Extended rank reduction formulas containing Wedderburn and Abaffy-Broyden-Spedicato rank reducing processes

摘要

The Wedderburn rank reduction formula and the Abaffy-Broyden-Spedicato (ABS) algorithms are powerful methods for developing matrix factorizations and many fundamental numerical linear algebra processes such as Gram-Schmidt, conjugate direction and Lanczos methods. We present a rank reduction formula for transforming the rows and columns of A, extending the Wedderburn rank reduction formula and the ABS approach. By repeatedly applying the formula to reduce the rank, an extended rank reducing process is derived. The biconjugation process associated with the Wedderburn rank reduction process and the scaled extended ABS class of algorithms are shown to be in our proposed rank reducing process, while the process is more general to produce several other effective reduction algorithms to compute various structured factorizations. The process provides a general finite iterative approach for constructing factorizations of A and A~T under a common framework of a general decomposition V ~TAP=Ω. We also show that the biconjugation process associated with the Wedderburn rank reduction process can be derived from the scaled ABS class of algorithms applied to A or A~T. Finally, we provide a list of some well-known reduction procedures as special cases of our extended rank reducing process. The approach is general enough to produce various structured decompositions as well.