Title:

Interval methods for nonlinear systems

In numerical mathematics, there is a need for methods which provide a user with the solution to his problem without requiring him to understand the mathematics underlying the method of solution. Such a method involves computable tests to determine whether or not a solution exists in a given region, and whether, if it exists, such a solution may be found by using the given method. Two valuable tools for the implementation of such methods are interval mathematics and symbolic computation. In. practice all computers have memories of finite size and cannot perform exact arithmetic. Therefore, in addition to the error which is inherent in a given numerical method, namely truncation error, there is also the error due to rounding. Using interval arithmetic, computable tests which guarantee the existence of a solution to a given problem in a given region, and the convergence of a particular iterative method to this solution, become practically realizable. This is not possible using real arithmetic due to the accumulation of rounding error on a computer. The advent of packages which allow symbolic computations to be carried out on a given computer is an important advance for computational numerical mathematics. In particular, the ability to compute derivatives automatically removes the need for a user to supply them, thus eliminating a major source of error in the use of methods requiring first or higher derivatives. In this thesis some methods which use interval arithmetic and symbolic computation for the solution of systems of nonlinear algebraic equations are presented. Some algorithms based on the symmetric singlestep algorithm are described. These methods however do not possess computable existence, uniqueness, and convergence tests. Algorithms which do possess such tests, based on the KrawczykMoore algorithm are also presented. A simple package which allows symbolic computations to be carried out is described. Several applications for such a package are given. In particular, an interval form of Brown's method is presented.
