 
  
  
  
  
 
  
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
 is upper triangular. T has the form

or

where  or
 or  is upper triangular.
 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
 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 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].