General three-point quadrature formulas of Euler type

Iva Franjic, Josip Pecaric, Ivan Peric

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

Keywords


general three-point quadrature formulas; corrected quadrature formulas; sharp estimates of error; Bernoulli polynomials; extended Euler formulaarp estimates of error, Bernoulli polynomials, extended Euler formula



DOI: http://dx.doi.org/10.21914/anziamj.v52i0.1964



Remember, for most actions you have to record/upload into this online system
and then inform the editor/author via clicking on an email icon or Completion button.
ANZIAM Journal, ISSN 1446-8735, copyright Australian Mathematical Society.