CADRE: AN ALGORITHM FOR NUMERICAL QUADRATURE

Abstract

Publisher Summary This chapter discusses the program CADRE—an algorithm for numerical quadrature. The program employs an adaptive scheme whereby CADRE is found as the sum of estimates for the integral of F ( x ) over suitably small subintervals of a given interval of integration. Starting with the interval of integration itself as the first such subinterval, the program attempts to find an acceptable estimate on a given subinterval by cautious Romberg extrapolation. If this attempt fails, the subinterval is divided into two subintervals of equal length. For the sake of economy, values of F ( x ), once calculated, are saved until they are successfully used in estimating the integral over some subinterval to which they belong. The program CADRE uses the composite trapezoid sum. For certain classes of integrands, the composite trapezoid sum exhibits a known and characteristic convergence behavior that the algorithm attempts to detect and to exploit through cautious extrapolation.