OPTIMAL QUADRATURE FORMULAS FOR FUNCTIONS WITH HIGHER-ORDER DERIVATIVES

Abstract

Abstract. This paper investigates the construction of optimal quadrature formulas for numerical integration of functions possessing higher-order derivatives. The study is formulated in Sobolev spaces and follows the Sard concept of optimality, in which the coefficients of a quadrature rule are chosen so as to minimize the norm of the associated error functional. A derivative-enriched quadrature model is considered on a uniform grid, and the exactness conditions, extremal error representation, and coefficient optimization problem are described. Special attention is paid to formulas obtained by supplementing nodal function values with odd-order derivatives at the endpoints. Such corrections lead to Euler–Maclaurin-type rules and substantially increase the algebraic order of accuracy for sufficiently smooth functions. A computational illustration for the integral of exp(x) on [0,1] confirms the theoretical convergence behavior: the standard trapezoidal rule exhibits second-order convergence, the first-derivative corrected formula exhibits fourth-order convergence, and the formula containing both first- and third-derivative corrections exhibits sixth-order convergence. The results demonstrate that information on higher-order derivatives can be used systematically to reduce the norm of the integration error while preserving a simple grid structure. The presented approach is relevant to high-accuracy numerical integration, approximation theory, and the numerical solution of differential and integral equations.