Newton's method and symbolic dynamics
Author:
Sherman Wong
Journal:
Proc. Amer. Math. Soc. 91 (1984), 245-253
MSC:
Primary 65H05; Secondary 58F12
DOI:
https://doi.org/10.1090/S0002-9939-1984-0740179-6
MathSciNet review:
740179
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: By the use of symbolic dynamics, this note proves a result of B. Barna concerning real polynomials of degree at least 4 and having all distinct simple real roots. Specifically, the set of initial points, for which Newton's method fails to converge to a root of the given polynomial, is homeomorphic to a Cantor set. Also the note shows that the requirement for simple roots may be relaxed, and one still has Barna's result being valid.
- [1] Béla Barna, Über die Divergenzpunkte des Newtonschen Verfahrens zur Bestimmung von Wurzeln algebraischer Gleichungen. I, Publ. Math. Debrecen 3 (1953), 109–118 (1954) (German). MR 61472
- [2] Béla Barna, Über die Divergenzpunkte des Newtonschen Verfahrens zur Bestimmung von Wurzeln algebraischen Gleichungen. II, Publ. Math. Debrecen 4 (1956), 384–397 (German). MR 78956
- [3] Louis Block, John Guckenheimer, Michał Misiurewicz, and Lai Sang Young, Periodic points and topological entropy of one-dimensional maps, Global theory of dynamical systems (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1979) Lecture Notes in Math., vol. 819, Springer, Berlin, 1980, pp. 18–34. MR 591173
- [4] Fred H. Croom, Basic concepts of algebraic topology, Springer-Verlag, New York-Heidelberg, 1978. Undergraduate Texts in Mathematics. MR 0478127
- [5] Manfred Denker, Christian Grillenberger, and Karl Sigmund, Ergodic theory on compact spaces, Lecture Notes in Mathematics, Vol. 527, Springer-Verlag, Berlin-New York, 1976. MR 0457675
- [6] Heinrich W. Guggenheimer, Differential geometry, Dover Publications, Inc., New York, 1977. Corrected reprint of the 1963 edition; Dover Books on Advanced Mathematics. MR 0493768
- [7] M. Hurley and C. Martin, Roots of polynomials, Newton's algorithm, and chaotic dynamical systems, Case Western Reserve University, 1982 (preprint).
- [8] D. Saari and J. Urenko, Newton's method, circle maps and chaotic motion, Northwestern University, 1982 (preprint).
- [9] Steve Smale, The fundamental theorem of algebra and complexity theory, Bull. Amer. Math. Soc. (N.S.) 4 (1981), no. 1, 1–36. MR 590817, https://doi.org/10.1090/S0273-0979-1981-14858-8
Retrieve articles in Proceedings of the American Mathematical Society with MSC: 65H05, 58F12
Retrieve articles in all journals with MSC: 65H05, 58F12
Additional Information
DOI:
https://doi.org/10.1090/S0002-9939-1984-0740179-6
Keywords:
Newton's method,
symbolic dynamics
Article copyright:
© Copyright 1984
American Mathematical Society