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A Simple Derivation of Newton-Cotes Formulas with Realistic Errors

DOI: 10.5539/jmr.v4n5p34

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Abstract:

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 $n,,(ngeq 2)$ 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 $ ilde E_n$ is constructed, so that the final rule $S_n=Q_n+ ilde E_n$ is interpolatory. The correction $ ilde E_n$, depending on the divided differences of the data, might be considered a {em realistic correction} for $Q_n$, in the sense that $ ilde 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} $ar E_{S_n}$ for the {it extended or corrected} rule $S_n$. The respective output $(Q_n, ilde E_n, S_n, ar E_{S_n})$ 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.

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