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An Overview of Results on the Quadratic Convergence of Scaled Iterates by Diagonalization Methods
by
Josip Matejač
University of Zagreb
Coauthors: Vjeran Hari (University of Zagreb)
An overview of results on the quadratic convergence of scaled iterates by diagonalization methods will be presented. Importance of scaled iterates was first noticed by Demmel and Veselić in 1992 in their paper on accuracy of symmetric eigenvalue decompositions. In proving the quadratic rate of convergence of scaled iterates one uses the structure of scaled almost diagonal matrices. This structure was estimated by Hari and Drmač for Hermitian matrices and by Matejaš and Hari for general matrices in connection with the SVD. The first quadratic convergence estimates of scaled iterates for the symmetric Jacobi method are proved in the author's Ph. D. thesis in 1996. In the meantime, new results have been proved for Hermitian and J-Hermitian Jacobi methods in the general case of multiple eigenvalues. Also, general estimates for Kogbetliantz method applying to complex triangular matrices have been obtained. All these results have nice applications, since in the convergence bounds there appear relative gaps instead of absolute gaps. The estimates hold under natural assumptions which are somewhat more stringent than the standard ones, which is due to nonunitary transformations linking the consecutive scaled iterates.
Date received: March 15, 2000
Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caex-05.