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
where is upper triangular. T has the form
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 . Further generalizations of the GQR factorization can be found in .