Abstract
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.
Rémi’s work is supported by NSF Grants # CCF-1563942 and # CCF-1708884.
Victor’s work is supported by NSF Grants # CCF-1116736 and # CCF-1563942 and by PSC CUNY Award 698130048.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
Notes
- 1.
- 2.
By effective, we refer to the pathway proposed in [XY19] to describe algorithms in three levels: abstract, interval, effective.
- 3.
MPsolve tries to isolate the roots unless the escape bound \(10^{-16}\) is reached.
References
Bini, D.A., Fiorentino, G.: Design, analysis, and implementation of a multiprecision polynomial rootfinder. Numer. Algorithms 23(2), 127–173 (2000)
Bini, D.A., Robol, L.: Solving secular and polynomial equations: a multiprecision algorithm. J. Comput. Appl. Math. 272, 276–292 (2014)
Becker, R., Sagraloff, M., Sharma, V., Xu, J., Yap, C.: Complexity analysis of root clustering for a complex polynomial. In: Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation, ISSAC 2016, pp. 71–78. ACM, New York (2016)
Becker, R., Sagraloff, M., Sharma, V., Yap, C.: A near-optimal subdivision algorithm for complex root isolation based on Pellet test and Newton iteration. J. Symb. Comput. 86, 51–96 (2018)
Henrici, P., Gargantini, I.: Uniformly convergent algorithms for the simultaneous approximation of all zeros of a polynomial. In: Constructive Aspects of the Fundamental Theorem of Algebra, pp. 77–113. Wiley-Interscience, New York (1969)
Imbach, R., Pan, V.Y., Yap, C.: Implementation of a near-optimal complex root clustering algorithm. Math. Soft. - ICMS 2018, 235–244 (2018)
Kobel, A., Rouillier, F., Sagraloff, M.: Computing real roots of real polynomials ... and now for real! In: Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation, ISSAC 2016, pp. 303–310. ACM, New York (2016)
Pan, V.Y.: Approximating complex polynomial zeros: modified Weyl’s quadtree construction and improved newton’s iteration. J. Complex. 16(1), 213–264 (2000)
Pan, V.Y.: Univariate polynomials: nearly optimal algorithms for numerical factorization and root-finding. J. Symb. Comput. 33(5), 701–733 (2002)
Pan, V.Y.: Old and new nearly optimal polynomial root-finders. ar**v preprint ar**v:1805.12042 (2018)
Pan, V.Y., Tsigaridas, E.P.: On the Boolean complexity of real root refinement. In: Proceedings of the 38th International Symposium on Symbolic and Algebraic Computation, ISSAC 2013, pp. 299–306. ACM, New York (2013)
Pan, V.Y., Tsigaridas, E.P.: Nearly optimal refinement of real roots of a univariate polynomial. J. Symb. Comput 74, 181–204 (2016)
Renegar, J.: On the worst-case arithmetic complexity of approximating zeros of polynomials. J. Complex. 3(2), 90–113 (1987)
Schönhage, A.: The fundamental theorem of algebra in terms of computational complexity. Manuscript. University of TĂ¼bingen, Germany (1982)
Sagraloff, M., Mehlhorn, K.: Computing real roots of real polynomials. J. Symb. Comput. 73, 46–86 (2016)
Xu, J., Yap, C.: Effective subdivision algorithm for isolating zeros of real systems of equations, with complexity analysis. ar**v preprint (2019). ar**v:1905.03505
Zaderman, V., Zhao, L.: Counting roots of a polynomial in a convex compact region by means of winding number calculation via sampling. ar**v preprint ar**v:1906.10805 (2019)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
Imbach, R., Pan, V.Y. (2020). New Practical Advances in Polynomial Root Clustering. In: Slamanig, D., Tsigaridas, E., Zafeirakopoulos, Z. (eds) Mathematical Aspects of Computer and Information Sciences. MACIS 2019. Lecture Notes in Computer Science(), vol 11989. Springer, Cham. https://doi.org/10.1007/978-3-030-43120-4_11
Download citation
DOI: https://doi.org/10.1007/978-3-030-43120-4_11
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-43119-8
Online ISBN: 978-3-030-43120-4
eBook Packages: Computer ScienceComputer Science (R0)