New Practical Advances in Polynomial Root Clustering

  • Rémi ImbachEmail author
  • Victor Y. Pan
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11989)


We report an ongoing work on clustering algorithms for complex roots of a univariate polynomial p of degree d with real or complex coefficients. As in their previous best subdivision algorithms our root-finders are robust even for multiple roots of a polynomial given by a black box for the approximation of its coefficients, and their complexity decreases at least proportionally to the number of roots in a region of interest (ROI) on the complex plane, such as a disc or a square, but we greatly strengthen the main ingredient of the previous algorithms. We build the foundation for a new counting test that essentially amounts to the evaluation of a polynomial p and its derivative \(p'\), which is a major benefit, e.g., for sparse polynomials p. Moreover with evaluation at about \(\log (d)\) points (versus the previous record of order d) we output correct number of roots in a disc whose contour has no roots of p nearby. Our second and less significant contribution concerns subdivision algorithms for polynomials with real coefficients. Our tests demonstrate the power of the proposed algorithms.


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© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  2. 2.Lehman College and the Graduate CenterCity University of New YorkNew YorkUSA

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