Extrapolation quadrature from equispaced samples of functions with jumps

Abstract

Based on the Euler–Maclaurin formula, the Romberg quadrature method extrapolates trapezoidal values to improve their accuracy when computing the integral of smooth functions from equispaced samples. It has been known at least since an article of Lyness in 1971 that the Euler–Maclaurin formula may be extended to accommodate functions with jumps. In the present work, we develop an extrapolation method, based on this extended formula, for the quadrature of such discontinuous functions. We illustrate the method with numerical examples, using one as well as several sample vectors.