Steplength Adaptations for the Predictor

Abstract

The convergence considerations of section 5.2 were carried out under the assumption that the steplength of the algorithm (5.1.1) was uniformly constant throughout. This is of course not efficient for any practical implementation. The discussion in chapter 5 did not indicate any means of choosing the steplength h > 0. To some extent of course, the steplength strategy depends upon the accuracy with which it is desired to numerically trace a solution curve. In any case, an efficient algorithm for this task needs to incorporate an automatic strategy for controlling the steplength. In this respect the PC methods are similar to the methods for numerically integrating initial value problems in ordinary differential equations. Indeed, one expedient way of tracing an implicitly defined curve c is to merely use a numerical initial value problem solver on the defining initial value problem (2.1.9). Although such an approach has been successfully used by some authors to solve a large variety of practical problems in science and engineering (for some examples, see the bibliography), the general opinion is that it is preferable to exploit the contractive properties of the zero set H⁻¹(0) relative to such iterative methods as those of Newton type.