Corrected Euler-Maclaurin's formulae
Quadrature formulas are proposed of the form \[ \int_a^b f(x)\,dx = D(a,b)+T_n(a,b)+R_n \] where \(D(a,b)=\lambda_1f(x_1)+\lambda_2f(x_2)+\lambda_3f(x_3)\) is an open formula with \(x_2=(a+b)/2\) and \(x_1,x_3 = x_2 \pm (b-a)/3\), while \(T_n=\sum_{k=1}^n \mu_k [f^{(k-1)}(b)-f^{(k-1)}(a)]\) gives a correction term in the end points involving derivatives. All the weights \(\lambda_k,\mu_k\) are positive. Bernoulli polynomials are used in the construction of the formulas. Depending on conditions for the integrand \(f\), several expressions and estimates for the error term \(R_n\) are given.
scientific article; zbMATH DE number 5121519