next up previous contents index
Next: QR Factorization with Column Up: Orthogonal Factorizations and Linear Previous: QR Factorization

LQ Factorization

The LQ factorization   is given by
where L is m-by-m lower triangular, Q is n-by-n orthogonal (or unitary), tex2html_wrap_inline13270 consists of the first m rows of Q, and tex2html_wrap_inline13276 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 tex2html_wrap_inline13300 (tex2html_wrap_inline13302 if Q is complex).

The LQ factorization of A is essentially the same as the QR factorization of tex2html_wrap_inline12722 (tex2html_wrap_inline12724 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.        

Susan Blackford
Tue May 13 09:21:01 EDT 1997