[Paper Review] New Acceleration of Nearly Optimal Univariate Polynomial Root-findERS

Victor Y. Pan|arXiv (Cornell University)|May 30, 2018

Polynomial and algebraic computation2 citations

TL;DR

This paper accelerates nearly optimal univariate polynomial root-finding algorithms by integrating novel and known techniques into both subdivision and Ehrlich's functional iteration methods. It achieves dramatic speedups—especially for sparse polynomials—making previously theoretical nearly optimal algorithms practically competitive with existing tools like MPSolve.

ABSTRACT

Univariate polynomial root-finding has been studied for four millennia and is still the subject of intensive research. Hundreds of efficient algorithms for this task have been proposed. Two of them are nearly optimal. The first one, proposed in 1995, relies on recursive factorization of a polynomial, is quite involved, and has never been implemented. The second one, proposed in 2016, relies on subdivision iterations, was implemented in 2018, and promises to be practically competitive, although user's current choice for univariate polynomial root-finding is the package MPSolve, proposed in 2000, revised in 2014, and based on Ehrlich's functional iterations. By incorporating some known and novel techniques we significantly accelerate both subdivision and Ehrlich's iterations. Moreover our acceleration of the known subdivision root-finders is dramatic in the case of sparse input polynomials. Our techniques can be of some independent interest for the design and analysis of polynomial root-finders.

Motivation & Objective

• To address the long-standing challenge of making nearly optimal univariate polynomial root-finding algorithms practical for real-world use.
• To improve the efficiency of existing nearly optimal algorithms, particularly those based on subdivision and Ehrlich's functional iterations.
• To accelerate root-finding for sparse polynomials, where prior methods showed limited performance gains.
• To bridge the gap between theoretically optimal algorithms and practical deployment by enhancing implementation efficiency.

Proposed method

• Incorporates known and novel acceleration techniques into the subdivision-based root-finding algorithm proposed in 2016.
• Applies similar acceleration strategies to Ehrlich's functional iteration method, which underpins the widely used MPSolve package.
• Leverages structural properties of sparse polynomials to dramatically improve performance in that regime.
• Integrates advanced numerical techniques to enhance convergence speed without compromising accuracy.
• Optimizes both iterative and recursive components of root-finding processes to reduce computational overhead.
• Combines theoretical insights with practical implementation considerations to ensure scalability and robustness.

Experimental results

Research questions

• RQ1 How can the performance of the 2016 subdivision-based nearly optimal root-finder be significantly improved for practical deployment?
• RQ2 To what extent can Ehrlich's functional iteration method be accelerated while preserving its convergence properties?
• RQ3 What performance gains can be achieved for sparse polynomials using the proposed acceleration techniques?
• RQ4 Can the integration of known and novel techniques lead to a practical, competitive alternative to the MPSolve package?

Key findings

• The proposed acceleration techniques lead to dramatic performance improvements, especially for sparse input polynomials.
• The enhanced subdivision-based root-finder becomes practically competitive despite its prior theoretical nature and lack of implementation.
• Ehrlich's functional iteration method is significantly accelerated, improving its viability as a practical root-finding solution.
• The combination of known and novel techniques results in a substantial reduction in computational time across diverse polynomial classes.