Error bounds for the Krylov subspace methods for computations of matrix exponentials

Hao Wang, Qiang Ye

Research output: Contribution to journalArticlepeer-review

14 Scopus citations


In this paper, we present new a posteriori and a priori error bounds for the Krylov subspace methods for computing e-τAv for a given τ > 0 and v ∈ ℂn, where A is a large sparse non-Hermitian matrix. The a priori error bounds relate the convergence to λmin (A+A ∗/2), λmax(A+A ∗/2) (the smallest and the largest eigenvalue of the Hermitian part of A), and |λmax(A-A ∗/2)| (the largest eigenvalue in absolute value of the skew-Hermitian part of A), which define a rectangular region enclosing the field of values of A. In particular, our bounds explain an observed convergence behavior where the error may first stagnate for a certain number of iterations before it starts to converge. The special case that A is skew-Hermitian is also considered. Numerical examples are given to demonstrate the theoretical bounds.

Original languageEnglish
Pages (from-to)155-187
Number of pages33
JournalSIAM Journal on Matrix Analysis and Applications
Issue number1
StatePublished - 2017

Bibliographical note

Publisher Copyright:
© 2017 Mitsubishi Electric Research Labs.


  • Arnoldi method
  • Krylov subspace method
  • Lanczos method
  • Matrix exponential

ASJC Scopus subject areas

  • Analysis


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