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).