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