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Twisted factorization of a banded matrix

Vömel, Christof ; Slemons, Jason

In: BIT Numerical Mathematics, 2009, vol. 49, no. 2, p. 433-447

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    Title
    • Twisted factorization of a banded matrix
    Author
    • Vömel, Christof. Institute of Computational Science, ETH Zürich, Universitätsstraße 6, CAB, 8092, Zürich, Switzerland
    • Slemons, Jason. Department of Applied Mathematics, University of Washington, 98195, Seattle, WA, USA
    Document Type
    • Postprint
    OAI-PMH Identifier
    • oai:doc.rero.ch:320998
    Summary
    The twisted factorization of a tridiagonal matrix T plays an important role in inverse iteration as featured in the MRRR algorithm. The twisted structure simplifies the computation of the eigenvector approximation and can also improve the accuracy. A tridiagonal twisted factorization is given by T=M k Δ k N k where Δ k is diagonal, M k ,N k have unit diagonals, and the k-th column of M k and the k-th row of N k correspond to the k-th column and row of the identity, that is $M_{k}e_{k}=e_{k},\;e_{k}^{t}N_{k}=e_{k}^{t}$ . This paper gives a constructive proof for the existence of the twisted factorizations of a general banded matrix A. We show that for a given twist index k, there actually are two such factorizations. We also investigate the implications on inverse iteration and discuss the role of pivoting