Automatic selection of the initial step size for an ODE solver

The choice of initial step size is critical for the reliable numerical solution of the initial value problem for a system of ordinary differential equations. Automatic selection of this step size may lead to a more robust and efficient integration than its provision by a user, and is always more convenient. It is especially important for the reliability of an ODE solver used as a module in a larger software package. Previous approaches to making the selection are combined with some new ideas to produce an effective scheme for the automatic choice of the initial step size. Numerical results illustrate the roles played by the individual phases of the algorithm and show that the whole algorithm is both robust and efficient.