On the Acceleration of an Interval-Arithmetic Iteration Method

Abstract

In the paper under consideration we describe a device for the acceleration of convergence of a monotonously enclosing iterative method for solving simultaneous systems of nonlinear equations. The method is a combination of the so called interval-arithmetic version of the Newton single step iteration method with forming intersections, which computes sequences of interval-vectors, and a modified Newton–ADI iterative method. Under certain conditions this new method is globally convergent to the solution of the nonlinear system. For some classes of nonlinear equations which originate from the discretization of partial differential equations these methods are monotonously convergent under weaker assumptions than known methods which also enclose the solution repeatedly. Furthermore these methods compute a sequence of approximations of the solution which is faster convergent to the solution than the lower and upper bounds of the enclosing interval-vectors. This paper ends with some numerical results which demonstrate these new iteration methods.