The **generalized** *QR***(GQR) factorization** of an *n*-by-*m* matrix *A* and
an *n*-by-*p* 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 is upper triangular. *T* has the form

or

where or is upper triangular.

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

without explicitly computing the matrix inverse or the product .

The routine PxGGQRF computes the GQR factorization by
computing first the *QR* factorization of *A* and then
the *RQ* 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 *QR* factorization
(see section 3.3.2).

The GQR factorization was introduced in [73, 100]. The implementation of the GQR factorization here follows that in [5]. Further generalizations of the GQR factorization can be found in [36].

Tue May 13 09:21:01 EDT 1997