General three-point quadrature formulas of Euler type

Abstract

General three-point quadrature formulas for the approximate evaluation of an integral of a function $f$ over$~[0,1]$, through the values $f(x)$, $f(1/2)$, $f(1-x)$, $f^\prime(0)$ and$~f'(1)$, are derived via the extended Euler formula. Such quadratures are sometimes called “corrected†or “quadratures with end corrections†and have a higher accuracy than the adjoint classical formulas, which only include the values $f(x)$, $f(1/2)$ and $f(1-x)$. The Gauss three-point, corrected Simpson, corrected dual Simpson, corrected Maclaurin and corrected Gauss two-point formulas are recaptured as special cases. Finally, sharp estimates of error are given for this type of quadrature formula. doi:10.1017/S1446181111000721