 
Summary: HYBRID GAUSSTRAPEZOIDAL QUADRATURE RULES
BRADLEY K. ALPERT
SIAM J. SCI. COMPUT. c 1999 Society for Industrial and Applied Mathematics
Vol. 20, No. 5, pp. 15511584
Abstract. A new class of quadrature rules for the integration of both regular and singular
functions is constructed and analyzed. For each rule the quadrature weights are positive and the
class includes rules of arbitrarily highorder convergence. The quadratures result from alterations to
the trapezoidal rule, in which a small number of nodes and weights at the ends of the integration
interval are replaced. The new nodes and weights are determined so that the asymptotic expansion
of the resulting rule, provided by a generalization of the EulerMaclaurin summation formula, has a
prescribed number of vanishing terms. The superior performance of the rules is demonstrated with
numerical examples and application to several problems is discussed.
Key words. EulerMaclaurin formula, Gaussian quadrature, highorder convergence, numerical
integration, positive weights, singularity
AMS subject classifications. 41A55, 41A60, 65B15, 65D32
PII. S1064827597325141
1. Introduction. Recent advances in algorithms for the numerical solution of
integral equations have stimulated renewed interest in integral equation formulations
of problems in potential theory, wave propagation, and other application areas. Fast
algorithms, including those by Rokhlin [1], [2], Greengard and Rokhlin [3], Hackbusch
