On a class of modified Newton—Cotes quadrature formulae based upon mixed-type interpolation

摘要

We present quadrature rules which integrate exactly not only polynomials up to a certain degree, but also the functions sin kx and cos kx (where k is a free parameter). The formulae we obtain are modified Newton—Cotes formulae. They are derived by replacing the integrand by an interpolation function of the form a cos kx + b sin kx + Σn-2j = 0cjxj based on equally spaced nodes. The total truncation error of the modified quadrature formulae is discussed and a rigorous proof of the error term is given for the modified Simpson's 38 rule. Numerical examples show the efficiency of the modified rules and the importance of the error term.