Convergence of Asynchronous Iterative Methods

Abstract

The objective of this chapter is to show how Jacobi-type iterative methods for the solution of linear and nonlinear equations and, more specifically asynchronous versions of these methods, can be analyzed in the framework introduced in this book. By regarding these methods as discrete-time interconnected systems with varying delays in the interconnections, the analysis based on a diagonal-type Liapunov function introduced in Chapter 3 leads to computable conditions for convergence based on nonnegative stable (and hence diagonally stable) matrices, as well as estimates for the speed of convergence of different iterative methods. A perspective on the analysis of interconnected systems using diagonal-type Liapunov functions that complements the approach in this chapter is to be found in Chapter 6.