Singular
Definition
Assembly: Novacta.Analytics (in Novacta.Analytics.dll) Version: 2.1.0+428f3840cfab98dda567bb0ed350b302533e273a
public static class SingularValueDecompositionPublic NotInheritable Class SingularValueDecompositionpublic ref class SingularValueDecomposition abstract sealed[<AbstractClassAttribute>]
[<SealedAttribute>]
type SingularValueDecomposition = class end- Inheritance
- Object SingularValueDecomposition
Remarks
Let 
be a matrix having
rows and
columns.
The Singular Value Decomposition (SVD) of 
is a factorization
having the following form:
where 
returns the conjugate transpose of
, 
and 
are unitary complex
matrices having sizes 
and 
,
respectively, and
is a diagonal 
real matrix.
Matrix 
has
columns known as left singular vectors.
Matrix 
has diagonal entries known as the
singular values of 
. Finally,
matrix 
has rows that represent
the conjugate transposed right singular vectors
of 
.
Method Decompose(DoubleMatrix, DoubleMatrix, DoubleMatrix)
and its overloads compute matrices
, 
, and 
for double or complex matrices.
Method GetSingularValues(DoubleMatrix) and its overloads return the singular values involved in a decomposition, without computing the corresponding singular vectors.
Example
In the following example, the SVD of a matrix is computed.
using System;
using Novacta.Analytics.Advanced;
namespace Novacta.Analytics.CodeExamples.Advanced
{
public class SingularValueDecompositionExample0
{
public void Main()
{
// Create a matrix.
var data = new double[6] {
1, 8,
2, 2,
-3, 9
};
var matrix = DoubleMatrix.Dense(3, 2, data, StorageOrder.RowMajor);
Console.WriteLine("Matrix to be decomposed:");
Console.WriteLine(matrix);
// Compute its SVD.
var singularValues = SingularValueDecomposition.Decompose(
matrix,
out DoubleMatrix leftSingularVectors,
out DoubleMatrix conjugateTransposedRightSingularVectors);
Console.WriteLine();
Console.WriteLine("Singular values:");
Console.WriteLine(singularValues);
Console.WriteLine();
Console.WriteLine("Left singular vectors:");
Console.WriteLine(leftSingularVectors);
Console.WriteLine();
Console.WriteLine("Conjugate transposed right singular vectors:");
Console.WriteLine(conjugateTransposedRightSingularVectors);
Console.WriteLine();
Console.WriteLine("Matrix reconstruction:");
var v = singularValues;
var l = leftSingularVectors;
var rh = conjugateTransposedRightSingularVectors;
Console.WriteLine(l * v * rh);
}
}
}
// Executing method Main() produces the following output:
//
// Matrix to be decomposed:
// 1 8
// 2 2
// -3 9
//
//
//
// Singular values:
// 12.273817 0
// 0 3.51474275
// 0 0
//
//
//
// Left singular vectors:
// 0.639011869 0.531180587 0.556337142
// 0.144195105 0.627724866 -0.76496357
// 0.755560456 -0.569041894 -0.32453
//
//
//
// Conjugate transposed right singular vectors:
// -0.109116772 0.994028938
// 0.994028938 0.109116772
//
//
//
// Matrix reconstruction:
// 1 8
// 2 2
// -3 9
//
//Methods
| Decompose(ComplexMatrix, ComplexMatrix, ComplexMatrix) | Computes the Singular Value Decomposition of the specified matrix. |
| Decompose(DoubleMatrix, DoubleMatrix, DoubleMatrix) | Computes the Singular Value Decomposition of the specified matrix. |
| Decompose(ReadOnlyComplexMatrix, ComplexMatrix, ComplexMatrix) | Computes the Singular Value Decomposition of the specified matrix. |
| Decompose(ReadOnlyDoubleMatrix, DoubleMatrix, DoubleMatrix) | Computes the Singular Value Decomposition of the specified matrix. |
| GetSingularValues(ComplexMatrix) | Computes the singular values of the specified matrix. |
| GetSingularValues(DoubleMatrix) | Computes the singular values of the specified matrix. |
| GetSingularValues(ReadOnlyComplexMatrix) | Computes the singular values of the specified matrix. |
| GetSingularValues(ReadOnlyDoubleMatrix) | Computes the singular values of the specified matrix. |