Predictor-Corrector Methods Using Updating

In iterative methods for numerically finding zero-points of a smooth map F : RN → RN it is often preferable to avoid the costly recalculation and decomposition of the Jacobian F′ at each iteration by using an approximation to F′. This results in sacrificing quadratic convergence in exchange for a superlinear convergence which is nearly as good, or a rather fast rate of linear convergence. When N is of at least moderate size, or F′ is otherwise cumbersome to calculate, this trade-off is usually to be preferred. It is reasonable to expect that the same situation should also hold in the corrector step of the predictor-corrector methods for numerically tracing H−1(0) where H : RN+1 → RN is a smooth map. Indeed, since the corrector process needs to be performed essentially at every predicted point, the possibility of saving computational effort in the corrector process becomes even more attractive for the predictor-corrector curve tracing methods.