On Optimality and Error Estimation of a Quadrature Formula With Derivative That Is Exact for Trigonometric Functions

In this paper, we investigate the determination of coefficients for an optimal quadrature formula involving derivatives by applying the ϕ-function technique. The ϕ-function approach enables the development of optimal quadrature rules for approximating definite integrals. In this work when the nodes are arbitrarily fixed, the conditions for the optimality of the quadrature rule are examined, and the approach for identifying the elements of the formula is discussed. Explicit expressions for the coefficients of the optimal quadrature formula are derived. Specifically, when the nodes are equally distributed, an Euler-Maclaurin type optimal quadrature rule is achieved.

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