A generalization of Simpson's Rule

Abstract

Let\(TA(f) = \int\limits_a^b {\frac{1}{2}(P_a^n (x) + P_b^n (x))dx} \) and let\(TA(f) = \int\limits_a^b {P_{(a + b)/2}^{n + 1} (x)dx} \), where Pcn denoles the Taylor polynomial to f at c of order n, where n is even. TA and TM are reach generalizations of the Trapezoidal rule and the midpoint rule, respectively, and are each exact for all polynomials of degree ≤n+1. We let L(f)=αTM(f)+(1−α)TA(f), where\(\alpha = \frac{{2^{n + 1} (n + 1)}}{{2^{n + 1} (n + 1) + 1}}\), to obtain a numerical integration rule L which is exact for all polynomials of degree≤n+3 (see Theorem 1). The case n=0 is just the classical Simpson's rule. We analyze in some detail the case n=2, where our formulae appear to be new. By replacing P(a+b)2/n+1(x) by the Hermite cubic interpolant at a and b, we obtain some known formulae by a different approach (see [1] and [2]). Finally we discuss some nonlinear numerical integration rules obtained by taking piecewise polynomials of odd degree, each piece being the Taylor polynomial of f at a and b, respectively. Of course all of our formulae can be compounded over subintervals of [a,b].