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Complete Orthogonal Factorization

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 tex2html_wrap_inline13551. However, by applying further orthogonal (or unitary) transformations  from the right to the upper trapezoidal matrix tex2html_wrap_inline13553, using the routine PxTZRZF, tex2html_wrap_inline13555 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 tex2html_wrap_inline13563 (tex2html_wrap_inline13565 if complex).

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