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A Flexible Extended Krylov Subspace Method for Approximating Markov Functions of Matrices

Author

Listed:
  • Shengjie Xu

    (School of Mathematical and Statistical Sciences, Clemson University, O-110 Martin Hall, Box 340975, Clemson, SC 29634, USA
    These authors contributed equally to this work.)

  • Fei Xue

    (School of Mathematical and Statistical Sciences, Clemson University, O-110 Martin Hall, Box 340975, Clemson, SC 29634, USA
    These authors contributed equally to this work.)

Abstract

A flexible extended Krylov subspace method ( F -EKSM) is considered for numerical approximation of the action of a matrix function f ( A ) to a vector b , where the function f is of Markov type. F -EKSM has the same framework as the extended Krylov subspace method (EKSM), replacing the zero pole in EKSM with a properly chosen fixed nonzero pole. For symmetric positive definite matrices, the optimal fixed pole is derived for F -EKSM to achieve the lowest possible upper bound on the asymptotic convergence factor, which is lower than that of EKSM. The analysis is based on properties of Faber polynomials of A and ( I − A / s ) − 1 . For large and sparse matrices that can be handled efficiently by LU factorizations, numerical experiments show that F -EKSM and a variant of RKSM based on a small number of fixed poles outperform EKSM in both storage and runtime, and usually have advantages over adaptive RKSM in runtime.

Suggested Citation

  • Shengjie Xu & Fei Xue, 2023. "A Flexible Extended Krylov Subspace Method for Approximating Markov Functions of Matrices," Mathematics, MDPI, vol. 11(20), pages 1-29, October.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:20:p:4341-:d:1262931
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