An iterative method for positive real systems

Abstract

In this paper we present a general form of an iterative method to approximate the solution of the linear system A x = b, when the matrix A + A^T is real and positive definite or equivalently A is positive real. Under some assumptions we generalise several results given by Li, Liang and Evans. This iterative method contains an auxiliary matrix D ₁ and, for a suitable choice of the matrix D ₁, we establish convergence conditions for this iterative method. As a particular case we have the point accelerated over-relaxation method (AOR) and we analyse its convergence. Some comments about the point case of this iterative method and the one presented by Li, Liang and Evans are made in the last remarks of this paper.

Keywords:

• Iterative method
• Spectral radius
• AOR method
• Positive real matrix

Acknowledgements

The authors are very grateful to the referees for their constructive and useful comments.