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Finding polynomial roots by dynamical systems – A case study
Institut de Mathématiques (UMR CNRS7373), Campus de Luminy 163 avenue de Luminy, Case 907 13288 Marseille 9, France |
We investigate two well known dynamical systems that are designed to find roots of univariate polynomials by iteration: the methods known by Newton and by Ehrlich–Aberth. Both are known to have found all roots of high degree polynomials with good complexity. Our goal is to determine in which cases which of the two algorithms is more efficient. We come to the conclusion that Newton is faster when the polynomials are given by recursion so they can be evaluated in logarithmic time with respect to the degree, or when all the roots are all near the boundary of their convex hull. Conversely, Ehrlich–Aberth has the advantage when no fast evaluation of the polynomials is available, and when roots are in the interior of the convex hull of other roots.
References:
[1] |
L. Arnold,
Über die Nullstellenverteilung zufälliger Polynome, Math. Z., 92 (1966), 12-18.
doi: 10.1007/BF01140538. |
[2] |
T. Bilarev, M. Aspenberg and D. Schleicher,
On the speed of convergence of Newton's method for complex polynomials, Math. Comp., 85 (2016), 693-705.
doi: 10.1090/mcom/2985. |
[3] |
D. A. Bini,
Numerical computation of polynomial zeros by means of Aberth's method, Numer. Algorithms, 13 (1996), 179-200.
doi: 10.1007/BF02207694. |
[4] |
D. A. Bini and G. Fiorentino,
On the parallel evaluation of a sparse polynomial at a point, Numer. Algorithms, 20 (1999), 323-329.
doi: 10.1023/A:1019116203957. |
[5] |
D. A. Bini and G. Fiorentino, Numerical computation of polynomials roots using MPSolve version 2.2, published online, 2000. |
[6] |
B. Bollobás, M. Lackmann and D. Schleicher,
A small probabilistic universal set of starting points for finding roots of complex polynomials by Newton's method, Math. Comp., 82 (2013), 443-457.
doi: 10.1090/S0025-5718-2012-02640-8. |
[7] |
J. Carrier, L. Greengard and V. Rokhlin,
A fast adaptive multipole algorithm for particle simulations, SIAM J. Sci. Statist. Comput., 9 (1988), 669-686.
doi: 10.1137/0909044. |
[8] |
A. Douady and J. H. Hubbard, Etude dynamique des polynômes complexes Ⅰ & Ⅱ, Publications Mathématiques d'Orsay, Université de Paris-Sud, Département de Mathématiques, Orsay, 1984. |
[9] |
L. W. Ehrlich,
A modified Newton method for polynomials, Comm. of ACM, 10 (1967), 107-108.
doi: 10.1145/363067.363115. |
[10] |
P. Erdös and P. Turán,
On the distribution of roots of polynomials, Ann. of Math., 51 (1950), 105-119.
doi: 10.2307/1969500. |
[11] |
J. Hubbard, D. Schleicher and S. Sutherland,
How to find all roots of complex polynomials with Newton's method, Invent. Math., 146 (2001), 1-33.
doi: 10.1007/s002220100149. |
[12] |
R. Lodge, Y. Mikulich and D. Schleicher, A classification of postcritically finite Newton maps, preprint, (2015), arXiv: 1510.02771. |
[13] |
P. D. Proinov, Unified convergence analysis for Picard iteration in $n$-dimensional vector spaces, Calcolo, 55 (2018), Paper No. 6, 1–21.
doi: 10.1007/s10092-018-0251-x. |
[14] |
M. Randig, D. Schleicher and R. Stoll, Newton's method in practice Ⅱ: The iterated refinement Newton method and near-optimal complexity for finding all roots of some polynomials of very large degrees, preprint, (2019), arXiv: 1703.05847.
doi: 10.1016/j.tcs.2017.03.025. |
[15] |
B. Reinke, D. Schleicher and M. Stoll, The Weierstrass root finder is not generally convergent, preprint, arXiv: 2004.04777 |
[16] |
L. Robol, A Rootfinding Algorithm for Polynomials and Secular Equation, Master's thesis, University of Pisa, 2013. |
[17] |
D. Schleicher, On the efficient global dynamics of Newton's method for complex polynomials, preprint, arXiv: 1108.5773. |
[18] |
D. Schleicher, The Dynamics of Iterated Polynomials, work in progress. |
[19] |
D. Schleicher and R. Stoll,
Newton's method in practice: Finding all roots of polynomials of degree one million efficiently, Theoret. Comput. Sci., 681 (2017), 146-166.
