 
Summary: Topics in Validated Computations
J. Herzberger (Editor)
1994 Elsevier Sciencc B.V. 7
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lnclusion methods for systems of nonlinear equations  the interval New
ton method and modifications
G. Alefeld
Institut für Angewandte Mathematik, Universität Karlsruhe, 76128 Karlsruhe, Germany
1. INTRODUCTION
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.
