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A two-sided short-recurrence extended Krylov subspace method for nonsymmetric matrices and its relation to rational moment matching

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Abstract

We present an extended Krylov subspace analogue of the two-sided Lanczos method, i.e., a method which, given a nonsingular matrix A and vectors b, c with \(\left \langle {{\mathbf {b}},{\mathbf {c}}}\right \rangle \neq 0\), constructs bi-orthonormal bases of the extended Krylov subspaces \({\mathcal {E}}_{m}(A,{\mathbf {b}})\) and \({\mathcal {E}}_{m}(A^{T}\!,{\mathbf {c}})\) via short recurrences. We investigate the connection of the proposed method to rational moment matching for bilinear forms c T f(A)b, similar to known results connecting the two-sided Lanczos method to moment matching. Numerical experiments demonstrate the quality of the resulting approximations and the numerical behavior of the new extended Krylov subspace method.

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Authors and Affiliations

  1. School of Mathematics and Natural Sciences, Bergische Universität Wuppertal, 42097, Wuppertal, Germany

    Marcel Schweitzer

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  1. Marcel Schweitzer
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Correspondence to Marcel Schweitzer.

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This work was supported by Deutsche Forschungsgemeinschaft through Collaborative Research Centre SFB TRR55 “Hadron Physics from Lattice QCD”.

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Schweitzer, M. A two-sided short-recurrence extended Krylov subspace method for nonsymmetric matrices and its relation to rational moment matching. Numer Algor 76, 1–31 (2017). https://doi.org/10.1007/s11075-016-0239-z

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