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Perturbation theory and algorithmic development for the symmetric eigenvalue problem
by
Zlatko Drmač
Department of Mathematics, University of Zagreb, Bijenicka 30, 10000 Zagreb, Croatia
To find only a few eigenpairs of large symmetric matrix one usually constructs a sequence of low-dimensional subspaces and hopes to approximate the target eigenpairs from these subspaces. To be successful in designing efficient algorithms, one needs to know (i) how good is the current subspace, e.g. how accurate are the Ritz pairs obtained from that subspace; (ii) how to enrich the current subspace with directions close to the target vectors, thus ensuring fast convergence; (iii) how to compute accurate Rayleigh quotient matrix and the Ritz pairs in finite precision (floating-point) arithmetic. So, for instance, if the task is to approximate the lowest eigenvalue \lambda of positive definite matrix, we wish to know how many correct digits are in our current (floating-point) approximation \lambda+\delta\lambda. Sharp computable error bound assures good stopping criterion and accurate approximations. These issues clearly belong to perturbation theory. We present computable bounds for |\delta\lambda/\lambda| and show how perturbation theory sheds a new light on some well-known eigensolvers. We also discuss how to use these results to construct better subspaces, which could lead to substantial improvement of some well-known algorithms.
Date received: March 12, 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 # cadz-61.