Quadrature formulas involving derivatives of the integrand

Abstract

and likewise formulas using higher derivatives. These formulas are asymmetrical and for some uses would be more appropriate than formula (1) or (2). It is considered that formula (1), (2), or (3) may be useful when the integrand function is represented as a variable integral for which derivatives may be easier to compute than the integrand function itself. It is also anticipated that these formulas may be used in the numerical solution of differential equations. Kopal [3] has devised formulas using the first derivative values. His approach leaves difficulties in establishing existence and reality of evaluation points and in some cases he has computed more than one formula of the same degree. The method we propose has direct connection with the established theory of orthogonal polynomials which gives the existence, reality, and distinctness of the evaluation points and the positiveness of the weights as . A linear transformation to give integration limits -h, h in formula (1) or (2) results in multiplying each derivative of order n by hn+l. We have not identified the orthogonal polynomial systems with any treated in detail in the literature. However, tables of the aj and xi for k = 1, 2, m = 1 (1) 10, and k = 3, 4, m = 1 (1) 9, have been computed by G. W. Struble in [5], where tables for the Aum) of the remainder terms are also to be found. For purposes of computation we give a standard type recursion formula permitting generation of each polynomial in a sequence from its two predecessors. Throughout the paper we assume that the integrand function has all-order derivatives appearing and that these are continuous.