Osculatory mechanical quadrature

Abstract

The literature contains many formulas of mechanical quadrature¹, most of which are expressible in the form where the A's are constants, f(a _v) represents the functional value of f(x) at each of the n + 1 points x=a _v (v=0,1,2,..., n), and R is the remainder term. Two general and important types of the above formula are the Newton-Cotes ² formula in which the points a _v are equally spaced from c to d, and the Gaussian ³ quadrature formula in which the a's are chosen so as to obtain the greatest accuracy. The Euler-Maclaurin ⁴ formula of summation and quadrature uses the functional values f(a _v ), and the odd ordered derivatives of f(x) at the end points of the interval of integration. Steffensen ⁵ developed a formula for approximate integration employing not only the functional values but the first derivatives, f'(a _v ).