A class of numerical integration rules with first order derivatives

Abstract

A novel approach to deriving a family of quadrature formulae is presented. The first member of the new family is the corrected trapezoidal rule. The second member, a two-segment rule, is obtained by interpolating the corrected trapezoidal rule and the Simpson one-third rule. The third member, a three-segment rule, is obtained by interpolating the corrected trapezoidal rule and the Simpson three-eights rule. The fourth member, a four-segment rule is obtained by interpolating the two-segment rule with the Boole rule. The process can be carried on to generate a whole class of integration rules by interpolating the proposed rules appropriately with the Newton-Cotes rules to cancel out an additional term in the Euler-MacLaurin error formula. The resulting rules integrate correctly polynomials of degrees less or equal to n+3 if n is even and n+2 if n is odd, where n is the number of segments of the single application rules. The proposed rules have excellent round-off properties, close to those of the trapezoidal rule. Members of the new family obtain with two additional functional evaluations the same order of errors as those obtained by doubling the number of segments in applying the Romberg integration to Newton-Cotes rules. Members of the proposed family are shown to be viable alternatives to Gaussian quadrature.