Fast evaluation and root finding for polynomials with floating-point coefficients
Pages 325 - 334
Abstract
Evaluating or finding the roots of a polynomial f(z) = f0 + ⋅⋅⋅ + fdzd with floating-point number coefficients is a ubiquitous problem. By using a piecewise approximation of f obtained with a careful use of the Newton polygon of f, we improve state-of-the-art upper bounds on the number of operations to evaluate and find the roots of a polynomial. In particular, if the coefficients of f are given with m significant bits, we provide for the first time an algorithm that finds all the roots of f with a relative condition number lower than 2m, using a number of bit operations quasi-linear in the bit-size of the floating-point representation of f. Notably, our new approach handles efficiently polynomials with coefficients ranging from 2− d to 2d, both in theory and in practice.
References
[1]
S. Basu, R. Pollack, and M.-R. Roy. 2006. Algorithms in Real Algebraic Geometry. Springer Berlin Heidelberg, Berlin, Heidelberg. 351–401 pages.
[2]
Ruben Becker, Michael Sagraloff, Vikram Sharma, and Chee Yap. 2018. A near-optimal subdivision algorithm for complex root isolation based on the Pellet test and Newton iteration. Journal of Symbolic Computation 86 (2018), 51–96.
[3]
Dario A. Bini and Giuseppe Fiorentino. 2000. Design, analysis, and implementation of a multiprecision polynomial rootfinder. Numerical Algorithms 23, 2 (01 Jun 2000), 127–173.
[4]
Dario A Bini and Leonardo Robol. 2014. Solving secular and polynomial equations: A multiprecision algorithm. J. of Computational and Applied Mathematics 272 (2014), 276–292.
[5]
Richard P. Brent. 1976. MULTIPLE-PRECISION ZERO-FINDING METHODS AND THE COMPLEXITY OF ELEMENTARY FUNCTION EVALUATION. In Analytic Computational Complexity, J.F. Traub (Ed.). Academic Press, 151–176.
[6]
Richard P Brent and Paul Zimmermann. 2010. Modern computer arithmetic. Vol. 18. Cambridge University Press.
[7]
Jean-Pierre Dedieu. 2006. Points fixes, zéros et la méthode de Newton. Springer Berlin Heidelberg, Berlin, Heidelberg. 75–110 pages.
[8]
Xavier Gourdon. 1993. Algorithmique du theoreme fondamental de l’algebre. Research Report RR-1852. INRIA. https://hal.inria.fr/inria-00074820
[9]
Ronald L Graham and F Frances Yao. 1983. Finding the convex hull of a simple polygon. Journal of Algorithms 4, 4 (1983), 324–331.
[10]
Stef Graillat. 2008. Accurate simple zeros of polynomials in floating point arithmetic. Computers & Mathematics with Applications 56, 4 (2008), 1114–1120.
[11]
A. A. Grau. 1963. On the Reduction of Number Range in the Use of the Graeffe Process. J. ACM 10, 4 (oct 1963), 538–544.
[12]
Peter Henrici. 1974. Applied and computational complex analysis, Vol. 1. Wiley, New York.
[13]
Peter Henrici. 1993. Applied and computational complex analysis, Volume 3: Discrete Fourier analysis, Cauchy integrals, construction of conformal maps, univalent functions. Vol. 41. John Wiley & Sons.
[14]
Nicholas J Higham. 2002. Accuracy and stability of numerical algorithms. SIAM.
[15]
Rémi Imbach and Victor Y Pan. 2021. Root radii and subdivision for polynomial root-finding. In Computer Algebra in Scientific Computing: 23rd International Workshop, CASC 2021, Sochi, Russia, September 13–17, 2021, Proceedings 23. Springer, 136–156.
[16]
Fredrik Johansson. 2014. Evaluating parametric holonomic sequences using rectangular splitting. In Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation. 256–263.
[17]
Alexander Kobel and Michael Sagraloff. 2013. Fast approximate polynomial multipoint evaluation and applications. arXiv preprint arXiv:1304.8069 (2013).
[18]
Gregorio Malajovich and Jorge P Zubelli. 2001. On the geometry of Graeffe iteration. journal of complexity 17, 3 (2001), 541–573.
[19]
Gregorio Malajovich and Jorge P Zubelli. 2001. Tangent graeffe iteration. Numer. Math. 89 (2001), 749–782.
