Which eigenvalues are found by the Lanczos method? (Q2706258)
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| Language | Label | Description | Also known as |
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| English | Which eigenvalues are found by the Lanczos method? |
scientific article |
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19 March 2001
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Ritz values
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equilibrium distribution
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Lanczos iteration
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eigenvalues
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symmetric matrices
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potential-theoretic characterization
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extremal problem
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Which eigenvalues are found by the Lanczos method? (English)
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\textit{L. N. Trefethen} and \textit{D. B. Bau} [Numerical linear algebra (1997; Zbl 0874.65013)] stated the thumb rule that the Lanczos iteration tends to converge to eigenvalues of real symmetric matrices that lie in regions of ``too little charge'' for an equilibrium distribution, so that outliers are well approximated, whereas eigenvalues in the bulk of the spectrum are poorly approximated.NEWLINENEWLINENEWLINEThe paper has the goal to provide a quantitative version of this rule, asymptotically for matrix size \(N\) and number of iterations \(n\). It uses results of \textit{E. A. Rakhmanov} [Sb. Math. 187, No. 8, 1213-1228 (1996; Zbl 0873.42014)] to obtain a potential-theoretic characterization of the problem as an extremal problem.
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