A Non-Adaptive Scheme for Numerical Integration

Abstract

In this paper we present an efficient scheme for the numerical approximation of integrals. The proposed method is an excellent alternative to adaptive quadrature methods. Like the adaptive quadrature approach, the method is based on using two different approximations on a sufficiently small interval to estimate the error and hence accept or reject the approximation on that interval. Adaptive quadrature methods, however, are based on the successive bisections of the interval of integration until a certain error tolerance criterion is met. In this paper we develop a variable step method which predicts the step size that has a good likelihood of achieving the desired error tolerance criterion. In some way, the proposed method resembles the variable step methods for approximating the initial value problem for ordinary differential equations. Several examples of highly oscillatory intgrands are given to demonstrate the accuracy and efficiency of the method. Comparisons are made with the adaptive quadrature method used in one of the MATLAB functions for approximating integrals. The results show a consistent pattern of performance in terms of both economy and accuracy.