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Mathematics of Computation

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Computing singular values of diagonally dominant matrices to high relative accuracy


Author: Qiang Ye
Journal: Math. Comp. 77 (2008), 2195-2230
MSC (2000): Primary 65F18, 65F05
DOI: https://doi.org/10.1090/S0025-5718-08-02112-1
Published electronically: May 5, 2008
MathSciNet review: 2429881
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Abstract: For a (row) diagonally dominant matrix, if all of its off-diagonal entries and its diagonally dominant parts (which are defined for each row as the absolute value of the diagonal entry subtracted by the sum of the absolute values of off-diagonal entries in that row) are accurately known, we develop an algorithm that computes all the singular values, including zero ones if any, with relative errors in the order of the machine precision. When the matrix is also symmetric with positive diagonals (i.e. a symmetric positive semi-definite diagonally dominant matrix), our algorithm computes all eigenvalues to high relative accuracy. Rounding error analysis will be given and numerical examples will be presented to demonstrate the high relative accuracy of the algorithm.


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Additional Information

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

DOI: https://doi.org/10.1090/S0025-5718-08-02112-1
Received by editor(s): April 9, 2007
Received by editor(s) in revised form: October 4, 2007
Published electronically: May 5, 2008
Additional Notes: This research was supported in part by NSF under Grant DMS-0411502.
Article copyright: © Copyright 2008 American Mathematical Society