## Abstract

In this paper, we present new a posteriori and a priori error bounds for the Krylov subspace methods for computing e-^{τA}v for a given τ > 0 and v ∈ ℂ^{n}, where A is a large sparse non-Hermitian matrix. The a priori error bounds relate the convergence to λ_{min} (A+A ∗/2), λ_{max}(A+A ∗/2) (the smallest and the largest eigenvalue of the Hermitian part of A), and |λ_{max}(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 language | English |
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Pages (from-to) | 155-187 |

Number of pages | 33 |

Journal | SIAM Journal on Matrix Analysis and Applications |

Volume | 38 |

Issue number | 1 |

DOIs | |

State | Published - 2017 |

### Bibliographical note

Funding Information:The work of the second author was supported in part by NSF under grants DMS-1317424, DMS-1318633, and DMS-1620082.

Publisher Copyright:

© 2017 Mitsubishi Electric Research Labs.

## Keywords

- Arnoldi method
- Krylov subspace method
- Lanczos method
- Matrix exponential

## ASJC Scopus subject areas

- Analysis