TY - JOUR

T1 - Optimal perturbation bounds for the Hermitian eigenvalue problem

AU - Barlow, Jesse L.

AU - Slapničar, Ivan

N1 - Funding Information:
∗ Corresponding author. E-mail addresses: barlow@cse.psu.edu (J.L. Barlow), ivan.slapnicar@fesb.hr (I. Slapnicˇar). 1 Research supported by the National Science Foundation under grants no. CCR-9424435 and no. CCR-9732081. 2 Research supported by the Croatian Ministry of Science and Technology under grant no. 037012. Part of this work was done while the author was visiting the Department of Computer Science and Engineering, The Pennsylvania State University, University Park, PA.

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.

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

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

M3 - Conference article

AN - SCOPUS:0346042887

SN - 0024-3795

VL - 309

SP - 19

EP - 43

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

IS - 1-3

T2 - The International Workshop on Accurate Solutions of Eigenvalue Problems

Y2 - 20 July 1998 through 23 July 1998

ER -