A Simple Derivation of Newton-Cotes Formulas with Realistic Errors

• Published in: arXiv.org 2012-02
• Main Author: Graça, Mário M
• Format: Article
• Language: English
• Subjects: Algorithms; Error analysis; Error correction; Mathematical models; Polynomials
• ISSN: 2331-8422

In order to approximate the integral \(I(f)=\int_a^b f(x) dx\), where \(f\) is a sufficiently smooth function, models for quadrature rules are developed using a given {\it panel} of \(n (n\geq 2)\) equally spaced points. These models arise from the undetermined coefficients method, using a Newton's basis for polynomials. Although part of the final product is algebraically equivalent to the well known closed Newton-Cotes rules, the algorithms obtained are not the classical ones. In the basic model the most simple quadrature rule \(Q_n\) is adopted (the so-called left rectangle rule) and a correction \(\tilde E_n\) is constructed, so that the final rule \(S_n=Q_n+\tilde E_n\) is interpolatory. The correction \(\tilde E_n\), depending on the divided differences of the data, might be considered a {\em realistic correction} for \(Q_n\), in the sense that \(\tilde E_n\) should be close to the magnitude of the true error of \(Q_n\), having also the correct sign. The analysis of the theoretical error of the rule \(S_n\) as well as some classical properties for divided differences suggest the inclusion of one or two new points in the given panel. When \(n\) is even it is included one point and two points otherwise. In both cases this approach enables the computation of a {\em realistic error} \(\bar E_{S_n}\) for the {\it extended or corrected} rule \(S_n\). The respective output \((Q_n,\tilde E_n, S_n, \bar E_{S_n})\) contains reliable information on the quality of the approximations \(Q_n\) and \(S_n\), provided certain conditions involving ratios for the derivatives of the function \(f\) are fulfilled. These simple rules are easily converted into {\it composite} ones. Numerical examples are presented showing that these quadrature rules are useful as a computational alternative to the classical Newton-Cotes formulas.