### Abstract

There is now a large literature on structured perturbation bounds for eigenvalue problems of the formvboxHx=λMx, where H and M are Hermitian. These results give relative error bounds on the ith eigenvalue, λ_{i}, of the form|λ_{i}-λ̃_{i}||λ_{i}|,and bound the error in the ith eigenvector in terms of the relative gap,min_{j≠i}|λ_{i}-λ _{j}||λ_{i}λ_{j}|^{1/2}. In general, this theory usually restricts H to be nonsingular and M to be positive definite. We relax this restriction by allowing H to be singular. For our results on eigenvalues weallow M to be positive semi-definite and for a few results we allow it to be more general. For these problems, for eigenvalues that are not zero or infinity under perturbation, it is possible to obtain local relative error bounds. Thus, a wider class of problems may be characterized by this theory. Although it is impossible to give meaningful relative error bounds on eigenvalues that are not bounded away from zero, we show that the error in the subspace associated with those eigenvalues can be characterized meaningfully.

Original language | English (US) |
---|---|

Pages (from-to) | 19-43 |

Number of pages | 25 |

Journal | Linear Algebra and Its Applications |

Volume | 309 |

Issue number | 1-3 |

DOIs | |

State | Published - Apr 15 2000 |

Event | The International Workshop on Accurate Solutions of Eigenvalue Problems - University Park, PA, United States Duration: Jul 20 1998 → Jul 23 1998 |

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### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory
- Numerical Analysis
- Geometry and Topology
- Discrete Mathematics and Combinatorics

### Cite this

*Linear Algebra and Its Applications*,

*309*(1-3), 19-43. https://doi.org/10.1016/S0024-3795(99)00268-2

}

*Linear Algebra and Its Applications*, vol. 309, no. 1-3, pp. 19-43. https://doi.org/10.1016/S0024-3795(99)00268-2

**Optimal perturbation bounds for the Hermitian eigenvalue problem.** / Barlow, Jesse Louis; Slapničar, Ivan.

Research output: Contribution to journal › Conference article

TY - JOUR

T1 - Optimal perturbation bounds for the Hermitian eigenvalue problem

AU - Barlow, Jesse Louis

AU - Slapničar, Ivan

PY - 2000/4/15

Y1 - 2000/4/15

N2 - There is now a large literature on structured perturbation bounds for eigenvalue problems of the formvboxHx=λMx, where H and M are Hermitian. These results give relative error bounds on the ith eigenvalue, λi, of the form|λi-λ̃i||λi|,and bound the error in the ith eigenvector in terms of the relative gap,minj≠i|λi-λ j||λiλj|1/2. In general, this theory usually restricts H to be nonsingular and M to be positive definite. We relax this restriction by allowing H to be singular. For our results on eigenvalues weallow M to be positive semi-definite and for a few results we allow it to be more general. For these problems, for eigenvalues that are not zero or infinity under perturbation, it is possible to obtain local relative error bounds. Thus, a wider class of problems may be characterized by this theory. Although it is impossible to give meaningful relative error bounds on eigenvalues that are not bounded away from zero, we show that the error in the subspace associated with those eigenvalues can be characterized meaningfully.

AB - There is now a large literature on structured perturbation bounds for eigenvalue problems of the formvboxHx=λMx, where H and M are Hermitian. These results give relative error bounds on the ith eigenvalue, λi, of the form|λi-λ̃i||λi|,and bound the error in the ith eigenvector in terms of the relative gap,minj≠i|λi-λ j||λiλj|1/2. In general, this theory usually restricts H to be nonsingular and M to be positive definite. We relax this restriction by allowing H to be singular. For our results on eigenvalues weallow M to be positive semi-definite and for a few results we allow it to be more general. For these problems, for eigenvalues that are not zero or infinity under perturbation, it is possible to obtain local relative error bounds. Thus, a wider class of problems may be characterized by this theory. Although it is impossible to give meaningful relative error bounds on eigenvalues that are not bounded away from zero, we show that the error in the subspace associated with those eigenvalues can be characterized meaningfully.

UR - http://www.scopus.com/inward/record.url?scp=0346042887&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0346042887&partnerID=8YFLogxK

U2 - 10.1016/S0024-3795(99)00268-2

DO - 10.1016/S0024-3795(99)00268-2

M3 - Conference article

VL - 309

SP - 19

EP - 43

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

IS - 1-3

ER -