APPLICATION AND ILLUSTRATIVE COMPUTATION

This chapter discusses the implementation of self-validating numerics on a computer and presents illustrative numerical experiments. It presents a review of the self-validating technique. The review is relatively self-contained and it emphasizes on the issues of implementation dealing with the passage from theoretical computation to computation on a screen, that is, to computation on a computer. The chapter also presents a commentary on the practical implementation of the ultra-arithmetic in functoids and in interval functoids. The operations of ultra-arithmetic are defined through semimorphism for functoids and for interval functoids. Practical implementation, however, requires that appropriate approximations to semimorphism be made. As computations in functoids and in interval functoids are iterative and proceed by refinement, it is sensible to gain higher accuracy through further iteration steps such as those of the iterative residual correction (IRC) processes. The chapter further also presents a set of illustrative example computations and computations illustrating the IRC process in both linear and nonlinear cases. It describes an effective treatment of Newton-type iteration operators that frequently occur in numerical methods. It also discusses bounds of high quality that are obtained by applying iterative residual correction (IRC) to the iteration.