Let *A* be a general real *m*-by-*n* matrix. The **singular value
decomposition (SVD)** of *A* is the factorization
, where *U* and *V* are orthogonal, and
, ,
with . If *A* is complex, then
its SVD is where *U* and *V* are unitary,
and is as before with real
diagonal elements.
The are called the **singular values**,
the first *r* columns of *V*
the **right singular vectors** and
the first *r* columns of *U*
the **left singular vectors**.

The SVD and symmetric eigendecompositions are entirely analogous,
so that any algorithm for one has a counterpart for the other. As soon
as the final version of the symmetric eigenvalue algorithm has been
developed, we will produce an SVD version. In the meantime, we plan
to release an SVD code based on serial QR iteration, where each processor
redundantly runs QR iteration on a bidiagonal matrix, but updates a
subset of the rows of the *U* and *V* in an embarrassingly parallel fashion.

Thu Jul 25 15:38:00 EDT 1996