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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

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 *R*_{12} or *R*_{21} is upper triangular. *T* has the form

or

where *T*_{11} 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 *AB*^{-1}:

**
***A B*^{-1} = ( *R T*^{-1} ) *Z*^{T}

without explicitly computing the matrix inverse *B*^{-1} or the product
*AB*^{-1}.
The routine xGGRQF computes the GRQ factorization
by first computing the *RQ* factorization of *A* and then
the *QR* factorization of *BQ*^{T}.
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 2.4.2).

The GRQ factorization can be used to solve the linear
equality-constrained least squares problem (LSE) (see (2.2) and
[55, page 567]).
We use the GRQ factorization of *B* and *A* (note that *B* and *A* have
swapped roles), written as

We write the linear equality constraints *Bx* = *d* as:

**
***T Q x* = *d*

which we partition as:

Therefore *x*_{2} is the solution of the upper triangular system

**
***T*_{12} *x*_{2} = *d*

Furthermore,

We partition this expression as:

where
,
which
can be computed by xORMQR (or xUNMQR).
To solve the LSE problem, we set

**
***R*_{11} *x*_{1} + *R*_{12} *x*_{2} - *c*_{1} = 0

which gives *x*_{1} as the solution of the upper triangular system

**
***R*_{11} *x*_{1} = *c*_{1} - *R*_{12} *x*_{2}.

Finally, the desired solution is given by

which can be computed
by xORMRQ (or xUNMRQ).

** Next:** Symmetric Eigenproblems
** Up:** Generalized Orthogonal Factorizations and
** Previous:** Generalized QR Factorization
** Contents**
** Index**
*Susan Blackford*

*1999-10-01*