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
Note that if B is square and nonsingular, the GQR factorization
of A and B implicitly gives the QR factorization of the matrix B-1A:
The routine xGGQRF computes the GQR factorization by first computing the QR factorization of A and then the RQ factorization of QTB. The orthogonal (or unitary) matrices Q and Z can either be formed explicitly or just used to multiply another given matrix in the same way as the orthogonal (or unitary) matrix in the QR factorization (see section 2.4.2).
The GQR factorization was introduced in [60,84]. The implementation of the GQR factorization here follows . Further generalizations of the GQR factorization can be found in .
The GQR factorization can be used to solve the general (Gauss-Markov) linear
model problem (GLM) (see (2.3) and
[55, page 252]).
Using the GQR factorization of A and B, we rewrite the equation
d = A x + B y from (2.3) as
The GLM problem is solved by setting