Interval Arithmetic Tools for Range Approxima- Tion and Inclusion of Zeros

In this paper we start in section 2 with an introduction to the basic facts of interval arithmetic: We introduce the arithmetic operations, explain how the range of a given function can be included and discuss the problem of overestimation of the range. Finally we demonstrate how range inclusion (of the first deriva.tive of a given function) can be used to compute zeros by a so-called enclosure method. An enclosure method usually starts with an interval vector which contains a solution and improves this inclusion iteratively. The quest ion which has to be discussed is under what conditions is the sequence of including interval vectors convergent to the solution. This will be discussed in section 3 for so-called Newton-like enclosure methods. An interesting feature of inclusion methods is that they cau also be used to prove tha.t there exists no solution in an interval vector. It will be shown that this proof needs only few steps if the test vector has already a small enough diameter. In the last section we demonstrate how for a given nonlinear system a. test vector cau be constructed which will very likely contain a solution. A very important point is, of course, the fact that all these ideas can be performed in a safe way (especially with respect to rounding errors) on a computer. We can not go into auy details in this pa.per and refer instead to the survey paper [14] by U. Kulisch and W. Miranker .