The **generalized** *RQ***(GRQ) factorization** of an *m*-by-*n* matrix *A* and
a *p*-by-*n* matrix *B* is given by the pair of factorizations

where *Q* and *Z* are respectively *n*-by-*n* and *p*-by-*p* orthogonal
matrices (or unitary matrices if *A* and *B* are complex).
*R* has the form

or

where or is upper triangular. *T* has the form

or

where is upper triangular.

Note that if *B* is square and nonsingular, the GRQ factorization of
*A* and *B* implicitly gives the *RQ* factorization of the matrix :

without explicitly computing the matrix inverse or the product
.

The routine PxGGRQF computes the GRQ factorization
by computing first the *RQ* factorization of *A* and then
the *QR* factorization of .
The orthogonal (or unitary) matrices *Q* and *Z*
can be formed explicitly or
can be used just to multiply another given matrix in the same way as the
orthogonal (or unitary) matrix
in the *RQ* factorization
(see section 3.3.2).

Tue May 13 09:21:01 EDT 1997