The *QR* factorization with column pivoting does not enable us to compute
a *minimum norm* solution to a rank-deficient linear least squares problem
unless . However,
by applying further orthogonal (or unitary) transformations
from the right to the upper trapezoidal matrix
,
using the routine PxTZRZF, can be eliminated:

This gives the
**complete orthogonal
factorization**

from which the minimum norm solution can be obtained as

The matrix *Z* is not
formed explicitly but is represented as a product of elementary
reflectors,
as described in section 3.4.
Users need not be aware of the details of this representation,
because associated routines are provided to work with *Z*:
PxORMRZ (or
PxUNMRZ ) can pre- or post-multiply
a given matrix by *Z* or
( if complex).

Tue May 13 09:21:01 EDT 1997