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In [39], effort has been taken in order to determine whether the averaged Gaussian formulas are an adequate alternative to the corresponding Gauss-Kronrod quadrature formulas to estimate the remainder term of a Gaussian rule.
SPALEVIC, On generalized averaged Gaussian formulas, Math.
39] --, A note on generalized averaged Gaussian formulas, Numer.
It is known that Gaussian formulas ensure a higher order of accuracy and have better stability properties than Newton-Cotes formulas [2].
2, we have compared our results with those obtained by the only other numerical approach we know, that is, DQ methods based on Newton-Cotes formulas (in particular the trapezoidal and Simpson 3/8 formulas) [7]: the errors of the two families of methods are very similar, in spite of the error of approximation due to the interpolation technique used for the Gaussian formulas.
We have studied the convergence properties of the constructed methods and we have proven that their order of convergence is the minimum between the order of convergence of the Gaussian formula and the degree of accuracy of the interpolating polynomial.
The closest parallelism with classical Gaussian formulas and orthogonal polynomials will be specially emphasized.
In both cases results given by Gaussian formulas are strongly improved.