// <copyright file="LU.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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/// <summary>
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/// <para>A class which encapsulates the functionality of an LU factorization.</para>
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/// <para>For a matrix A, the LU factorization is a pair of lower triangular matrix L and
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/// upper triangular matrix U so that A = L*U.</para>
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/// <para>In the Math.Net implementation we also store a set of pivot elements for increased
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/// numerical stability. The pivot elements encode a permutation matrix P such that P*A = L*U.</para>
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/// </summary>
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/// <remarks>
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/// The computation of the LU factorization is done at construction time.
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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 LU<T> : ISolver<T>
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where T : struct, IEquatable<T>, IFormattable
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{
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static readonly T One = BuilderInstance<T>.Matrix.One;
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readonly Lazy<Matrix<T>> _lazyL;
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readonly Lazy<Matrix<T>> _lazyU;
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readonly Lazy<Permutation> _lazyP;
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protected readonly Matrix<T> Factors;
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protected readonly int[] Pivots;
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protected LU(Matrix<T> factors, int[] pivots)
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{
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Factors = factors;
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Pivots = pivots;
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_lazyL = new Lazy<Matrix<T>>(ComputeL);
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_lazyU = new Lazy<Matrix<T>>(Factors.UpperTriangle);
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_lazyP = new Lazy<Permutation>(() => Permutation.FromInversions(Pivots));
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}
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Matrix<T> ComputeL()
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{
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var result = Factors.LowerTriangle();
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for (var i = 0; i < result.RowCount; i++)
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{
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result.At(i, i, One);
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}
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return result;
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}
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/// <summary>
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/// Gets the lower triangular factor.
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/// </summary>
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public Matrix<T> L => _lazyL.Value;
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/// <summary>
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/// Gets the upper triangular factor.
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/// </summary>
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public Matrix<T> U => _lazyU.Value;
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/// <summary>
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/// Gets the permutation applied to LU factorization.
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/// </summary>
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public Permutation P => _lazyP.Value;
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/// <summary>
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/// Gets the determinant of the matrix for which the LU factorization 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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/// Solves a system of linear equations, <b>AX = B</b>, with A LU 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(input, input.RowCount, 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 LU 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 LU 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(input, input.Count);
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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 LU 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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/// <summary>
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/// Returns the inverse of this matrix. The inverse is calculated using LU decomposition.
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/// </summary>
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/// <returns>The inverse of this matrix.</returns>
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public abstract Matrix<T> Inverse();
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}
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}
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