// <copyright file="UserQR.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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using System.Linq;
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using IStation.Numerics.LinearAlgebra.Factorization;
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using IStation.Numerics.Threading;
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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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/// <para>A class which encapsulates the functionality of the QR decomposition.</para>
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/// <para>Any real square matrix A may be decomposed as A = QR where Q is an orthogonal matrix
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/// (its columns are orthogonal unit vectors meaning QTQ = I) and R is an upper triangular matrix
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/// (also called right triangular matrix).</para>
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/// </summary>
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/// <remarks>
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/// The computation of the QR decomposition is done at construction time by Householder transformation.
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/// </remarks>
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internal sealed class UserQR : QR
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{
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/// <summary>
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/// Initializes a new instance of the <see cref="UserQR"/> class. This object will compute the
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/// QR factorization when the constructor is called and cache it's factorization.
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/// </summary>
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/// <param name="matrix">The matrix to factor.</param>
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/// <param name="method">The QR factorization method to use.</param>
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/// <exception cref="ArgumentNullException">If <paramref name="matrix"/> is <c>null</c>.</exception>
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public static UserQR Create(Matrix<Complex> matrix, QRMethod method = QRMethod.Full)
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{
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if (matrix.RowCount < matrix.ColumnCount)
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{
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throw Matrix.DimensionsDontMatch<ArgumentException>(matrix);
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}
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Matrix<Complex> q;
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Matrix<Complex> r;
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var minmn = Math.Min(matrix.RowCount, matrix.ColumnCount);
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var u = new Complex[minmn][];
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if (method == QRMethod.Full)
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{
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r = matrix.Clone();
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q = Matrix<Complex>.Build.SameAs(matrix, matrix.RowCount, matrix.RowCount, fullyMutable: true);
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for (var i = 0; i < matrix.RowCount; i++)
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{
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q.At(i, i, 1.0f);
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}
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for (var i = 0; i < minmn; i++)
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{
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u[i] = GenerateColumn(r, i, i);
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ComputeQR(u[i], r, i, matrix.RowCount, i + 1, matrix.ColumnCount, Control.MaxDegreeOfParallelism);
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}
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for (var i = minmn - 1; i >= 0; i--)
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{
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ComputeQR(u[i], q, i, matrix.RowCount, i, matrix.RowCount, Control.MaxDegreeOfParallelism);
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}
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}
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else
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{
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q = matrix.Clone();
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for (var i = 0; i < minmn; i++)
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{
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u[i] = GenerateColumn(q, i, i);
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ComputeQR(u[i], q, i, matrix.RowCount, i + 1, matrix.ColumnCount, Control.MaxDegreeOfParallelism);
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}
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r = q.SubMatrix(0, matrix.ColumnCount, 0, matrix.ColumnCount);
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q.Clear();
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for (var i = 0; i < matrix.ColumnCount; i++)
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{
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q.At(i, i, 1.0f);
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}
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for (var i = minmn - 1; i >= 0; i--)
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{
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ComputeQR(u[i], q, i, matrix.RowCount, i, matrix.ColumnCount, Control.MaxDegreeOfParallelism);
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}
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}
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return new UserQR(q, r, method);
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}
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UserQR(Matrix<Complex> q, Matrix<Complex> rFull, QRMethod method)
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: base(q, rFull, method)
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{
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}
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/// <summary>
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/// Generate column from initial matrix to work array
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/// </summary>
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/// <param name="a">Initial matrix</param>
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/// <param name="row">The first row</param>
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/// <param name="column">Column index</param>
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/// <returns>Generated vector</returns>
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static Complex[] GenerateColumn(Matrix<Complex> a, int row, int column)
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{
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var ru = a.RowCount - row;
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var u = new Complex[ru];
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for (var i = row; i < a.RowCount; i++)
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{
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u[i - row] = a.At(i, column);
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a.At(i, column, 0.0);
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}
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var norm = u.Aggregate(Complex.Zero, (current, t) => current + (t.Magnitude*t.Magnitude));
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norm = norm.SquareRoot();
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if (row == a.RowCount - 1 || norm.Magnitude == 0)
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{
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a.At(row, column, -u[0]);
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u[0] = Constants.Sqrt2;
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return u;
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}
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if (u[0].Magnitude != 0.0)
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{
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norm = norm.Magnitude*(u[0]/u[0].Magnitude);
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}
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a.At(row, column, -norm);
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for (var i = 0; i < ru; i++)
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{
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u[i] /= norm;
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}
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u[0] += 1.0;
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var s = (1.0/u[0]).SquareRoot();
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for (var i = 0; i < ru; i++)
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{
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u[i] = u[i].Conjugate()*s;
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}
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return u;
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}
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/// <summary>
