Theoretical and practical efficiency measures for symmetric interpolatory quadrature formulas

Abstract

Abstract.We study two criteria to evaluate quadrature formulas when used in automatic quadrature programs. The former consists of the computation of a quantity depending on both the truncation error behavior 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 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 in good agreement.

An extended doubly-adaptive quadrature method based on the combination of the Ninomiya and the FLR schemes

“…This scheme chooses either to apply the order adaptive method to the current subinterval or to further split the subinterval, by detecting the local regularity of the integrand. An improved version QXG (QXGS) due to Favati, Lotti and Romani [5] (we call FLR henceforth) of QAG (QAGS) is a doublyadaptive algorithm based on recursive monotone stable (RMS) formulas [4,6]. Ninomiya's method is less effective for oscillatory integrands than the FLR method.…”