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Extended nonsymmetric global Lanczos method for matrix function approximation

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Abstract

Extended Krylov subspace methods are attractive methods for computing approximations of matrix functions and other problems producing large-scale matrices. In this work, we propose the extended nonsymmetric global Lanczos method for solving some matrix approximation problems. The derived algorithm uses short recursive relations to generate bi-orthonormal bases, with respect to the Frobenius inner product, of the corresponding extended Krylov subspaces \({K^{e}_{m}}(A,V)\) and \({K^{e}_{m}}(A^{T},W)\). Here, A is a large nonsymmetric matrix; V and \(W\in \mathbb {R}^{n\times s}\) are two blocks. New algebraic properties of the proposed method are developed and applications to approximation of both WTf(A)V and trace(WTf(A)V ) are given. Numerical examples are presented to show the performance of the extended nonsymmetric global Lanczos for these problems.

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Acknowledgments

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

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Authors and Affiliations

  1. Laboratory LAMAI, Faculty of Sciences and Technologies, Cadi Ayyad University, Marrakech, Morocco

    A. H. Bentbib & M. El Ghomari

  2. Laboratory LMPA, University ULCO, 50 Rue F. Buisson, Calais cedex, France

    K. Jbilou

  3. Laboratory CSEHS, University UM6P, Ben Guérir, Morocco

    K. Jbilou

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  1. A. H. Bentbib
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  2. M. El Ghomari
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  3. K. Jbilou
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Correspondence to M. El Ghomari.

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Bentbib, A.H., El Ghomari, M. & Jbilou, K. Extended nonsymmetric global Lanczos method for matrix function approximation. Numer Algor 84, 1459–1479 (2020). https://doi.org/10.1007/s11075-020-00896-8

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Mathematics Subject Classification (2010)