lnclusion methods for systems of nonlinear equations - the interval New - ton method and modifications

In this paper we give a survey of methods which can be used for including solutions of a nonlinear system of equations. These methods are called inclusion methods or enclosure methods. An inclusion method usuaIly starts with an interval vector which contains a solution of a given system and improves this inclusion iteratively. The question which has to be discussed is under what conditions is the sequence of including interval vectors convergent to the solution. More often an including interval vector is not known and one tries to compute an interval vector containing a solution by some operator which forms the basis of an inclusion method. In other words, we prove the existence of a solution. Both concepts are discussed and illustrated in this article. An interesting feature of inclusion methods is that they can also be used to prove that there exists no solution in an interval vector. Our methods are based on interval arithmetic took As is weIl known from the literature there exists a great variety of such inclusion methods (see [2] and [13], e.g.). Since the purpose of this article consists in discussing the main principles of inclusion methods we limit ourselves to only a few methods. Enclosure methods relying on other ideas are not considered. The paper is organized as folIows: In chapter 2.1 we repeat the weIl known results for the one-dimensional interval Newton method. Chapter 2.2 contains aseries of properties of the Gaussian algorithm applied to linear systems with interval data. These results are used in chapter 2.3 where the interval Newton operator is introduced. In chapter 2.4 convergence and divergence statements for the so-caIled interval Newton method which is based on the interval Newton operator are investigated. Chapter 2.5 contains results about the speed of convergence and divergence, respectively. In chapter 3 the Krawczyk operator is introduced and in the final chapter 4 it is shown how test intervals can be efficiently constructed.