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Generalized QR Factorization


   The generalized QR (GQR) factorization of an n-by-m matrix A and an n-by-p matrix B is given by the pair of factorizations
where Q and Z are respectively n-by-n and p-by-p orthogonal matrices (or unitary matrices if A and B are complex). R has the form
where tex2html_wrap_inline13492 is upper triangular. T has the form
where tex2html_wrap_inline13706 or tex2html_wrap_inline13708 is upper triangular.

Note that if B is square and nonsingular, the GQR factorization of A and B implicitly gives the QR factorization of the matrix tex2html_wrap_inline13718:
without explicitly computing the matrix inverse tex2html_wrap_inline13720 or the product tex2html_wrap_inline13718.

The routine PxGGQRF computes the GQR  factorization by     computing first the QR factorization of A and then the RQ factorization of tex2html_wrap_inline13730. The orthogonal (or unitary) matrices Q and Z can be formed explicitly or can be used just to multiply another given matrix in the same way as the orthogonal (or unitary) matrix in the QR factorization (see section 3.3.2).

The GQR factorization was introduced in [73, 100]. The implementation of the GQR factorization here follows that in [5]. Further generalizations of the GQR  factorization can be found in [36].

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