SVD Class

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

For a list of all members of this type, see SVD Members.

public class SVD

Thread Safety

Public static (Shared in Visual Basic) members of this type are safe for multithreaded operations. Instance members are not guaranteed to be thread-safe.

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

Requirements

Namespace: Imsl.Math

Assembly: ImslCS (in ImslCS.dll)

SVD Class

Thread Safety

Remarks

Requirements

See Also