Explicit Coefficients and Error Estimate for an Optimal Quadrature Formula

Accurate numerical integration is essential in many computational applications, particularly when the integrands possess higher-order smoothness. Optimal quadrature formulas in Sobolev spaces provide a rigorous framework for controlling approximation errors, while classical formulas such as the Euler–Maclaurin formula may be less flexible for general distributions of quadrature nodes. This motivates the development of accurate quadrature formulas with explicit error estimates and applicability beyond uniform partitions. This paper aims to construct an optimal quadrature formula in the Sobolev space ????(5)2(????, ????) and to establish explicit expressions for its coefficients and error characteristics using the ????-function method. The proposed approach represents the error of the quadrature formula through a ????-function and determines the coefficients of the quadrature formula by solving the corresponding optimality conditions. This procedure yields explicit analytical expressions for the quadrature coefficients and an explicit estimate of the approximation error. In addition, a closed-form expression for the norm of the error of the quadrature formula is obtained, together with a simplified form for equally spaced quadrature nodes. The construction is formulated for arbitrary node distributions and therefore does not require a uniform partition of the integration interval. The resulting quadrature formula provides a rigorous error estimate and exhibits high numerical accuracy. Numerical experiments conducted for several test functions demonstrate that the proposed formula produces smaller absolute approximation errors than the classical Euler–Maclaurin quadrature formula. The numerical findings are also consistent with the theoretically established error estimates. The obtained results extend the range of applicable optimal quadrature formulas in Sobolev spaces and provide a flexible and accurate alternative to classical numerical integration methods, particularly when non-uniform quadrature nodes are required.