SVD Class

SVD Class

Singular Value Decomposition (SVD) of a rectangular matrix of type double.

Inheritance Hierarchy

SystemObject
Imsl.MathSVD

Namespace: Imsl.Math
Assembly: ImslCS (in ImslCS.dll) Version: 6.5.2.0

Syntax

C++

Copy

[SerializableAttribute]
public class SVD

<SerializableAttribute>
Public Class SVD

[SerializableAttribute]
public ref class SVD

[<SerializableAttribute>]
type SVD =  class end

The SVD type exposes the following members.

Constructors

	Name	Description
	SVD(Double)	Construct the singular value decomposition of a rectangular matrix with default tolerance.
	SVD(Double, Double)	Construct the singular value decomposition of a rectangular matrix with a given tolerance.

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Methods

	Name	Description
	Equals	Determines whether the specified object is equal to the current object. (Inherited from Object.)
	Finalize	Allows an object to try to free resources and perform other cleanup operations before it is reclaimed by garbage collection. (Inherited from Object.)
	GetHashCode	Serves as a hash function for a particular type. (Inherited from Object.)
	GetS	Returns the singular values.
	GetType	Gets the Type of the current instance. (Inherited from Object.)
	GetU	Returns the left singular vectors.
	GetV	Returns the right singular vectors.
	Inverse	Compute the Moore-Penrose generalized inverse of a real matrix.
	MemberwiseClone	Creates a shallow copy of the current Object. (Inherited from Object.)
	ToString	Returns a string that represents the current object. (Inherited from Object.)

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Properties

	Name	Description
	Info	Returns the index of the first singular value for which the algorithm converged.
	NumberOfProcessors	Perform the parallel calculations with the maximum possible number of processors set to NumberOfProcessors.
	Rank	Returns the rank of the matrix used to construct this instance.

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Remarks

SVD is based on the LINPACK routine SSVDC; see Dongarra et al. (1979).

Let n be the number of rows in A and let p be the number of columns in A. For any

n x p matrix A, there exists an n x n orthogonal matrix U and a p x p orthogonal matrix V such that

$U^T A V = \left\{ \begin{array}{cl} \left[ \begin{array}{l} \Sigma \\ 0 \end{array} \right] & \mbox{if $n \ge p$} \\ \left[ \Sigma \,\, 0 \right] & \mbox{if $n \le p$} \end{array} \right.$

where $\Sigma = {\rm diag}(\sigma_1, \ldots, \sigma_m)$ , and $m = \min(n, p)$ . The scalars $\sigma_1 \geq \sigma_2 \geq \ldots \geq \sigma_m \geq 0$ are called the singular values of A. The columns of U are called the left singular vectors of A. The columns of V are called the right singular vectors of A.

The estimated rank of A is the number of $\sigma_k$ that is larger than a tolerance $\eta$ . If $\tau$ is the parameter tol in the program, then

$\eta = \left\{ \begin{array}{cl} \tau \hfill & \mbox {if $\tau \gt 0$} \\ {\left| \tau \right|\left\| A \right\|_\infty } \hfill & \mbox {if $\tau \lt 0$} \end{array} \right.$

The Moore-Penrose generalized inverse of the matrix is computed by partitioning the matrices U, V and $\Sigma$ as , and $\Sigma_1 = {\rm diag}(\sigma_1,\ldots,\sigma_k)$ where the "1" matrices are k by k. The Moore-Penrose generalized inverse is $V_1 \Sigma_1^{-1} U_1^T$ .

Reference

Imsl.Math Namespace

Other Resources

Example