## Abstract

In this paper, strong relative perturbation bounds are developed for a number of linear algebra problems involving diagonally dominant matrices. The key point is to parameterize diagonally dominant matrices using their off-diagonal entries and diagonally dominant parts and to consider small relative componentwise perturbations of these parameters. This allows us to obtain new relative perturbation bounds for the inverse, the solution to linear systems, the symmetric indefinite eigenvalue problem, the singular value problem, and the nonsymmetric eigenvalue problem. These bounds are much stronger than traditional perturbation results, since they are independent of either the standard condition number or the magnitude of eigenvalues/singular values. Together with previously derived perturbation bounds for the LDU factorization and the symmetric positive definite eigenvalue problem, this paper presents a complete and detailed account of relative structured perturbation theory for diagonally dominant matrices.

Original language | English |
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Pages (from-to) | 1303-1328 |

Number of pages | 26 |

Journal | SIAM Journal on Matrix Analysis and Applications |

Volume | 35 |

Issue number | 4 |

DOIs | |

State | Published - 2014 |

### Bibliographical note

Publisher Copyright:© 2014 Society for Industrial and Applied Mathematics.

## Keywords

- Accurate computations
- Diagonally dominant matrices
- Diagonally dominant parts
- Eigenvalues
- Inverses
- Linear systems
- Relative perturbation theory
- Singular values

## ASJC Scopus subject areas

- Analysis