## Abstract

Many methods for the computation of selected eigenpairs of generalized eigenproblems for matrix pairs use a shift-and-invert technique. When applied to large-scale problems, this requires the solution of large linear systems of equations. This paper proposes an application of an Arnoldi method described in Voss (2004) [21] to the computation of a few extreme eigenpairs of a matrix pair. An advantage of this approach, when compared to methods that use the shift-and-invert technique, is that no large systems of equations have to be solved. We compare this approach to using a technique for simultaneously reducing a pair of large matrices to a pair of small matrices by a generalized Arnoldi process described in Li and Ye (2003) [12] and Hoffnung et al. (2006) [6]. The latter technique does not require the solution of large linear systems of equations either. Computed examples show the proposed method to yield approximations of the desired eigenpairs of higher accuracy when using about the same amount of computer storage space.

Original language | English |
---|---|

Pages (from-to) | 108-117 |

Number of pages | 10 |

Journal | Applied Numerical Mathematics |

Volume | 183 |

DOIs | |

State | Published - Jan 2023 |

### Bibliographical note

Publisher Copyright:© 2022 IMACS

### Funding

The authors would like to thank a referee for insightful comments. M.A. would like to thank the Deanship of Scientific Research at Qassim University for funding his research for this project. Research of L.R. was supported in part by NSF grant DMS-1720259 , and research of Q.Y. was supported in part by NSF grant DMS-1821144 .

Funders | Funder number |
---|---|

National Science Foundation Arctic Social Science Program | DMS-1720259, DMS-1821144 |

National Science Foundation Arctic Social Science Program |

## Keywords

- Generalized Krylov subspace method
- Generalized eigenvalue problem
- Large-scale problem

## ASJC Scopus subject areas

- Numerical Analysis
- Computational Mathematics
- Applied Mathematics