A black-box rational Arnoldi variant for Cauchy–Stieltjes matrix functions

Rational Arnoldi is a powerful method for approximating functions of large sparse matrices times a vector. The selection of asymptotically optimal parameters for this method is crucial for its fast convergence. We present and investigate a novel strategy for the automated parameter selection when th...

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Bibliographic Details
Published in:BIT Numerical Mathematics 2013-09, Vol.53 (3), p.595-616
Main Authors: Güttel, Stefan, Knizhnerman, Leonid
Format: Article
Language:English
Subjects:
Computational Mathematics and Numerical Analysis
Mathematics
Mathematics and Statistics
Numeric Computing
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Summary:Rational Arnoldi is a powerful method for approximating functions of large sparse matrices times a vector. The selection of asymptotically optimal parameters for this method is crucial for its fast convergence. We present and investigate a novel strategy for the automated parameter selection when the function to be approximated is of Cauchy–Stieltjes (or Markov) type, such as the matrix square root or the logarithm. The performance of this approach is demonstrated by numerical examples involving symmetric and nonsymmetric matrices. These examples suggest that our black-box method performs at least as well, and typically better, as the standard rational Arnoldi method with parameters being manually optimized for a given matrix.
ISSN:0006-3835
1572-9125
DOI:10.1007/s10543-013-0420-x