Optimal Quadrature Formulas for Curvilinear Integrals of the First Kind

Abstract: We consider the problem on optimal quadrature formulas for curvilinear integrals ofthe first kind that are exact for constant functions. This problem is reduced to the minimizationproblem for a quadratic form in many variables whose matrix is symmetric and positive definite.We prove that the objective quadratic function attains its minimum at a single point ofthe corresponding multi-dimensional space. Hence, for a prescribed set of nodes, there existsa unique optimal quadrature formula over a closed smooth contour, i.e., a formula with the leastpossible norm of the error functional in the conjugate space. We show that the tuple of weights ofthe optimal quadrature formula is a solution of a special nondegenerate system of linear algebraicequations. © Pleiades Publishing, Ltd. 2024.