Behavior of the Numerical Integration Error

Tchavdar Marinov, Joe Omojola, Quintel Washington, LaQunia Banks

Department of Natural Sciences, Southern University at New Orleans, New Orleans, USA.

DOI: 10.4236/am.2014.510133                

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.

Keywords

Numerical Integration, Algorithms with Automatic Result Verification, Roundoff Error

Conflicts of Interest

The authors declare no conflicts of interest.

References

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