Algebraic-Trigonometric Optimal Formulas for Numerical Integration
DOI:
https://doi.org/10.71310/pcam.5_69.2025.08Keywords:
quadrature formula, definite integral, nodal points, optimalityAbstract
In this article, we present a detailed study on the determination of the coefficients of a quadrature formula that involves derivatives, making use of the so-called the phi – function method. This method provides a systematic framework for constructing optimal quadrature formulas in the context of approximate integration. In particular, the phi – function approach not only simplifies the process of finding the required coefficients, but also ensures that the resulting formulas achieve a high degree of accuracy. Furthermore, the functional error associated with the constructed quadrature formula is carefully analyzed. The error analysis is supported by precise mathematical expressions, which confirm the reliability and effectiveness of the derived formula. By considering quadrature formulas with arbitrarily fixed nodes, the optimality conditions are rigorously examined, and the procedure for determining the corresponding components and coefficients is described in detail. As a key outcome of this investigation, explicit analytical expressions for the coefficients of the optimal quadrature formula are successfully obtained. Of particular interest is the case of equally spaced nodes, for which the derived quadrature formula naturally reduces to a classical Euler-Maclaurin type formula, demonstrating both the generality and the practical significance of the phi – function method.
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