Abstract
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 G x = 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 η 2 ≤ 1 − 1 / κ c , where c = 2 + log 2 n . The main consequence from this bound is that small condition number forces fast convergence while large condition number allows slow convergence.
Highlights
The SOR method is one of the basic iterative algorithms for solving a large sparse linear system of the formGx = b, ð1Þ where G ∈ Rn×n, b ∈ Rn, and x ∈ Rn denotes the vector of unknowns
We investigate the SOR rate of convergence in the special case when G is a real symmetric positive semidefinite matrix
Since G is positive semidefinite, a zero diagonal entry implies that the corresponding row and column are null and can be deleted
Summary
We investigate the SOR rate of convergence in the special case when G is a real symmetric positive semidefinite matrix This means that G has at least one zero eigenvalue and that the system (1) can be inconsistent. The convergence properties of iterative methods for solving consistent positive semidefinite linear systems have attracted the attention of several authors. Since G has t zero eigenvalues, the eigenvalues of Hw satisfy λj = 1 for j = 1, ⋯, t, ð16Þ 1 > jλt+1j≥⋯≥jλnj: the Jordan canonical form of Hw shows that the rate of convergence is determined by the size of jλt+1j, which is sometimes called the “convergence factor,” e.g., [26] This situation means that we need an upper bound on jλt+1j. In the last section, we compare our approach with former attempts to derive such bounds
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