// <copyright file="Svd.cs" company="Math.NET">
|
// Math.NET Numerics, part of the Math.NET Project
|
// http://numerics.mathdotnet.com
|
// http://github.com/mathnet/mathnet-numerics
|
//
|
// Copyright (c) 2009-2015 Math.NET
|
//
|
// Permission is hereby granted, free of charge, to any person
|
// obtaining a copy of this software and associated documentation
|
// files (the "Software"), to deal in the Software without
|
// restriction, including without limitation the rights to use,
|
// copy, modify, merge, publish, distribute, sublicense, and/or sell
|
// copies of the Software, and to permit persons to whom the
|
// Software is furnished to do so, subject to the following
|
// conditions:
|
//
|
// The above copyright notice and this permission notice shall be
|
// included in all copies or substantial portions of the Software.
|
//
|
// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
|
// EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES
|
// OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
|
// NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT
|
// HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY,
|
// WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
|
// FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
|
// OTHER DEALINGS IN THE SOFTWARE.
|
// </copyright>
|
|
using System;
|
using System.Linq;
|
using IStation.Numerics.LinearAlgebra.Factorization;
|
|
namespace IStation.Numerics.LinearAlgebra.Complex.Factorization
|
{
|
using Complex = System.Numerics.Complex;
|
|
/// <summary>
|
/// <para>A class which encapsulates the functionality of the singular value decomposition (SVD).</para>
|
/// <para>Suppose M is an m-by-n matrix whose entries are real numbers.
|
/// Then there exists a factorization of the form M = UΣVT where:
|
/// - U is an m-by-m unitary matrix;
|
/// - Σ is m-by-n diagonal matrix with nonnegative real numbers on the diagonal;
|
/// - VT denotes transpose of V, an n-by-n unitary matrix;
|
/// Such a factorization is called a singular-value decomposition of M. A common convention is to order the diagonal
|
/// entries Σ(i,i) in descending order. In this case, the diagonal matrix Σ is uniquely determined
|
/// by M (though the matrices U and V are not). The diagonal entries of Σ are known as the singular values of M.</para>
|
/// </summary>
|
/// <remarks>
|
/// The computation of the singular value decomposition is done at construction time.
|
/// </remarks>
|
internal abstract class Svd : Svd<Complex>
|
{
|
protected Svd(Vector<Complex> s, Matrix<Complex> u, Matrix<Complex> vt, bool vectorsComputed)
|
: base(s, u, vt, vectorsComputed)
|
{
|
}
|
|
/// <summary>
|
/// Gets the effective numerical matrix rank.
|
/// </summary>
|
/// <value>The number of non-negligible singular values.</value>
|
public override int Rank
|
{
|
get
|
{
|
double tolerance = Precision.EpsilonOf(S.AbsoluteMaximum().Magnitude)*Math.Max(U.RowCount, VT.RowCount);
|
return S.Count(t => t.Magnitude > tolerance);
|
}
|
}
|
|
/// <summary>
|
/// Gets the two norm of the <see cref="Matrix{T}"/>.
|
/// </summary>
|
/// <returns>The 2-norm of the <see cref="Matrix{T}"/>.</returns>
|
public override double L2Norm => S[0].Magnitude;
|
|
/// <summary>
|
/// Gets the condition number <b>max(S) / min(S)</b>
|
/// </summary>
|
/// <returns>The condition number.</returns>
|
public override Complex ConditionNumber
|
{
|
get
|
{
|
var tmp = Math.Min(U.RowCount, VT.ColumnCount) - 1;
|
return S[0].Magnitude / S[tmp].Magnitude;
|
}
|
}
|
|
/// <summary>
|
/// Gets the determinant of the square matrix for which the SVD was computed.
|
/// </summary>
|
public override Complex Determinant
|
{
|
get
|
{
|
if (U.RowCount != VT.ColumnCount)
|
{
|
throw new ArgumentException("Matrix must be square.");
|
}
|
|
var det = Complex.One;
|
foreach (var value in S)
|
{
|
det *= value;
|
if (value.Magnitude.AlmostEqual(0.0))
|
{
|
return 0;
|
}
|
}
|
|
return det.Magnitude;
|
}
|
}
|
}
|
}
|
