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
A = RQ and B = ZTQ
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 or is upper triangular. T has the form
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 xGGRQF computes the GRQ factorization by first computing the RQ factorization of A and then the QR factorization of . 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 RQ factorization (see section 2.3.2).
The GRQ factorization can be used to solve the linear equality-constrained least squares problem (LSE) (see (2.2) and [page 567]GVL2). We use the GRQ factorization of B and A (note that B and A have swapped roles), written as
B = TQ and A = ZRQ
We write the linear equality constraints Bx = d as:
TQx = d
which we partition as:
Therefore is the solution of the upper triangular system
We partition this expression as:
where , which can be computed by xORMQR (or xUNMQR).
To solve the LSE problem, we set
which gives as the solution of the upper triangular system
Finally, the desired solution is given by
which can be computed by xORMRQ (or xUNMRQ).