The *LQ***factorization**
is given by

where *L* is *m*-by-*m* lower triangular, *Q* is *n*-by-*n*
orthogonal (or unitary), consists of the first *m* rows of *Q*,
and consists of the remaining *n*-*m* rows.

This factorization is computed by the routine PxGELQF, and again *Q* is
represented as a product of elementary reflectors; PxORGLQ
(or PxUNGLQ in the complex case) can generate
all or part of *Q*, and PxORMLQ (or PxUNMLQ ) can pre- or post-multiply a given
matrix
by *Q* or ( if *Q* is complex).

The *LQ* factorization of *A* is essentially the same as the *QR* factorization
of ( if *A* is complex), since

The *LQ* factorization may be used to find a minimum norm solution of
an underdetermined system of linear equations *A x* = *b*, where *A* is
*m*-by-*n* with *m* < *n* and has rank *m*. The solution is given by

and may be computed by calls to PxTRTRS and PxORMLQ.

Tue May 13 09:21:01 EDT 1997