Quadrature methods based on the Euler-Maclaurin formula and on the Clenshaw-Curtis method of integration

Abstract

The Euler-Maclaurin formula [11 may be used to derive a wide range of quadrature formulas including the Newton-Cotes formulas [2]. Some of these are derived, including two which are more accurate than Simpson's rule and which share some of its advantages. These are especially useful in the automatic evaluation of integrals which are badly behaved at their end points. The Clenshaw-Curtis method [3] is also examined and it is shown to be considerably more accurate in practice than the equivalent trapezoidal rule even though the Clenshaw-Curtis method appears to converge to the trapezoidal rule as the number of abscissas approaches infinity [41. When the Clenshaw-Curtis formula is used in a way different from that put forward by the original authors it is suggested that it may have significant advantages over other methods of numerical integration in many problems.