Using Weight Functions in Self-Validating Quadrature

Abstract

We describe how weight functions are implemented in SVALAQ, Self-Validating Adaptive Quadrature, to provide a self-validating computation of the definite integral equation

$$I\!f=\int^{B}_{A}f(x)dx,$$

SVALAQ computes an interval [c,d] in which If is guaranteed to lie. The inclusion is validated by SVALAQ, provided that f(x) can be evaluated on all of [A,B] or if f(x) has certain singularities.

Rules for evaluating definite integrals using weight functions in a self-validating environment were also developed. These rules, based on Taylor’s polynomials, cover some of the difficulties associated with certain algebraic, logarithmic and sinusoidal weight functions in validated quadrature.