Geodesic
Optimal quadrature formulas for curvilinear integrals of the first kind
Matematičeskie trudy, Tome 26 (2023) no. 2, pp. 44-61.

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We consider the problem on optimal quadrature formulas for curvilinear integrals of the first kind that are exact for constant functions. This problem is reduced to the minimization problem 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 of the corresponding multi-dimensional space. Hence, for a prescribed set of nodes, there exists a unique optimal quadrature formula over a closed smooth contour, i.e., a formula with the least possible norm of the error functional in the conjugate space. We show that the tuple of weights of the optimal quadrature formula is a solution of a special nondegenerate system of linear algebraic equations. 
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V. L. Vaskevich; I. M. Turgunov. Optimal quadrature formulas for curvilinear integrals of the first kind. Matematičeskie trudy, Tome 26 (2023) no. 2, pp. 44-61. http://geodesic.mathdoc.fr/item/MT_2023_26_2_a2/

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