Bounds on the largest singular value of a matrix and the convergence of simultaneous and block-iterative algorithms for sparse linear systems
We obtain the following upper bounds for the eigenvalues of the matrix A†A. For any a in the interval [0, 2] let and ca and ra the maxima of the caj and rai, respectively. Then no eigenvalue of the matrix A†A exceeds the maximum of over all i, nor the maximum of over all j. Therefore, no eigenvalue...
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| Vydané v: | International transactions in operational research Ročník 16; číslo 4; s. 465 - 479 |
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| Hlavný autor: | |
| Médium: | Journal Article |
| Jazyk: | English |
| Vydavateľské údaje: |
Oxford, UK
Blackwell Publishing Ltd
01.07.2009
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| Predmet: | |
| ISSN: | 0969-6016, 1475-3995 |
| On-line prístup: | Získať plný text |
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| Shrnutí: | We obtain the following upper bounds for the eigenvalues of the matrix A†A. For any a in the interval [0, 2] let
and ca and ra the maxima of the caj and rai, respectively. Then no eigenvalue of the matrix A†A exceeds the maximum of
over all i, nor the maximum of
over all j. Therefore, no eigenvalue of A†A exceeds cara.
Using these bounds, it follows that, for the matrix G with entries
no eigenvalue of G†G exceeds one, provided that, for some a in the interval [0, 2], we have
and
Using this result, we obtain convergence theorems for several iterative algorithms for solving the problem Ax=b, including the CAV, BICAV, CARP1, SART, SIRT, and the block‐iterative DROP and SART methods. |
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| Bibliografia: | ark:/67375/WNG-4G16V2LG-T ArticleID:ITOR692 istex:81C69F834E25120A443565DE711048C62BCF0B8D SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 14 |
| ISSN: | 0969-6016 1475-3995 |
| DOI: | 10.1111/j.1475-3995.2009.00692.x |