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GMRES/CR and Arnoldi/Lanczos as Matrix Approximation Problems

Author
Greenbaum, Anne; Trefethen, Lloyd N.
Abstract
The GMRES and Arnoldi algorithms, which reduce to the CR and Lanczos algorithms in the symmetric case, both minimize ||p(A)b|| over polynomials p of degree n. The difference is that p is nor- malized at z=0 for GMRES and at z=infinity for Arnoldi. Analogous "ideal GMRES" and "ideal Arnoldi" problems are obtained if one removes b from the discussion and minimizes ||p(A)|| instead. Investigation of these true and ideal approximation problems gives insight into how fast GMRES converges and how the Arnoldi iteration locates eigenvalues.
Date Issued
1992-06Publisher
Cornell University
Subject
theory center
Previously Published As
http://techreports.library.cornell.edu:8081/Dienst/UI/1.0/Display/cul.tc/92-096
Type
technical report