An optimal quadrature formula with derivatives for arbitrarily fixed nodes in the Sobolev space

The numerical computation of definite integrals plays a crucial role in various applied and theoretical disciplines. In many cases, exact analytical evaluation of integrals is in feasible due to the complexity of the integrand or the nature of the integration limits. Quadrature formulas provide an efficient approach for approximating definite integrals, relying on weighted sums of function values at selected nodes. Traditional quadrature formulas, such as those of Newton-Cotes, Gauss, aim to optimize accuracy by care fully choosing nodes and weights. However, optimal quadrature formulas can also be constructed in the sense of Sard, minimizing the error functional within a given func tion space. In this paper, we focus on constructing an optimal quadrature formula in the Sobolev space with arbitrarily fixed nodes. Unlike conventional approaches where coefficients are determined sequentially, we simultaneously optimize both function and derivative coefficients, improving overall accuracy and stability. The derivation employs the method of- functions, which allows us to express the quadrature formula’s co efficients explicitly and analyze its error properties. The obtained quadrature formula minimizes the error norm in the chosen function space, ensuring an improved approxi mation of definite integrals. Furthermore, when the nodes are equally spaced, our results generalize the well-known Euler-Maclaurin formula, demonstrating the effectiveness of our approach.