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Mathematics of Computation

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Accurate inverses for computing eigenvalues of extremely ill-conditioned matrices and differential operators

Author: Qiang Ye
Journal: Math. Comp. 87 (2018), 237-259
MSC (2010): Primary 65F15, 65F35, 65N06, 65N25
Published electronically: May 11, 2017
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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.

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Additional Information

Qiang Ye
Affiliation: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506

Keywords: Eigenvalue, ill-conditioned matrix, accuracy, Lanczos method, differential eigenvalue problem, biharmonic operator
Received by editor(s): December 28, 2015
Received by editor(s) in revised form: June 14, 2016, and August 24, 2016
Published electronically: May 11, 2017
Additional Notes: This research was supported in part by NSF Grants DMS-1317424, DMS-1318633 and DMS-1620082
Article copyright: © Copyright 2017 American Mathematical Society

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