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

or

where  is upper triangular.
  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
 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).
. 
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).