An efficient step size control for continuation methods

Abstract

Abstract.In solving a nonlinear equation by the use of a continuation method one of the crucial problems is the choice of the step sizes. We present a model for the total computational cost of a standard numerical continuation process and solve the problem of optimal step size control for this model. Using the theoretical results as a basis, we develop an adaptive step size algorithm for Newton's method. This procedure is computationally inexpensive and it gives quite satisfactory results compared to some other numerical experiments found in the literature.

Affine invariant convergence results for Newton's method

“…We remark further that if we assume equality in (3.5) we obtain the fixed point iteration 6i+~ = 9 (6). This result, though weaker than theorem 3.7, finds application in the selection of an optimal steplength in predictor-corrector continuation procedures [6].…”

“…This result, though weaker than theorem 3.7, finds application in the selection of an optimal steplength in predictor-corrector continuation procedures [6].…”

“…The results in this paper are largely alY~ne invariant anal0gues.of known results concerning the radius of convergence of Newton's method. Such results are useful in the context of predictor-corrector continuation procedures ( [2]- [4], [6]- [7], [l 1]- [13]). Some of these results have previously been given ( [7], [15]).…”

“…In predictor-corrector continuation procedures ( [2]- [4], [6]- [7], [11]-[13] a posteriori estimates of the radius of convergence are useful, although the inherent dangers of such estimation are considerable [2]. We derive some estimates by modifying the techniques of [11]- [12].…”

Following paths through turning points

“…This result has implications in the selection of an efficient stepsize in predictorcorrector continuation [6].…”