doi: 10.1016/j.tcs.2017.03.025. |
[20] |
Sergey Shemyakov, Newton's Method for Complex Polynomials with Roots on a Unit Circle, Bachelor thesis, Jacobs University, 2018. |
show all references
References:
[1] |
L. Arnold,
Über die Nullstellenverteilung zufälliger Polynome, Math. Z., 92 (1966), 12-18.
doi: 10.1007/BF01140538. |
[2] |
T. Bilarev, M. Aspenberg and D. Schleicher,
On the speed of convergence of Newton's method for complex polynomials, Math. Comp., 85 (2016), 693-705.
doi: 10.1090/mcom/2985. |
[3] |
D. A. Bini,
Numerical computation of polynomial zeros by means of Aberth's method, Numer. Algorithms, 13 (1996), 179-200.
doi: 10.1007/BF02207694. |
[4] |
D. A. Bini and G. Fiorentino,
On the parallel evaluation of a sparse polynomial at a point, Numer. Algorithms, 20 (1999), 323-329.
doi: 10.1023/A:1019116203957. |
[5] |
D. A. Bini and G. Fiorentino, Numerical computation of polynomials roots using MPSolve version 2.2, published online, 2000. |
[6] |
B. Bollobás, M. Lackmann and D. Schleicher,
A small probabilistic universal set of starting points for finding roots of complex polynomials by Newton's method, Math. Comp., 82 (2013), 443-457.
doi: 10.1090/S0025-5718-2012-02640-8. |
[7] |
J. Carrier, L. Greengard and V. Rokhlin,
A fast adaptive multipole algorithm for particle simulations, SIAM J. Sci. Statist. Comput., 9 (1988), 669-686.
doi: 10.1137/0909044. |
[8] |
A. Douady and J. H. Hubbard, Etude dynamique des polynômes complexes Ⅰ & Ⅱ, Publications Mathématiques d'Orsay, Université de Paris-Sud, Département de Mathématiques, Orsay, 1984. |
[9] |
L. W. Ehrlich,
A modified Newton method for polynomials, Comm. of ACM, 10 (1967), 107-108.
doi: 10.1145/363067.363115. |
[10] |
P. Erdös and P. Turán,
On the distribution of roots of polynomials, Ann. of Math., 51 (1950), 105-119.
doi: 10.2307/1969500. |
[11] |
J. Hubbard, D. Schleicher and S. Sutherland,
How to find all roots of complex polynomials with Newton's method, Invent. Math., 146 (2001), 1-33.
doi: 10.1007/s002220100149. |
[12] |
R. Lodge, Y. Mikulich and D. Schleicher, A classification of postcritically finite Newton maps, preprint, (2015), arXiv: 1510.02771. |
[13] |
P. D. Proinov, Unified convergence analysis for Picard iteration in $n$-dimensional vector spaces, Calcolo, 55 (2018), Paper No. 6, 1–21.
doi: 10.1007/s10092-018-0251-x. |
[14] |
M. Randig, D. Schleicher and R. Stoll, Newton's method in practice Ⅱ: The iterated refinement Newton method and near-optimal complexity for finding all roots of some polynomials of very large degrees, preprint, (2019), arXiv: 1703.05847.
doi: 10.1016/j.tcs.2017.03.025. |
[15] |
B. Reinke, D. Schleicher and M. Stoll, The Weierstrass root finder is not generally convergent, preprint, arXiv: 2004.04777 |
[16] |
L. Robol, A Rootfinding Algorithm for Polynomials and Secular Equation, Master's thesis, University of Pisa, 2013. |
[17] |
D. Schleicher, On the efficient global dynamics of Newton's method for complex polynomials, preprint, arXiv: 1108.5773. |
[18] |
D. Schleicher, The Dynamics of Iterated Polynomials, work in progress. |
[19] |
D. Schleicher and R. Stoll,
Newton's method in practice: Finding all roots of polynomials of degree one million efficiently, Theoret. Comput. Sci., 681 (2017), 146-166.
doi: 10.1016/j.tcs.2017.03.025. |
[20] |
Sergey Shemyakov, Newton's Method for Complex Polynomials with Roots on a Unit Circle, Bachelor thesis, Jacobs University, 2018. |









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