Numerically Integrating Irregularly-spaced (x, y) Data

Abstract

(ProQuest: ... denotes formulae omitted.)1. IntroductionA survey of basic techniques of numerical integration is a common element of college-level calculus classes. Virtually all such students can expect to be exposed to the Trapezoidal rule and the somewhat more accurate Simpson's rule, both of which are specific cases of a broader class of techniques known as Newton-Cotes formulas (Press, et al., 1986) More advanced students may encounter more sophisticated techniques such as Gaussian quadrature.Most textbook examples of these techniques utilize data points that are equally-spaced in the abscissa coordinate. This is not a fundamental requirement, but has the advantage that very compact expressions can be developed for the integral in such cases; a paper previously published in this journal describes how to program such routines into a spreadsheet (El-Gebeily & Yushau, 2007). In many experimental circumstances, however, the {x,y) values are not equally spaced in x. How then can you estimate the area under a "graphical" y(x) curve? Surprisingly, textbooks tend to be silent on this very practical issue. For readers familiar with more advanced numerical methods, one tactic might be to apply an interpolation scheme such as a cubic spline fit. Aside from the issue that such techniques are usually directed more at determining values of the dependent variable at specified values of the independent variable, they demand knowledge of the values of the derivatives ofy(x) at the end points of the data - information unlikely to be known in an experimental circumstance.While the simplest approach to determining the desired integral would be a trapezoidal or "picket fence" - type summation, such a procedure would be aesthetically unsatisfying: physical phenomena are not normally discontinuous. Any sensible approach needs to incorporate some "smoothing," presumably based on some sort of interpolation.The purpose of this article is to offer an easy-to-use scheme for dealing with such circumstances. The essence of the method, which is an extension of Simpson's rule, is to fit a series of parabolic segments to groups of three successive data points and accumulate the areas under the segments.Before describing the details of the computation, there is a philosophical issue here that deserves some discussion. This is that if an N-th order polynomial can always be fit exactly through N points, why not build the method to fit higher-order polynomial segments to the data? The answer offered here is that "simplest is best." If there is no model equation for the data, then there is no justification for using a polynomial of any specific order, or, for that matter, any particular function at all on which to base computing the integral. Quadratic segments are the lowest-order ones which allow one to build in some "curvature" to the run ofy(x). Simpson's rule is based on fitting parabolic segments to the often presumed equally-spaced data points, so the method developed here can be considered an extension of this time-honored technique.As sketched in Figure 1, consider three successive (x ,y) points in your data table; call them (xi, yi), (x2,y2), and (x3, yj). It is assumed that your data are ordered in terms of monotonically increasing or decreasing values of x, and do not include any "degenerate" points, that is, there can be no duplicate values of x. A unique interpolating parabola can always be fit through three non-vertical points in a plane:... (1)where the coefficients are given by inverting a 3 by 3 matrix:. …