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

Authors:
Attahiru Sule Alfa;Jungong Xue;Qiang Ye
Affiliations:
Department of Industrial and Manufacturing Systems Engineering, University of Windsor, Windsor, Ontario, Canada;Department of Industrial and Manufacturing Systems Engineering, University of Windsor, Windsor, Ontario, Canada;Department of Mathematics, University of Kentucky, Lexington, Kentucky
Venue:
Mathematics of Computation
Year:
2002

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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.