Nowadays many problems are solved by computers. Algorithms are implemented on computers in order to find a solution of a given problem. However, truncation errors may result in solutions of dubious quality and the problem of qualitatively evaluating the accuracy of a numerical solution is often neglected.

Interval methods offer the possibility to avoid these problems. Here the correct solution is enclosed by an interval. A drawback of interval methods is the potential overestimation of a solution. In many cases the interval computed by such techniques is much larger than the correct solution. This is the reason why we introduce methods calculating smaller intervals that guarantee the correct solution.

This diploma thesis is concerned with the use of Taylor models for interval methods. A Taylor model is a pair of a polynomial and a remainder interval. In order to get an enclosure of a problem, an enclosure of a polynomial is needed. There are several methods for finding an enclosure of a polynomial. Here, we analyse two options of finding an enclosure with Taylor models. Both approaches are compared with well-known techniques of interval methods. Finally the use of Taylor models for a verified solution of systems of equations is investigated employing these results.

Understanding and control of chemical processes are very important issues in process engineering. On that score such chemical processes are modeled to prevent poor quality of substances or even explosions of whole installations. In the analysis of the resulting models it is necessary to find all roots of a non-linear system of equations. In close proximity to these roots small alteration of parameters could have harmful consequences for the model. To solve the non-linear systems one can use result-verifying algorithms, which need enclosures for the range of functions and derivatives.

Practically there are many ways to compute such enclosures. In this thesis so called first and second order interval extensions are presented, which provide desired enclosures for the range of functions and derivatives. In fact, different interval extensions yield different results. Of course one is interested in high quality enclosures. One goal of this thesis is the exact analysis of the quality of the presented interval extensions. The second main goal is the implementation of the variety of interval extensions in a computer program. This implementation is realized in the programming language Pascal-XSC.