[20]
Kurt Mehlhorn, Michael Sagraloff, and Pengming Wang. 2015. From approximate factorization to root isolation with application to cylindrical algebraic decomposition. Journal of Symbolic Computation 66 (2015), 34–69.
[21]
Guillaume Moroz. 2022. New data structure for univariate polynomial approximation and applications to root isolation, numerical multipoint evaluation, and other problems. In 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS). IEEE, 1090–1099.
[22]
Jean-Michel Muller, Nicolas Brisebarre, Florent De Dinechin, Claude-Pierre Jeannerod, Vincent Lefevre, Guillaume Melquiond, Nathalie Revol, Damien Stehlé, Serge Torres, 2018. Handbook of floating-point arithmetic. Springer.
[23]
Alexandre Ostrowski. 1940. Recherches sur la méthode de graeffe et les zéros des polynomes et des séries de laurent. Acta Mathematica 72, 1 (1940), 157–257.
[24]
Victor Y Pan. 2000. Approximating complex polynomial zeros: modified Weyl’s quadtree construction and improved Newton’s iteration. J. of Complexity 16, 1 (2000), 213–264.
[25]
Victor Y. Pan. 2002. Univariate Polynomials: Nearly Optimal Algorithms for Numerical Factorization and Root-finding. Journal of Symbolic Computation 33, 5 (2002), 701–733.
[26]
Peter Ritzmann. 1986. A fast numerical algorithm for the composition of power series with complex coefficients. Theoretical Computer Science 44 (1986), 1–16.
[27]
Arnold Schönhage. 1982. The fundamental theorem of algebra in terms of computational complexity. Manuscript. Univ. of Tübingen, Germany (1982).
[28]
Joris van der Hoeven. 2008. Fast composition of numeric power series. Technical Report 2008-09. Université Paris-Sud, Orsay, France.
Recommendations
On coefficients of Carlitz cyclotomic polynomials
Let n Z + , and n ( x ) be the nth classical cyclotomic polynomial. In 4, Theorem 1, D. Lehmer showed that the geometric mean of { s ( 1 ) : s , n Z + , s n } e 2.71828 , as n ∞ . Replacing Z by F q T , and the nth elementary cylcotomic polynomial n ( x ...
Computing recurrence coefficients of multiple orthogonal polynomials
Multiple orthogonal polynomials satisfy a number of recurrence relations, in particular there is a (r+2)-term recurrence relation connecting the type II multiple orthogonal polynomials near the diagonal (the so-called step-line recurrence relation) and ...
Connection and linearization coefficients of the Askey-Wilson polynomials
The linearization problem is the problem of finding the coefficients C"k(m,n) in the expansion of the product P"n(x)Q"m(x) of two polynomial systems in terms of a third sequence of polynomials R"k(x),P"n(x)Q"m(x)=@?k=0n+mC"k(m,n)R"k(x). The polynomials ...
Comments
Information & Contributors
Information
Published In
July 2023
567 pages
Copyright © 2023 ACM.
Publication rights licensed to ACM. ACM acknowledges that this contribution was authored or co-authored by an employee, contractor or affiliate of a national government. As such, the Government retains a nonexclusive, royalty-free right to publish or reproduce this article, or to allow others to do so, for Government purposes only.
Publisher
Association for Computing Machinery
New York, NY, United States
Publication History
Published: 24 July 2023
Check for updates
Qualifiers
- Research-article
- Research
- Refereed limited
Conference
ISSAC 2023
ISSAC 2023: International Symposium on Symbolic and Algebraic Computation 2023
July 24 - 27, 2023
Tromsø, Norway
Acceptance Rates
Overall Acceptance Rate 395 of 838 submissions, 47%
Contributors
Other Metrics
Bibliometrics & Citations
Bibliometrics
Article Metrics
- 0Total Citations
- 41Total Downloads
- Downloads (Last 12 months)37
- Downloads (Last 6 weeks)2
Reflects downloads up to 12 Aug 2024
Other Metrics
Citations
View Options
Get Access
Login options
Check if you have access through your login credentials or your institution to get full access on this article.
Sign inFull Access
View options
View or Download as a PDF file.
PDFeReader
View online with eReader.
eReaderHTML Format
View this article in HTML Format.
HTML Format