Numerical Quadrature by Extrapolation with Automatic Result Verification

Abstract

This chapter discusses adaptive algorithms for a verified computation of an enclosure of the values of definite integrals in tight bounds with automatic error control. The integral is considered the sum of an approximation term and a remainder term. Starting from a Romberg extrapolation, the recursive computation of the T-table elements is replaced by one direct evaluation of an accurate scalar product. The remainder term is verified numerically via automatic differentiation algorithms. Concerning interval arithmetic, the disadvantages of the Bulirsch sequence are overcome by introducing a decimal sequence. By choosing different step-size sequences, one can generate a table of coefficients of remainder terms. Using this table and depending on the required accuracy, a fast search algorithm determines the method that involves least computational effort. A local adaptive refinement makes it possible to efficiently reduce the global error to the required size because an additional computation is carried out only where necessary. In comparison to alternative enclosure algorithms, theoretical considerations and numerical results demonstrate the advantages of the new method. This algorithm provides guaranteed intervals with tight bounds, even at points where the approximation method delivers numbers with an incorrect sign. The chapter discusses the application of this method to multidimensional problems.