An error analysis for numerical multiple integration. I
Robert E. Barnhill · Mathematics of Computation · 1968
Introduction.Error analyses for numerical methods dealing with functions of more than one variable are not abundant in the literature.The purpose of this paper is to popularize and to extend an idea due to Davis [12] for estimating the error made in approximating analytic functions of more than one variable.Two types of asymptotic results are given for the new cubatures, as well as numerical examples.Sard [24] has obtained error estimates that involve various partial derivatives of the function/ to be integrated.His kernel theorem for functions of two variables is established by noting the effect of an error functional on the remainder in a Taylor's expansion of /.He obtains sharp bounds for the appropriate function spaces, but these bounds are frequently inconvenient to apply because of the difficulty in computing them.For the one-dimensional case, see Stroud and Secrest [29, p. 65].In one dimension, Davis [12] has stated a method of estimation for analytic functions that has the advantage of being comparatively easy to compute.An interesting feature of Davis' work is that it can be generalized to deal with analytic functions of more than one variable.He noted this in one paper [12] and it has been mentioned again by Valentin [30].In this paper, Davis' method is extended and, in a future paper, it will be compared with Sard's method.Other authors have studied error bounds for some special cases, which we now discuss.Error estimates for cross-product rules have been given by several authors [20], [25], [29] and the general idea has been to express the error of the cross-product rule as the product of the errors of lower-dimensional rules.Variations of this procedure include, for example, Hammer's conical product rules [20].Stenger [27] has recently considered error estimates for the cross-products of Gaussian quadratures.Von Mises has established a certain error bound for cubatures, which is discussed by Stroud [28] and involves bounding certain partial derivatives after a transformation into spherical coordinates.Lyness [22] has discussed symmetric integration rules and his work contains error estimates.During the work leading to this paper, the author conjectured that theorems similar to those proved in Krylov [21] for the trapezoidal and Simpson's rules and in Meinguet [23] for Romberg one-dimensional integration could be proved for symmetric one-dimensional rules and extended to symmetric multidimensional rules by using Lyness' work.This conjecture has not been resolved.There is also a growing literature on approximation of functions of more than one variable by spline functions, references to which can be found in an article by Birkhoff and de Boor [9].Although only cubatures will be discussed in this paper, the same methods can be used for other linear approximations, some of which will be discussed in a future