Improvement of the accuracy for the Euler-Maclaurin quadrature formulas
Abstract
The present work is devoted to extension of the Euler-Maclaurin formula in the Hilbert space W2(2k,2k−1). The optimal quadrature formula is obtained by minimizing the error of the formula by coefficients at values of the (2k – 1)-th derivative of a function. Using the discrete analogue of the operator d2dx2−1 the explicit formulas for the coefficients of the optimal quadrature formula are obtained. Furthermore, it is proved that the obtained quadrature formula is exact for any function of the set F = span {1, 2, …x2k−2, e−x}. Finally, in the space W2(2k,2k−1) (k = 1, 2, 3), the square of the norm of the error functional for the constructed quadrature formula is calculated. It is shown that the errors of the obtained optimal quadrature formulas are less than the errors of the Euler-Maclaurin quadrature formulas on the space L2(2k) (k = 1, 2, 3).