Algorithm 451: chi-square quantiles [G1]

Abstract

This paper outlines a program that searches for the predominant terms of the asymptotic error expansion of initial value problems in ordinary differential equations and uses this information in a self-modifying extrapolation process. During the integration process, using a ratio that Carl de Boor (1971) used in an integration program, the method seeks to recognize trends of change in the error expansion of the differential equation and to adjust the method of extrapolation. A basic algorithm used in the modifying process is presented along with a brief explanation. Also, a comparison made with the well-known rational extrapolation method shows rational extrapolation to be generally less efficient in terms of function evaluations but also demonstrates that the self-modifying method is generally not able to reduce its error to the level of rational extrapolation. A note, though, shows the self-modifying method to be superior to the regular Romberg extrapolation.