The **singular value decomposition (SVD)** of an *m*-by-*n* matrix *A* is given by

where *U* and *V* are orthogonal (unitary)
and is an *m*-by-*n* diagonal matrix with real
diagonal elements, , such that

The are the **singular values** of *A* and the
first min(*m*,*n*) columns of *U* and *V*
are the **left** and **right singular vectors** of *A*.

The singular values and singular vectors satisfy

where and are the *i*th columns of *U* and *V*, respectively.

A single driver routine, PxGESVD , computes the ``economy size'' or
``thin'' singular value decomposition of a general nonsymmetric matrix
(see table 3.4). Thus, if *A* is *m*-by-*n* with
*m*>*n*, then only the first *n* columns of *U* are computed and is an
*n*-by-*n* matrix. For a detailed discussion of the ``thin'' singular
value decomposition, refer to [71, p. 72,].

Currently, only PSGESVD and PDGESVD are provided.

**Table 3.4:** Driver routines for standard eigenvalue and singular value problems

Tue May 13 09:21:01 EDT 1997