// <copyright file="DenseEvd.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) 2009-2020 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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// 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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using IStation.Numerics.Providers.LinearAlgebra;
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namespace IStation.Numerics.LinearAlgebra.Double.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 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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internal sealed class DenseEvd : Evd
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{
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/// <summary>
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/// Initializes a new instance of the <see cref="DenseEvd"/> 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 DenseEvd Create(DenseMatrix 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 = new DenseMatrix(order);
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var blockDiagonal = new DenseMatrix(order);
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var eigenValues = new LinearAlgebra.Complex.DenseVector(order);
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bool isSymmetric;
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switch (symmetricity)
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{
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case Symmetricity.Symmetric:
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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.IsSymmetric();
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break;
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}
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LinearAlgebraControl.Provider.EigenDecomp(isSymmetric, order, matrix.Values, eigenVectors.Values, eigenValues.Values, blockDiagonal.Values);
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return new DenseEvd(eigenVectors, eigenValues, blockDiagonal, isSymmetric);
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}
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DenseEvd(Matrix<double> eigenVectors, Vector<Complex> eigenValues, Matrix<double> 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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/// 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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// 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("Matrix column dimensions must agree.");
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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 (EigenValues.Count != input.RowCount)
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{
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throw new ArgumentException("Matrix row dimensions must agree.");
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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 (EigenValues.Count != result.RowCount)
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{
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throw new ArgumentException("Matrix column dimensions must agree.");
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}
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if (IsSymmetric)
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{
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var order = EigenValues.Count;
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var tmp = new double[order];
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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 += ((DenseMatrix) EigenVectors).Values[(j*order) + i]*input.At(i, k);
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}
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value /= EigenValues[j].Real;
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}
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tmp[j] = value;
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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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for (var i = 0; i < order; i++)
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{
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value += ((DenseMatrix) EigenVectors).Values[(i*order) + j]*tmp[i];
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}
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result.At(j, k, value);
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}
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}
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}
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else
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{
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throw new ArgumentException("Matrix must be symmetric.");
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}
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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 override void Solve(Vector<double> input, Vector<double> result)
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{
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// Ax=b where A is an m x m matrix
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// Check that b is a column vector with m entries
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if (EigenValues.Count != input.Count)
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{
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throw new ArgumentException("All vectors must have the same dimensionality.");
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}
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// Check that x is a column vector with n entries
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if (EigenValues.Count != result.Count)
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{
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throw new ArgumentException("Matrix dimensions must agree.");
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}
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if (IsSymmetric)
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{
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// Symmetric case -> x = V * inv(λ) * VT * b;
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var order = EigenValues.Count;
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var tmp = new double[order];
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double value;
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for (var j = 0; j < order; j++)
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{
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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 += ((DenseMatrix) EigenVectors).Values[(j*order) + i]*input[i];
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}
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value /= EigenValues[j].Real;
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}
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tmp[j] = value;
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}
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for (var j = 0; j < order; j++)
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{
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value = 0;
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for (var i = 0; i < order; i++)
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{
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value += ((DenseMatrix) EigenVectors).Values[(i*order) + j]*tmp[i];
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}
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result[j] = value;
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}
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}
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else
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{
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throw new ArgumentException("Matrix must be symmetric.");
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}
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}
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}
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}
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