| Abstract: | The paper considers cubature formulas for calculating integrals of functions f(X), X = (x 1, …, x n ) which are defined on the n-dimensional unit hypercube K n = [0, 1] n and have integrable mixed derivatives of the kind (partial _{begin{array}{*{20}c} {alpha _1 alpha _n } {x_1 , ldots , x_n } end{array} } f(X)), 0 ≤ α j ≤ 2. We estimate the errors R[f] = (smallint _{K^n } ) f(X)dX ? Σ k = 1 N c k f(X(k)) of cubature formulas (c k > 0) as functions of the weights c k of nodes X(k) and properties of integrable functions. The error is estimated in terms of the integrals of the derivatives of f over r-dimensional faces (r≤n) of the hypercube K n : |R(f)| ≤ (sum _{alpha _j } ) G(α j )(int_{K^r } {left| {partial _{begin{array}{*{20}c} {alpha _1 alpha _n } {x_1 , ldots , x_n } end{array} } f(X)} right|} ) dX r , where coefficients G(α j ) are criteria which depend only on parameters c k and X(k). We present an algorithm to calculate these criteria in the two- and n-dimensional cases. Examples are given. A particular case of the criteria is the discrepancy, and the algorithm proposed is a generalization of those used to compute the discrepancy. The results obtained can be used for optimization of cubature formulas as functions of c k and X(k). |
