Syntax

C#
[SerializableAttribute] public class SVD

Visual Basic (Declaration)
<SerializableAttribute> _ Public Class SVD

Visual C++
[SerializableAttribute] public ref class SVD

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 $V_1 \Sigma_1^{-1} U_1^T$ .

Inheritance Hierarchy

System..::.Object
Imsl.Math..::.SVD

Syntax

Remarks

Inheritance Hierarchy

See Also