The Rate of Convergence of the SOR Method in the Positive Semidefinite Case

In this paper, we derive upper bounds that characterize the rate of convergence of the SOR method for solving a linear system of the form Gx=b, where G is a real symmetric positive semidefinite n×n matrix. The bounds are given in terms of the condition number of G, which is the ratio κ=α/β, where α...

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Bibliographic Details
Published in:Computational and mathematical methods 2022-03, Vol.2022, p.1-8
Main Author: Dax, Achiya
Format: Article
Language:English
Subjects:
Convergence
Eigenvalues
Linear systems
Matrices (mathematics)
Upper bounds
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Summary:In this paper, we derive upper bounds that characterize the rate of convergence of the SOR method for solving a linear system of the form Gx=b, where G is a real symmetric positive semidefinite n×n matrix. The bounds are given in terms of the condition number of G, which is the ratio κ=α/β, where α is the largest eigenvalue of G and β is the smallest nonzero eigenvalue of G. Let H denote the related iteration matrix. Then, since G has a zero eigenvalue, the spectral radius of H equals 1, and the rate of convergence is determined by the size of η, the largest eigenvalue of H whose modulus differs from 1. The bound has the form η2≤1−1/κc, where c=2+log2n. The main consequence from this bound is that small condition number forces fast convergence while large condition number allows slow convergence.
ISSN:2577-7408
2577-7408
DOI:10.1155/2022/6143444