Convergence Proofs for Numerical IVP Software

Abstract

The study of the running times of algorithms in computer science can be broken down into two broad types: worst-case and average-case analyses. For many problems this distinction is very important as the orders of magnitude (in terms of some measure of the problem size) of the running times may differ significantly in each case, providing useful information about the merits of the algorithm. Historically average-case analyses were first done with respect to a measure on the input data; to counter the argument that it is often difficult to find a natural measure on the data, randomised algorithms were then developed. In this paper similar questions are studied for adaptive software used to integrate initial value problems for ODEs. In this worst case these algorithms may fail completely giving O(1) errors. We consider the probability of failure for generic vector fields with random initial data chosen from a ball and perform average-case and worst-case analyses. We then perform a different average-case analysis where, having fixed the initial data, it is the algorithm that is chosen at random from some suitable class. This last analysis suggests a modified deterministic algorithm which cannot fail for generic vector fields.