Performance of Self-validating Adaptive Quadrature

Abstract

SVALAQ is a suite of programs for self-validated quadrature. For the usual quadrature problem on a finite interval,

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

we compute an interval [c,d] in which I f ε is guaranteed to lie. The inclusion If ε [c,d] is automatically validated by the computer program provided only that f can be evaluated at every point of [A,B]. This paper addresses two issues related to the performance of self-validated quadrature:

• How much does validation cost? and

• How accurate can it be?

The answers are that the cost in CPU time for validation typically varies from a factor of 3–5 for very stringent accuracy requests to 3–15 for modest accuracy requests compared to the CPU time required by the general purpose routine QAGS from QUADPACK. Accuracies of a few units in the last place (ULP) can be achieved, while for modest accuracy requests, validation assures that the request has been met without costly excess accuracy.

SVALAQ runs in any IBM System 370 environment using ACRITH. The algorithms could be implemented in any environment which supports interval calculations and an accurate scalar product, such as Pascal-SC.

Supported in the part by IBM Deutschland GmbH.