Behavior of the Numerical Integration Error

Abstract

In this work, we consider different numerical methods for the approximation of definite integrals. The three basic methods used here are the Midpoint, the Trapezoidal, and Simpson's rules. We trace the behavior of the error when we refine the mesh and show that Richardson's extrapolation improves the rate of convergence of the basic methods when the integrands are sufficiently differentiable many times. However, Richardson's extrapolation does not work when we approximate improper integrals or even proper integrals from functions without smooth derivatives. In order to save computational resources, we construct an adaptive recursive procedure. We also show that there is a lower limit to the error during computations with floating point arithmetic.

A New Analysis of Approximate Solutions for Numerical Integration Problems with Quadrature-based Methods

“…Smyth, G. K. focused on the process of approximating a definite integral from values of the integrand when exact mathematical integration is not available [20]. Furthermore, Marinov, T. et al used the three basic methods which are the Midpoint, the Trapezoidal, and Simpson's rules [21]. Kwasi A. et al proposed a numerical integration method using polynomial interpolation that provides improved estimates as compared to the Newton-Cotes methods of integration [22].…”