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SIAM J. on Scientific Computing / Volume 34 / Issue 1 / Methods and Algorithms for Scientific Computing

SIAM J. Sci. Comput. 34, pp. A485-A496 (12 pages)

Computing All or Some Eigenvalues of Symmetric $\mathcal{H}_{\ell}$-Matrices

Peter Benner and Thomas Mach

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We use a bisection method [B. Parlett, The Symmetric Eigenvalue Problem, Prentice-Hall, Englewood Cliffs, NJ, 1980, p. 51] to compute the eigenvalues of a symmetric $\mathcal{H}_{\ell}$-matrix $M$. The number of negative eigenvalues of $M-\mu I$ is computed via the LDL$^T$ factorization of $M-\mu I$. For dense matrices, the LDL$^T$ factorization is too expensive to yield an efficient eigenvalue algorithm in general, but not so for $\mathcal{H}_{\ell}$-matrices. In the special structure of $\mathcal{H}_{\ell}$-matrices there is an LDL$^T$ factorization with linear-polylogarithmic complexity. The bisection method requires only matrix-size independent many iterations to find an eigenvalue up to the desired accuracy, so that an eigenvalue can be found in linear-polylogarithmic time. For all $n$ eigenvalues, $\mathcal{O}(n^{2}(\log n )^{4}\log( \Vert {M}\Vert_{2}/\epsilon_{\text{ev}}))$ flops are needed to compute all eigenvalues with an accuracy $\epsilon_{\text{ev}}$. It is also possible to compute only eigenvalues in a specific interval or the $j$th smallest one. Numerical experiments demonstrate the efficiency of the algorithm, in particular for the case when some interior eigenvalues are required. (A corrected PDF is attached to this article.)

© 2012 Society for Industrial and Applied Mathematics

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KEYWORDS

AMS Subject Headings

65F15, 65F50, 15A18

PUBLICATION DATA

ISSN

1064-8275 (print)  
1095-7197 (online)

Publisher

$abstract.society.value is a Member of CrossRef Society for Industrial and Applied Mathematics

ARTICLE DATA

History
Received November 18, 2010
Accepted October 26, 2011
Published online February 14, 2012
Digital Object Identifier
http://dx.doi.org/10.1137/100815323

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