Notes on the Adaptive Simpson Quadrature Routine

A definif, ive account of the principles of automatic integration appears in the book Numerical Integration by Davis and Rabinowitz [2]. Several algorithms and routines are described which can be used to obtain an approximation having a prescribed tolerance e to a given definite integral. In general, the user provides only the limits of integration A and B, the desired tolerance e, and a subroutine FUN(X) which calculates the integrand. The automatic integration scheme, in the form of a subroutine, provides a result, thus "relieving the user of any need to think." One of the earlier, and certainly one of the most successful, routines of this type is the adaptive Simpson routine, published in algorithm form in 1962 (McKeeman [6]). Modifications of this routine (McKeeman [7], [9] and McKeeman and Tesler [8]) have been widely used in computing centers ever since. However, to the author's knowledge there is no account of this routine or error analysis of it in the open literature. In this paper the method on which the routine is based is investigated with a view to making improvements. It is perhaps pertinent to mention the form improvements in an automatic integration routine might take. In general, the required tolerance or accuracy e is given and a "good" routine attempts to attain only this accuracy using as few function evaluations as possible. In practice, the routine produces a result of greater accuracy. I t is a defect if greater accuracy is obtained at a cost of additional function evaluations, even if the increase in accuracy is considerable and the number of additional function evaluations is marginal. In fact one modification mentioned above was introduced by MeKeeman in all at tempt to correct to some extent a defect of this type. Bearing this in mind, improvements result if any changes of the following nature can be made: (i) a change which produces a result of lower accuracy, but still within the prescribed accuracy e, with a consequent reduction in the number of function evaluations required;