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Title: GMRES vs. ideal GMRES
Authors: Toh, K.-C. 
Keywords: Arnoldi
Ideal Arnoldi
Issue Date: Jan-1997
Source: Toh, K.-C. (1997-01). GMRES vs. ideal GMRES. SIAM Journal on Matrix Analysis and Applications 18 (1) : 30-36. ScholarBank@NUS Repository.
Abstract: The GMRES algorithm minimizes ∥p(A)b∥ over polynomials p of degree n normalized at z = 0. The ideal GMRES problem is obtained if one considers minimization of ∥p(A)∥ instead. The ideal problem forms an upper bound for the worst-case true problem, where the GMRES norm ∥pb(A)b∥ is maximized over b. In work not yet published, Faber, Joubert, Knill, and Manteuffel have shown that this upper bound need not be attained, constructing a 4 × 4 example in which the ratio of the true to ideal GMRES norms is 0.9999. Here, we present a simpler 4 × 4 example in which the ratio approaches zero when a certain parameter tends to zero. The same example also leads to the same conclusion for Arnoldi vs. ideal Arnoldi norms.
Source Title: SIAM Journal on Matrix Analysis and Applications
ISSN: 08954798
Appears in Collections:Staff Publications

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