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An Extended-Rational Arnoldi Method for Large Matrix Exponential Evaluations

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Abstract

The numerical computation of a matrix function such as \(\exp {(-tA)V}\), where A is an \(n\times n\) large and sparse matrix, V is an \(n \times p\) block with \(p\ll n\), and \(t>0\) arises in various applications including network analysis, the solution of time-dependent partial differential equations (PDE’s) and others. In this work, we propose the use of the global extended-rational Arnoldi method for computing approximations of such functions. The derived method projects the initial problem onto the global extended-rational Krylov subspace \(\mathcal {RK}^{e}_m(A,V)=\text {span}\{\prod \limits \nolimits _{i=1}^m(A+s_iI_n)^{-1}V,\ldots ,(A+s_1I_n)^{-1}V,V\) \(,AV, \ldots ,A^{m-1}V\}\) of a low dimension. An adaptive procedure of getting the shifts \(\{s_1,\ldots ,s_m\}\) during the algorithmic process is given and analyzed. Applications to the solution of time-dependent PDE’s and to network analysis are presented. Numerical examples are presented to show the performance of the global extended-rational Arnoldi process.

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Acknowledgements

We would like to thank the referees for their valuable remarks and suggestions allowing us to improve the quality of the paper.

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Correspondence to K. Jbilou.

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Bentbib, A.H., Ghomari, M.E. & Jbilou, K. An Extended-Rational Arnoldi Method for Large Matrix Exponential Evaluations. J Sci Comput 91, 36 (2022). https://doi.org/10.1007/s10915-022-01808-9

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  • DOI: https://doi.org/10.1007/s10915-022-01808-9

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