SOLVING FUNCTION SPACE PROBLEMS WITH GUARANTEED CLOSE BOUNDS

Abstract

This chapter focuses on solving function space problems with guaranteed close bounds. Using arithmetical routines and methods, it becomes possible to compute very close and reliable bounds economically for the solutions of functional, differential, and integral equations. These methods are called E-methods corresponding to their properties. The methods are suitable for software packages. They provide automatically error control for the first time. Without any additional demand on the part of the user, an incomparably high level of software security and reliability can be achieved. Based on these methods, it is at present economically possible to contain the solutions of functional equations within guaranteed functional bounds. The computation of a reliable error estimate for the solution involves the determination of a set of vector functions X(t) containing the solution x(t) for all t. The computation of X(t) requires new computer arithmetic, such as segment arithmetic and functional ultra arithmetic. All these must be executed automatically and intrinsically on a computer without intervention by the users. Consequently, such methods are called error controlling methods. In most cases, an error controlling method necessarily proves the existence of the solution x(t) in X(t) automatically.