TITLE:
Behavior of the Numerical Integration Error
AUTHORS:
Tchavdar Marinov, Joe Omojola, Quintel Washington, LaQunia Banks
KEYWORDS:
Numerical Integration, Algorithms with Automatic Result Verification, Roundoff Error
JOURNAL NAME:
Applied Mathematics,
Vol.5 No.10,
May
23,
2014
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.