## Abstract

This paper is concerned with computations of a few smallest eigenvalues (in absolute value) of a large extremely ill-conditioned matrix. It is shown that a few smallest eigenvalues can be accurately computed for a diagonally dominant matrix or a product of diagonally dominant matrices by combining a standard iterative method with the accurate inversion algorithms that have been developed for such matrices. Applications to the finite difference discretization of differential operators are discussed. In particular, a new discretization is derived for the 1-dimensional biharmonic operator that can be written as a product of diagonally dominant matrices. Numerical examples are presented to demonstrate the accuracy achieved by the new algorithms.

Original language | English |
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Pages (from-to) | 237-259 |

Number of pages | 23 |

Journal | Mathematics of Computation |

Volume | 87 |

Issue number | 309 |

DOIs | |

State | Published - 2018 |

### Bibliographical note

Publisher Copyright:© 2017 American Mathematical Society.

### Funding

Received by the editor December 28, 2015 and, in revised form, June 14, 2016 and August 24, 2016. 2010 Mathematics Subject Classification. Primary 65F15, 65F35, 65N06, 65N25. Key words and phrases. Eigenvalue, ill-conditioned matrix, accuracy, Lanczos method, differential eigenvalue problem, biharmonic operator. This research was supported in part by NSF Grants DMS-1317424, DMS-1318633 and DMS-1620082.

Funders | Funder number |
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National Science Foundation Arctic Social Science Program | 1620082, DMS-1318633, DMS-1317424, DMS-1620082 |

Directorate for Mathematical and Physical Sciences | 1318633, 1317424 |

## Keywords

- Accuracy
- Biharmonic operator
- Differential eigenvalue problem
- Eigenvalue
- Ill-conditioned matrix
- Lanczos method

## ASJC Scopus subject areas

- Algebra and Number Theory
- Computational Mathematics
- Applied Mathematics