Homotopy Continuation and Newton’s Method
A general approach to solving a problem consists in reducing it to another problem for which a solution can be found. The first section in this chapter is an example of this approach for the zero-finding problem. Yet, in most occurrences of this strategy, this auxiliary problem is different from the original one, as in the reduction of a nonlinear problem to one or more linear ones. In contrast with this, the treatment we consider reduces the situation at hand to the consideration of a number of instances of the same problem with different data. The key remark is that for these instances, either we know the corresponding solution or we can compute it with little effort.
We mentioned in the introduction of the previous chapter that even for functions as simple as univariate polynomials, there is no hope of computing their zeros, and the best we can do is to compute accurate approximations. A goal of the second section in this chapter is to provide a notion of approximation (of a zero) that does not depend on preestablished accuracies. It has an intrinsic character. In doing so, we rely on a pearl of numerical analysis, Newton’s method, and on the study of it pioneered by Kantorovich and Smale.