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 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 ( 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 Ax = 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.