)
|
{
|
matrixH[n, j] *= s.Conjugate();
|
}
|
}
|
|
// Inverse operation (columns).
|
for (var j = l + 1; j <= n; j++)
|
{
|
x = eigenValues[j - 1];
|
for (var i = 0; i <= j; i++)
|
{
|
z = matrixH[i, j];
|
if (i != j)
|
{
|
y = matrixH[i, j - 1];
|
matrixH[i, j - 1] = (x*y) + (matrixH[j, j - 1].Imaginary*z);
|
}
|
else
|
{
|
y = matrixH[i, j - 1].Real;
|
matrixH[i, j - 1] = new Complex((x.Real*y.Real) - (x.Imaginary*y.Imaginary) + (matrixH[j, j - 1].Imaginary*z.Real), matrixH[i, j - 1].Imaginary);
|
}
|
|
matrixH[i, j] = (x.Conjugate()*z) - (matrixH[j, j - 1].Imaginary*y);
|
}
|
|
for (var i = 0; i < order; i++)
|
{
|
y = eigenVectors.At(i, j - 1);
|
z = eigenVectors.At(i, j);
|
eigenVectors.At(i, j - 1, (x*y) + (matrixH[j, j - 1].Imaginary*z));
|
eigenVectors.At(i, j, (x.Conjugate()*z) - (matrixH[j, j - 1].Imaginary*y));
|
}
|
}
|
|
if (s.Imaginary != 0.0)
|
{
|
for (var i = 0; i <= n; i++)
|
{
|
matrixH[i, n] *= s;
|
}
|
|
for (var i = 0; i < order; i++)
|
{
|
eigenVectors.At(i, n, eigenVectors.At(i, n)*s);
|
}
|
}
|
}
|
}
|
|
// All roots found.
|
// Backsubstitute to find vectors of upper triangular form
|
norm = 0.0;
|
for (var i = 0; i < order; i++)
|
{
|
for (var j = i; j < order; j++)
|
{
|
norm = Math.Max(norm, Math.Abs(matrixH[i, j].Real) + Math.Abs(matrixH[i, j].Imaginary));
|
}
|
}
|
|
if (order == 1)
|
{
|
return;
|
}
|
|
if (norm == 0.0)
|
{
|
return;
|
}
|
|
for (n = order - 1; n > 0; n--)
|
{
|
x = eigenValues[n];
|
matrixH[n, n] = 1.0;
|
|
for (var i = n - 1; i >= 0; i--)
|
{
|
z = 0.0;
|
for (var j = i + 1; j <= n; j++)
|
{
|
z += matrixH[i, j]*matrixH[j, n];
|
}
|
|
y = x - eigenValues[i];
|
if (y.Real == 0.0 && y.Imaginary == 0.0)
|
{
|
y = eps*norm;
|
}
|
|
matrixH[i, n] = z/y;
|
|
// Overflow control
|
var tr = Math.Abs(matrixH[i, n].Real) + Math.Abs(matrixH[i, n].Imaginary);
|
if ((eps*tr)*tr > 1)
|
{
|
for (var j = i; j <= n; j++)
|
{
|
matrixH[j, n] = matrixH[j, n]/tr;
|
}
|
}
|
}
|
}
|
|
// Back transformation to get eigenvectors of original matrix
|
for (var j = order - 1; j > 0; j--)
|
{
|
for (var i = 0; i < order; i++)
|
{
|
z = Complex.Zero;
|
for (var k = 0; k <= j; k++)
|
{
|
z += eigenVectors.At(i, k)*matrixH[k, j];
|
}
|
|
eigenVectors.At(i, j, z);
|
}
|
}
|
}
|
|
/// <summary>
|
/// Solves a system of linear equations, <b>AX = B</b>, with A SVD factorized.
|
/// </summary>
|
/// <param name="input">The right hand side <see cref="Matrix{T}"/>, <b>B</b>.</param>
|
/// <param name="result">The left hand side <see cref="Matrix{T}"/>, <b>X</b>.</param>
|
public override void Solve(Matrix<Complex> input, Matrix<Complex> result)
|
{
|
// The solution X should have the same number of columns as B
|
if (input.ColumnCount != result.ColumnCount)
|
{
|
throw new ArgumentException("Matrix column dimensions must agree.");
|
}
|
|
// The dimension compatibility conditions for X = A\B require the two matrices A and B to have the same number of rows
|
if (EigenValues.Count != input.RowCount)
|
{
|
throw new ArgumentException("Matrix row dimensions must agree.");
|
}
|
|
// The solution X row dimension is equal to the column dimension of A
|
if (EigenValues.Count != result.RowCount)
|
{
|
throw new ArgumentException("Matrix column dimensions must agree.");
|
}
|
|
if (IsSymmetric)
|
{
|
var order = EigenValues.Count;
|
var tmp = new Complex[order];
|
|
for (var k = 0; k < order; k++)
|
{
|
for (var j = 0; j < order; j++)
|
{
|
Complex value = 0.0;
|
if (j < order)
|
{
|
for (var i = 0; i < order; i++)
|
{
|
value += EigenVectors.At(i, j).Conjugate()*input.At(i, k);
|
}
|
|
value /= EigenValues[j].Real;
|
}
|
|
tmp[j] = value;
|
}
|
|
for (var j = 0; j < order; j++)
|
{
|
Complex value = 0.0;
|
for (var i = 0; i < order; i++)
|
{
|
value += EigenVectors.At(j, i)*tmp[i];
|
}
|
|
result.At(j, k, value);
|
}
|
}
|
}
|
else
|
{
|
throw new ArgumentException("Matrix must be symmetric.");
|
}
|
}
|
|
/// <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<Complex> input, Vector<Complex> 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)
|
{
|
throw new ArgumentException("All vectors must have the same dimensionality.");
|
}
|
|
// Check that x is a column vector with n entries
|
if (EigenValues.Count != result.Count)
|
{
|
throw Matrix.DimensionsDontMatch<ArgumentException>(EigenValues, result);
|
}
|
|
if (IsSymmetric)
|
{
|
// Symmetric case -> x = V * inv(λ) * VH * b;
|
var order = EigenValues.Count;
|
var tmp = new Complex[order];
|
Complex value;
|
|
for (var j = 0; j < order; j++)
|
{
|
value = 0;
|
if (j < order)
|
{
|
for (var i = 0; i < order; i++)
|
{
|
value += EigenVectors.At(i, j).Conjugate()*input[i];
|
}
|
|
value /= EigenValues[j].Real;
|
}
|
|
tmp[j] = value;
|
}
|
|
for (var j = 0; j < order; j++)
|
{
|
value = 0;
|
for (int i = 0; i < order; i++)
|
{
|
value += EigenVectors.At(j, i)*tmp[i];
|
}
|
|
result[j] = value;
|
}
|
}
|
else
|
{
|
throw new ArgumentException("Matrix must be symmetric.");
|
}
|
}
|
}
|
}
|
