Optimal quadrature for definite integrals with polynomial weight

Abstract

Classical quadrature rules do not always account simultaneously for the weight, node geometry, and functional class of the integrand, which motivates formulas optimized with respect to the norm of the error functional. The purpose of this study is to construct a Sard-optimal quadrature formula for an integral with polynomial weight in a space of differentiable functions with square-integrable first derivative for a fixed system of nodes. The ????-function method is employed. The remainder is represented in terms of the derivative of the integrand, reducing minimization of the error-functional norm to independent minimization problems on the partial intervals. The integration constants are determined from the minimum conditions for the resulting quadratic functionals. Explicit optimal coefficients are obtained for an arbitrary nonuniform grid, together with a simplified form for a uniform grid. Numerical experiments with a cubic polynomial weight are performed for three test functions and compared with the composite trapezoidal rule on the same nodes. For the nonlinear test functions, the optimal formula produces smaller absolute errors; for the linear function, it reproduces the integral to machine precision. The observed error decay is approximately of second order. The coefficients can be applied directly on adaptive and nonuniform grids and incorporate the polynomial weight without a preliminary transformation of the integral.