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

where *L* is *m*-by-*m* lower triangular, *Q* is *n*-by-*n*
orthogonal (or unitary), *Q*_{1} consists of the first *m* rows of *Q*,
and *Q*_{2} the remaining *n*-*m* rows.
This factorization is computed by the routine xGELQF, and again *Q* is
represented as a product of elementary reflectors; xORGLQ
(or xUNGLQ in the complex case) can generate
all or part of *Q*, and xORMLQ (or xUNMLQ
) can pre- or post-multiply a given
matrix
by *Q* or *Q*^{T} (*Q*^{H} if *Q* is complex).

The *LQ* factorization of *A* is essentially the same as the *QR* factorization
of *A*^{T} (*A*^{H} 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 xTRTRS and xORMLQ.

** Next:** QR Factorization with Column
** Up:** Orthogonal Factorizations and Linear
** Previous:** QR Factorization
** Contents**
** Index**
*Susan Blackford*

*1999-10-01*