A Note on a Verified Automatic Integration Algorithm

A verified integration algorithm is proposed for calculating s-dimensional integrals over a finite domain using numerical computations. To construct an efficient verified numerical integrator, the truncation error and the rounding error need to be considered. It has been known that interval arithmetic is one of the most efficient methods of evaluating the rounding error. However, it is much slower than pure floating-point arithmetic, so that in an inclusion algorithm for integrals, the computational effort by the interval arithmetic tends to become a large part. To overcome this problem, an algorithm for evaluating the rounding error using floating-point computations is proposed. The proposed algorithm is based on calculating a priori error bounds for function evaluations and an accurate sum algorithm. With the use of the proposed algorithm and a inclusion algorithm for evaluating the truncation error, we propose an automatic inclusion algorithm. Numerical examples are presented for illustrating the effectiveness of the proposed algorithm.