Numerical Integration in Two Dimensions with Automatic Result Verification

Abstract

Publisher Summary This chapter discusses numerical integration in two dimensions with automatic result verification. For calculating an enclosure of two-dimensional integrals, two different methods with automatic result verification are developed in the chapter. Both procedures are based on Romberg extrapolation. They determine an enclosure of the approximation of the integral and an enclosure of the corresponding remainder term using interval arithmetic. In both algorithms, the quality of the remainder term determines the error of the result, that is, the width of the enclosure of the integral. The chapter examines the representations of the remainder terms in dependency on the chosen step-size sequences. The two extrapolation methods can be combined. First of all, in an analogy to the single Romberg extrapolation, the Taylor coefficients of a triangular matrix are determined; subsequently, for both extrapolation methods, all remainder terms that can be determined using the previously computed Taylor coefficients are calculated.