 
  
  
  
  
 
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 first m rows of Q, 
and  consists of the remaining n-m rows.
 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).
 if Q is complex).
The LQ factorization of A is essentially the same as the QR factorization
of  (
 ( if A is complex), since
 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.