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GMRES vs. ideal GMRES

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\begin{abstract} \noindent The GMRES algorithm minimizes \normp(A)b over polynomials p of degree n normalized at z=0. The ideal GMRES problem is obtained if one considers minimization of \normp(A) instead. The ideal problem forms an upper bound for the worst-case true problem, where the GMRES norm \normpb(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. \end{abstract}

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1994-05

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Cornell University

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computer science; technical report

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http://techreports.library.cornell.edu:8081/Dienst/UI/1.0/Display/cul.cs/TR94-1472

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technical report

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