A note on the convergence of the AOR method

Abstract

In Gao and Huang [Z.X. Gao, T.Z Huang, Convergence of AOR method, Appl. Math. Comput. 176 (2006) 134–140] some practical sufficient conditions for the convergence of the AOR (accelerated overrelaxation) method for solving linear system $\mathit{Ax}=b$, with A being doubly diagonally dominant matrix, are presented. Using a different approach we will give some improvements in both cases, when the matrix A is either strictly diagonally dominant (SDD) or doubly diagonally dominant. Using the same simple example as in Gao and Huang (2006), we will illustrate how the new approach can significantly improve convergence area.

Introduction

Considering linear system $\mathit{Ax}=b$, where $A=[a_{\mathit{ij}}]\in {\mathbf{C}}^{n\text{,}n}$ is a nonsingular matrix with all diagonal entries equal to one, and $x=[x_i]\text{,}b=[b_i]\in {\mathbf{C}}^n$, we split the system matrix as $A=I-L-U$, where I is identity matrix and $-L$ and $-U$ are strictly lower and strictly upper triangular parts of A, respectively.

The AOR (accelerated overrelaxation) method, introduced in [5], for solving system $\mathit{Ax}=b$ is a two-parameter relaxation method:

$$x^{k+1}=M_{\sigma \text{,}\omega }{x}^k+d\text{,}k=0\text{,}1\text{,}\dots \text{;}{x}^0\in {\mathbf{C}}^n\text{,}$$

where

$$M_{\sigma \text{,}\omega }=(I-\sigma L{)}^{-1}[(1-\omega )I+(\omega -\sigma )L+\omega U]\text{,}d=\omega (I-\sigma L{)}^{-1}\text{,}$$

and $\omega \text{,}\sigma \in \mathbf{R}\text{,}\omega \ne 0$.

For each nonempty subset S of indices $N≔\{1\text{,}2\text{,}\dots \text{,}n\}$ we denote

$$r_i^S(A)≔\sum _{k\in S⧹\{i\}}|a_{\mathit{ik}}|\mathrm{and}{\ell }_i^S(A)≔\sum _{k\in S\text{,}k<i}|a_{\mathit{ik}}|\text{.}$$

Also, we denote $r_i(A)≔r_i^N(A)$ and ${\ell }_i(A)≔{\ell }_i^N(A)$.

Finally, with $\rho (A)$ we denote spectral radius of the matrix A.