Condition pp 283-294 | Cite as

Homotopy Continuation and Newton’s Method

  • Peter Bürgisser
  • Felipe Cucker
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 349)


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.


Auxiliary Problem Strong Approximation Intrinsic Character Univariate Polynomial Homotopy Method 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Peter Bürgisser
    • 1
  • Felipe Cucker
    • 2
  1. 1.Institut für MathematikTechnische Universität BerlinBerlinGermany
  2. 2.Department of MathematicsCity University of Hong KongHong KongHong Kong SAR

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