Open AccessGMRES/CR and Arnoldi/Lanczos as Matrix Approximation Problems
Author(s) -
Anne Greenbaum,
Lloyd N. Trefethen
Publication year - 1994
Publication title -
siam journal on scientific computing
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.674
H-Index - 147
eISSN - 1095-7197
pISSN - 1064-8275
DOI - 10.1137/0915025
Subject(s) - generalized minimal residual method , mathematics , lanczos resampling , arnoldi iteration , eigenvalues and eigenvectors , ideal (ethics) , lanczos algorithm , matrix (chemical analysis) , iterative method , mathematical optimization , chemistry , philosophy , physics , epistemology , chromatography , quantum mechanics
TheGMRES and Arnoldi algorithms, which reduce to theCR andLanczos algorithms in the symmetric case, both minimize p(A)b over polynomials p of degree n. The difference is that p is normalized at z 0 for GMRES and at z x for Arnoldi. Analogous "ideal GMRES" and "ideal Arnoldi" problems are obtained if one removes b from the discussion and minimizes p(/l)II instead. Investigation of these true and ideal approximation problems gives insight into how fast GMRES converges and how the Arnoldi iteration locates eigenvalues.
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