Abstract

Given a limited amount of memory and a target accuracy, we propose and compare several polynomial Krylov methods for the approximation of $f(A){b}$, the action of a Stieltjes matrix function of a large Hermitian matrix on a vector. Using new error bounds and estimates, as well as existing results, we derive predictions of the practical performance of the methods and rank them accordingly. As byproducts, we derive new results on inexact Krylov iterations for matrix functions in order to allow for a fair comparison of rational Krylov methods with polynomial inner solves.

Keywords

  1. matrix function
  2. Krylov method
  3. shift-and-invert method
  4. restarted method
  5. Stieltjes function
  6. inexact Krylov method
  7. outer-inner iteration

MSC codes

  1. 65F60
  2. 65F50
  3. 65F10
  4. 65F30

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Information & Authors

Information

Published In

cover image SIAM Journal on Matrix Analysis and Applications
SIAM Journal on Matrix Analysis and Applications
Pages: 83 - 107
ISSN (online): 1095-7162

History

Submitted: 7 July 2020
Accepted: 2 November 2020
Published online: 21 January 2021

Keywords

  1. matrix function
  2. Krylov method
  3. shift-and-invert method
  4. restarted method
  5. Stieltjes function
  6. inexact Krylov method
  7. outer-inner iteration

MSC codes

  1. 65F60
  2. 65F50
  3. 65F10
  4. 65F30

Authors

Affiliations

Funding Information

Engineering and Physical Sciences Research Council https://doi.org/10.13039/501100000266 : EP/N510129/1

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