Inproceedings,
Global Homotopies and Newton Methods
Recent Advances in Numerical Analysis, page 73-94. Academic Press, (1978)
DOI: 10.1016/B978-0-12-208360-0.50009-7
Abstract
This chapter describes the global homotopies and Newton methods. A key to devising global methods is to give up the monotone convergence and to consider more general homotopies. It turns out that singular matrices on the path cause no difficulties in the proof of Smales result. They cause trouble in attempts to implement this and most other global Newton methods numerically. Small steps must be taken in the neighborhood of vanishing Jacobians. This feature is not always pointed out in descriptions of the implementations but it is easily detected. These difficulties can be eliminated by using a somewhat different homotopy. The chapter discusses a pseudo-arc length continuation procedure in which the parameter is distance along a local tangent ray to the path. Using this parameter, this chapter discusses how to accurately locate the roots and the limit points on the path. These latter points are of great interest in many physical applications.
