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Accurate computation of the smallest eigenvalue of a diagonally dominant $M$-matrix


Authors: Attahiru Sule Alfa, Jungong Xue and Qiang Ye
Journal: Math. Comp. 71 (2002), 217-236
MSC (2000): Primary 65F18, 65F05
DOI: https://doi.org/10.1090/S0025-5718-01-01325-4
Published electronically: May 14, 2001
MathSciNet review: 1862996
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Abstract:

If each off-diagonal entry and the sum of each row of a diagonally dominant $M$-matrix are known to certain relative accuracy, then its smallest eigenvalue and the entries of its inverse are known to the same order relative accuracy independent of any condition numbers. In this paper, we devise algorithms that compute these quantities with relative errors in the magnitude of the machine precision. Rounding error analysis and numerical examples are presented to demonstrate the numerical behaviour of the algorithms.


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Additional Information

Attahiru Sule Alfa
Affiliation: Department of Industrial and Manufacturing Systems Engineering, University of Windsor, Windsor, Ontario, Canada N9B 3P4
Email: alfa@uwindsor.ca

Jungong Xue
Affiliation: Fakultaet fuer Mathematik, Technishe Universitaet Chemnitz, Reichenhainer Str. 41, 09126 Chemnitz, Germany
Address at time of publication: Department of Industrial and Manufacturing Systems Engineering, University of Windsor, Windsor, Ontario, Canada N9B 3P4
Email: jxue@server.uwindsor.ca

Qiang Ye
Affiliation: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506-0027
Email: qye@ms.uky.edu

DOI: https://doi.org/10.1090/S0025-5718-01-01325-4
Keywords: Entrywise perturbation, diagonal dominant matrix, $M$-matrix, eigenvalue
Received by editor(s): March 22, 1999
Received by editor(s) in revised form: March 14, 2000
Published electronically: May 14, 2001
Additional Notes: Research of the first author was supported by grant No. OGP0006854 from Natural Sciences and Engineering Research Council of Canada
Research of the second author was supported by Natural Sciences Foundation of China and Alexander von Humboldt Foundation of Germany.
Research of the third author was supported by grants from University of Manitoba Research Development Fund and Natural Sciences and Engineering Research Council of Canada while this author was with University of Manitoba, Winnipeg, Manitoba, Canada
Article copyright: © Copyright 2001 American Mathematical Society

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