The efficient, scale-invariant determination of starting step sizes for high-order integrators

Many routines for solving differential equations require the user to specify an estimate of the initial step size. We introduce a means of estimating an appropriate value automatically. Strong emphasis is placed on the need for scale invariance in making such estimates, and we therefore review the importance of scale and matters equivalent to scale in estimating the order of magnitude of the appropriate starting step size. In this context, we also discuss the response to scale of integrators employing the error per unit step concept of error control and are forced to conclude that our procedure should not be used in such integrators, if the latter should be used at all. We note a special case where the procedure could be badly deceived and suggest an option by which the user can protect himself against this kind of mishap. Test results are cited.