factorization
The generalized (GRQ) factorization of an 
-by-
matrix 
and
a 
-by- matrix 
is given by the pair of factorizations
where 
and 
are respectively -by- and -by- orthogonal
matrices (or unitary matrices if and are complex).
has the form
or
where 
or 
is upper triangular. 
has the form
or
where 
is upper triangular.
Note that if is square and nonsingular, the GRQ factorization of
and implicitly gives the factorization of the matrix 
:
without explicitly computing the matrix inverse 
or the product
.
The routine xGGRQF computes the GRQ factorization
by first computing the factorization of and then
the 
factorization of 
.
The orthogonal (or unitary) matrices and
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 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 and (note that and have
swapped roles), written as
We write the linear equality constraints 
as:
which we partition as:
Therefore 
is the solution of the upper triangular system
Furthermore,
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).