Root-finding with Implicit Deflation

TL;DR

This paper explores implicit deflation techniques for polynomial root-finding, enhancing efficiency by combining variable transformations with advanced algorithms like multipoint evaluation and Fast Multipole Method.

Contribution

It introduces an implicit deflation approach combined with polynomial transformations and demonstrates significant efficiency improvements using fast algorithms.

Findings

Implicit deflation effectively isolates remaining roots.

Incorporating fast algorithms accelerates root-finding process.

The method improves local efficiency in polynomial root-finding.

Abstract

Functional iterations such as Newton's are a popular tool for polynomial root-finding. We consider realistic situation where some (e.g., better-conditioned) roots have already been approximated and where further computations is directed to approximation of the remaining roots. Such situation is also realistic for root-finding by means of subdivision. A natural approach of applying explicit deflation has been much studied and recently advanced by one of the authors of this paper, but presently we contribute to the alternative approach of applying implicit deflation, which we combine with mapping the variable and reversion of an input polynomial. We also show another unexplored direction for substantial further progress in this long and extensively studied area. Namely we dramatically increase their local efficiency by means of the incorporation of fast algorithms for multipoint polynomial evaluation and Fast Multipole Method.