A Modified Simpson's Rule and Fortran Subroutine for Cumulative Numerical Integration of a Function Defined by Data Points

Abstract

Abstract : Formulas are derived for finding the areas between each pair of points under the second-degree-polynomial curve defined by three equispaced points in an x-y (cartesian) coordinate system. These formulas are a modification of Simpson's numerical-integration rule which gives only the total area lying under the curve between the initial and final points. The formulas, implemented by a Fortran computer subroutine named SIMCUM, are useful in problems where it is necessary to find integrals under a curve defined by a limited number of data points, and the cumulative integral is desired at each data point rather than at every second data point as would be possible with the ordinary form of Simpson's rule. With a fixed number of data points, the method gives improved accuracy, compared with the alternative of using the trapezoidal rule, when the 'true' curve is continuous, not a straight line, and is reasonably well defined by the data points. For a specified integration accuracy, a considerable cost saving can often be effected by using this method, instead of the trapezoidal rule with a considerably greater number of data points. (Author)