Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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

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.


References [Enhancements On Off] (What's this?)

  • [1] B. Barna, über die Divergenzpunkte des Newtonschen Verfahrens zur Bestimmung von Wurzeln algebraischer Gleichungen. I, Publ. Math. Debrecen 3 (1953), 109-118. MR 0061472 (15:831f)
  • [2] -, Über die Divergenzpunkte des Newtonschen Verfahrens zur Bestimmung von Wurzeln algebraischer Gleichungen. II, Publ. Math. Debrecen 4 (1956), 384-397. MR 0078956 (18:5f)
  • [3] L. Block, J. Guckenheimer, M. Misiurewicz and L.-S. Young, Periodic points and topological entropy of one-dimensional maps, Global Theory of Dynamical Systems, Lecture Notes in Math., vol. 819, Springer-Verlag, New York, 1980, pp. 18-34. MR 591173 (82j:58097)
  • [4] F. Croom, Basic concepts of algebraic topology, Springer-Verlag, New York, 1978. MR 0478127 (57:17616)
  • [5] M. Denker, C. Grillenberger and K. Sigmund, Ergodic theory of compact spaces, Lecture Notes in Math., vol. 527, Springer-Verlag, New York, 1976. MR 0457675 (56:15879)
  • [6] H. Guggenheimer, Differential geometry, Dover, New York, 1977. MR 0493768 (58:12737)
  • [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] S. Smale, The functional theorem of algebra and complexity theory, Bull. Amer. Math. Soc. (N.S.) 4 (1981), 1-36. MR 590817 (83i:65044)

Similar Articles

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

American Mathematical Society