Block Krylov Subspace Methods for Functions of Matrices II: Modified Block FOM

Frommer, Andreas; Lund, Kathryn; Szyld, Daniel B.

doi:10.1137/19m1255847

Title: Block Krylov Subspace Methods for Functions of Matrices II: Modified Block FOM

Journal Article · Wed May 20 00:00:00 EDT 2020 · SIAM Journal on Matrix Analysis and Applications

DOI:https://doi.org/10.1137/19m1255847· OSTI ID:1832747

Frommer, Andreas ^[1]; Lund, Kathryn ^[2];

^[3]

Univ. of Wuppertal (Germany)
Charles Univ., Prague (Czech Republic)
Temple Univ., Philadelphia, PA (United States)

We analyze an expansion of the generalized block Krylov subspace framework of [Electron. Trans. Numer. Anal., 47 (2017), pp. 100--126]. This expansion allows the use of low-rank modifications of the matrix projected onto the block Krylov subspace and contains, as special cases, the block GMRES method and the new block Radau--Arnoldi method. Within this general setting, we present results that extend the interpolation property from the nonblock case to a matrix polynomial interpolation property for the block case, and we relate the eigenvalues of the projected matrix to the latent roots of these matrix polynomials. Some error bounds for these modified block FOM methods for solving linear systems are presented. We then show how cospatial residuals can be preserved in the case of families of shifted linear block systems. This result is used to derive computationally practical restarted algorithms for block Krylov approximations that compute the action of a matrix function on a set of several vectors simultaneously. Finally, we prove some error bounds and present numerical results showing that two modifications of FOM, the block harmonic and the block Radau--Arnoldi methods for matrix functions, can significantly improve the convergence behavior.

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Research Organization:: Temple Univ., Philadelphia, PA (United States)

Sponsoring Organization:: USDOE; National Science Foundation (NSF); Swiss National Science Foundation (SNSF); German Research Foundation (DFG)

Grant/Contract Number:: SC0016578

OSTI ID:: 1832747

Journal Information:: SIAM Journal on Matrix Analysis and Applications, Vol. 41, Issue 2; ISSN 0895-4798

Publisher:: Society for Industrial and Applied Mathematics (SIAM)Copyright Statement

Country of Publication:: United States

Language:: English

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97 MATHEMATICS AND COMPUTING
generalized block Krylov methods
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block GMRES
restarts
families of shifted linear systems
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matrix polynomials
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Title: Block Krylov Subspace Methods for Functions of Matrices II: Modified Block FOM

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