// <copyright file="Svd.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) 2009-2015 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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// 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 the singular value decomposition (SVD).</para>
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/// <para>Suppose M is an m-by-n matrix whose entries are real numbers.
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/// Then there exists a factorization of the form M = UΣVT where:
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/// - U is an m-by-m unitary matrix;
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/// - Σ is m-by-n diagonal matrix with nonnegative real numbers on the diagonal;
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/// - VT denotes transpose of V, an n-by-n unitary matrix;
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/// Such a factorization is called a singular-value decomposition of M. A common convention is to order the diagonal
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/// entries Σ(i,i) in descending order. In this case, the diagonal matrix Σ is uniquely determined
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/// by M (though the matrices U and V are not). The diagonal entries of Σ are known as the singular values of M.</para>
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/// </summary>
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/// <remarks>
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/// The computation of the singular value decomposition 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 Svd<T> : ISolver<T>
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where T : struct, IEquatable<T>, IFormattable
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{
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readonly Lazy<Matrix<T>> _lazyW;
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/// <summary>Indicating whether U and VT matrices have been computed during SVD factorization.</summary>
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protected readonly bool VectorsComputed;
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protected Svd(Vector<T> s, Matrix<T> u, Matrix<T> vt, bool vectorsComputed)
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{
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S = s;
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U = u;
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VT = vt;
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VectorsComputed = vectorsComputed;
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_lazyW = new Lazy<Matrix<T>>(ComputeW);
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}
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Matrix<T> ComputeW()
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{
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var rows = U.RowCount;
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var columns = VT.ColumnCount;
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var result = Matrix<T>.Build.SameAs(U, rows, columns);
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for (var i = 0; i < rows; i++)
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{
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for (var j = 0; j < columns; j++)
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{
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if (i == j)
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{
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result.At(i, i, S[i]);
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}
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}
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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 singular values (Σ) of matrix in ascending value.
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/// </summary>
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public Vector<T> S { get; }
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/// <summary>
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/// Gets the left singular vectors (U - m-by-m unitary matrix)
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/// </summary>
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public Matrix<T> U { get; }
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/// <summary>
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/// Gets the transpose right singular vectors (transpose of V, an n-by-n unitary matrix)
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/// </summary>
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public Matrix<T> VT { get; }
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/// <summary>
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/// Returns the singular values as a diagonal <see cref="Matrix{T}"/>.
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/// </summary>
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/// <returns>The singular values as a diagonal <see cref="Matrix{T}"/>.</returns>
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public Matrix<T> W => _lazyW.Value;
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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 the two norm of the <see cref="Matrix{T}"/>.
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/// </summary>
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/// <returns>The 2-norm of the <see cref="Matrix{T}"/>.</returns>
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public abstract double L2Norm { get; }
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/// <summary>
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/// Gets the condition number <b>max(S) / min(S)</b>
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/// </summary>
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/// <returns>The condition number.</returns>
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public abstract T ConditionNumber { get; }
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/// <summary>
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/// Gets the determinant of the square matrix for which the SVD 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 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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/// <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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if (!VectorsComputed)
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{
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throw new InvalidOperationException("The singular vectors were not computed.");
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}
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var x = Matrix<T>.Build.SameAs(U, VT.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 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 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 SVD 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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if (!VectorsComputed)
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{
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throw new InvalidOperationException("The singular vectors were not computed.");
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}
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var x = Vector<T>.Build.SameAs(U, VT.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 SVD 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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