Estimation and testing of symmetric interpolatory quadrature formulas

We study two criteria to evaluate quadrature formulas, when used in automatic quadrature programs. The former one is a theoretical load factor which takes into consideration both the truncation error behaviour and the geometric properties of the nodes of the rule. This measure allows estimating the asymptotical computational cost in various abstract models of automatic quadrature. The latter one is a testing technique which can be used to measure the efficiency of the formulas under consideration in a real environment. The relationships between the two criteria are investigated and the two approaches seem to agree significant1y. Moreover, we introduce two families of symmetric, interpolatory integration formulas on the interval [-1, 1]. These formulas, related to the class of recursive monotone stable (RMS) formulas, allow the application of higher order or compound rules with an efficient reuse of computed function values. One family (SM) uses function values computed outside the integration interval, the other one (HR) uses derivative data. These formulas are evaluated using the techniques introduced in the first part of the paper.