// <copyright file="Evd.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.Factorization
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{
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using Numerics;
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using Complex = System.Numerics.Complex;
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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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/// <typeparam name="T">Supported data types are double, single, <see cref="Complex"/>, and <see cref="Complex32"/>.</typeparam>
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public abstract class Evd<T> : ISolver<T>
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where T : struct, IEquatable<T>, IFormattable
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{
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protected Evd(Matrix<T> eigenVectors, Vector<Complex> eigenValues, Matrix<T> blockDiagonal, bool isSymmetric)
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{
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EigenVectors = eigenVectors;
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EigenValues = eigenValues;
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D = blockDiagonal;
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IsSymmetric = isSymmetric;
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}
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/// <summary>
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/// Gets or sets a value indicating whether matrix is symmetric or not
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/// </summary>
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public bool IsSymmetric { get; private set; }
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/// <summary>
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/// Gets the absolute value of determinant of the square matrix for which the EVD was computed.
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/// </summary>
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public abstract T Determinant { get; }
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/// <summary>
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/// Gets the effective numerical matrix rank.
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/// </summary>
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/// <value>The number of non-negligible singular values.</value>
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public abstract int Rank { get; }
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/// <summary>
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/// Gets a value indicating whether the matrix is full rank or not.
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/// </summary>
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/// <value><c>true</c> if the matrix is full rank; otherwise <c>false</c>.</value>
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public abstract bool IsFullRank { get; }
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/// <summary>
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/// Gets or sets the eigen values (λ) of matrix in ascending value.
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/// </summary>
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public Vector<Complex> EigenValues { get; private set; }
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/// <summary>
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/// Gets or sets eigenvectors.
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/// </summary>
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public Matrix<T> EigenVectors { get; private set; }
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/// <summary>
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/// Gets or sets the block diagonal eigenvalue matrix.
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/// </summary>
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public Matrix<T> D { get; private set; }
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/// <summary>
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/// Solves a system of linear equations, <b>AX = B</b>, with A EVD 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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/// <returns>The left hand side <see cref="Matrix{T}"/>, <b>X</b>.</returns>
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public virtual Matrix<T> Solve(Matrix<T> input)
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{
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var x = Matrix<T>.Build.SameAs(EigenVectors, EigenVectors.ColumnCount, input.ColumnCount, fullyMutable: true);
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Solve(input, x);
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return x;
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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 EVD 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 abstract void Solve(Matrix<T> input, Matrix<T> result);
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/// <summary>
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/// Solves a system of linear equations, <b>Ax = b</b>, with A EVD factorized.
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/// </summary>
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/// <param name="input">The right hand side vector, <b>b</b>.</param>
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/// <returns>The left hand side <see cref="Vector{T}"/>, <b>x</b>.</returns>
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public virtual Vector<T> Solve(Vector<T> input)
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{
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var x = Vector<T>.Build.SameAs(EigenVectors, EigenVectors.ColumnCount);
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Solve(input, x);
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return x;
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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 EVD factorized.
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/// </summary>
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/// <param name="input">The right hand side vector, <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 abstract void Solve(Vector<T> input, Vector<T> result);
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}
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}
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