Cambridge International Science Publishing Cambridge Cb1 6az Great Britain the Cost/reliability Trade-oo in Verifying Approximate Solutions to Diierential Equations 1

It is now standard practice in computational science for large scale simulations to be implemented and investigated in a problem solving environment (PSE) such as MATLAB or MAPLE. In such an environment, a scientist or engineer will formulate a mathematical model, approximate its solution using an appropriate numerical method, visualize the approximate solution and verify (or validate) the quality of the approximate solution. Traditionally we have been most concerned with the development of eeective numerical software for generating the approximate solution and several eecient and reliable numerical libraries are now available for use within the most widely used PSEs. On the other hand, the visualization and veriication tasks have received little attention, even though each often requires as much computational eeort as is involved in generating the approximate solution. In this paper we will investigate the eeectiveness of a suite of tools that we have recently introduced in the MATLAB PSE to verify approximate solutions of ordinary diierential equations. In particular we will identify and illustrate the inherent trade-oo between reliability and eeciency that arises. We will use the notion ofèeectivity index', widely used by researchers in the adaptive mesh PDE community, to quantify the quality of our veriication tools and illustrate the performance of these tools on a two test problems.