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/// Perform calculation of Q or R
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/// </summary>
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/// <param name="u">Work array</param>
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/// <param name="a">Q or R matrices</param>
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/// <param name="rowStart">The first row</param>
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/// <param name="rowDim">The last row</param>
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/// <param name="columnStart">The first column</param>
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/// <param name="columnDim">The last column</param>
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/// <param name="availableCores">Number of available CPUs</param>
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static void ComputeQR(Complex[] u, Matrix<Complex> a, int rowStart, int rowDim, int columnStart, int columnDim, int availableCores)
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{
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if (rowDim < rowStart || columnDim < columnStart)
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{
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return;
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}
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var tmpColCount = columnDim - columnStart;
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if ((availableCores > 1) && (tmpColCount > 200))
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{
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var tmpSplit = columnStart + (tmpColCount/2);
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var tmpCores = availableCores/2;
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CommonParallel.Invoke(
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() => ComputeQR(u, a, rowStart, rowDim, columnStart, tmpSplit, tmpCores),
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() => ComputeQR(u, a, rowStart, rowDim, tmpSplit, columnDim, tmpCores));
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}
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else
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{
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for (var j = columnStart; j < columnDim; j++)
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{
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var scale = Complex.Zero;
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for (var i = rowStart; i < rowDim; i++)
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{
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scale += u[i - rowStart]*a.At(i, j);
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}
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for (var i = rowStart; i < rowDim; i++)
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{
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a.At(i, j, a.At(i, j) - (u[i - rowStart].Conjugate()*scale));
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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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/// Solves a system of linear equations, <b>AX = B</b>, with A QR 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<Complex> input, Matrix<Complex> 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 (FullR.RowCount != 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 (FullR.ColumnCount != 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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var inputCopy = input.Clone();
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// Compute Y = transpose(Q)*B
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var column = new Complex[FullR.RowCount];
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for (var j = 0; j < input.ColumnCount; j++)
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{
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for (var k = 0; k < FullR.RowCount; k++)
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{
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column[k] = inputCopy.At(k, j);
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}
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for (var i = 0; i < FullR.RowCount; i++)
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{
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var s = Complex.Zero;
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for (var k = 0; k < FullR.RowCount; k++)
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{
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s += Q.At(k, i).Conjugate()*column[k];
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}
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inputCopy.At(i, j, s);
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}
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}
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// Solve R*X = Y;
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for (var k = FullR.ColumnCount - 1; k >= 0; k--)
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{
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for (var j = 0; j < input.ColumnCount; j++)
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{
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inputCopy.At(k, j, inputCopy.At(k, j)/FullR.At(k, k));
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}
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for (var i = 0; i < k; i++)
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{
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for (var j = 0; j < input.ColumnCount; j++)
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{
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inputCopy.At(i, j, inputCopy.At(i, j) - (inputCopy.At(k, j)*FullR.At(i, k)));
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}
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}
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}
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for (var i = 0; i < FullR.ColumnCount; i++)
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{
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for (var j = 0; j < inputCopy.ColumnCount; j++)
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{
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result.At(i, j, inputCopy.At(i, j));
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}
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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 QR 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<Complex> input, Vector<Complex> result)
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{
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// Ax=b where A is an m x n matrix
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// Check that b is a column vector with m entries
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if (FullR.RowCount != 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 (FullR.ColumnCount != result.Count)
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{
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throw Matrix.DimensionsDontMatch<ArgumentException>(FullR, result);
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}
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var inputCopy = input.Clone();
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// Compute Y = transpose(Q)*B
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var column = new Complex[FullR.RowCount];
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for (var k = 0; k < FullR.RowCount; k++)
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{
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column[k] = inputCopy[k];
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}
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for (var i = 0; i < FullR.RowCount; i++)
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{
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var s = Complex.Zero;
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for (var k = 0; k < FullR.RowCount; k++)
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{
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s += Q.At(k, i).Conjugate()*column[k];
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}
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inputCopy[i] = s;
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}
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// Solve R*X = Y;
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for (var k = FullR.ColumnCount - 1; k >= 0; k--)
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{
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inputCopy[k] /= FullR.At(k, k);
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for (var i = 0; i < k; i++)
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{
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inputCopy[i] -= inputCopy[k]*FullR.At(i, k);
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}
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}
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for (var i = 0; i < FullR.ColumnCount; i++)
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{
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result[i] = inputCopy[i];
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}
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}
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}
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}
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