 
  
  
  
  
 
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
. However,
by applying further orthogonal (or unitary) transformations 
from the right to the upper trapezoidal matrix 
 , 
using the routine PxTZRZF,
, 
using the routine PxTZRZF,  can be eliminated:
 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).
 if complex).