[Paper Review] Computing Real Roots of Real Polynomials - An Efficient Method Based on Descartes' Rule of Signs and Newton Iteration

Michael Sagraloff, Kurt Mehlhorn|arXiv (Cornell University)|Aug 19, 2013

Numerical Methods and Algorithms24 references10 citations

TL;DR

This paper presents an efficient algorithm for isolating and refining real roots of square-free real polynomials using a combination of Descartes' Rule of Signs and Newton iteration. It achieves optimal bit complexity with a simpler design than prior methods, enabling high-precision root refinement with nearly optimal complexity.

ABSTRACT

Computing the real roots of a polynomial is a fundamental problem of computational algebra. We describe a variant of the Descartes method that isolates the real roots of any real square-free polynomial given through coefficient oracles. A coefficient oracle provides arbitrarily good approximations of the coefficients. The bit complexity of the algorithm matches the complexity of the best algorithm known, and the algorithm is simpler than this algorithm. The algorithm derives its speed from the combination of Descartes method with Newton iteration. Our algorithm can also be used to further refine the isolating intervals to an arbitrary small size. The complexity of root refinement is nearly optimal.

Motivation & Objective

• To develop a faster and simpler algorithm for isolating real roots of square-free real polynomials.
• To achieve optimal bit complexity comparable to the best-known algorithms.
• To enable arbitrary-precision refinement of isolating intervals for real roots.
• To combine Descartes' Rule of Signs with Newton iteration for improved efficiency.

Proposed method

• The algorithm uses coefficient oracles to access arbitrarily precise approximations of polynomial coefficients.
• It applies Descartes' Rule of Signs to isolate real roots within intervals.
• Newton iteration is integrated to refine isolating intervals to arbitrary precision.
• The method maintains high efficiency by leveraging the sign variation counting of Descartes' Rule.
• It ensures correctness through square-free polynomial assumptions and interval refinement.
• The algorithm's structure is simplified compared to previous optimal-complexity approaches.

Experimental results

Research questions

• RQ1 Can a simpler algorithm achieve the same bit complexity as the best-known methods for real root isolation?
• RQ2 How can Descartes' Rule of Signs be effectively combined with Newton iteration for real root computation?
• RQ3 What is the complexity of refining isolating intervals to arbitrary precision using this hybrid approach?
• RQ4 Can the method maintain optimal complexity while simplifying implementation?

Key findings

• The algorithm achieves the same bit complexity as the best-known algorithms for real root isolation.
• The method is simpler in design than existing optimal-complexity algorithms.
• Root refinement to arbitrary precision is nearly optimal in terms of complexity.
• The combination of Descartes' Rule and Newton iteration enables both efficient isolation and high-precision refinement.