Higher Order Predictors in Numerical Path Following Schemes

Abstract

A new class of polynomial higher order predictors for numerical path following schemes is presented. Explicit computation of first order derivatives only is required, no process of numerical differentiation is involved to obtain the additional higher order terms. The new predictor method presented here offers a variety of advantages and possible applications. The method is hierarchical and adaptive : predictors of varying order may be used on following a solution curve ; the method allows to detect the presence as well as to efficiently compute the exact locations of limit points ; it allows to monitor the approximation quality by means of a forward—backward strategy.

To improve quality in the subsequent corrector process a new ellipse normalization condition is suggested which provides an automatic step length and direction adjustment in the iterative process ; it can be used in combination with the common predictors based on tangent directions as well.