Abstract
We discuss Newton's method with respect to obtaining convergence to a fixed point with orders of convergence greater than 2. We identify the role played by the Schwarzian derivative in controlling the convergence of Newton's map to a super stable fixed point.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Barna, B. (1953).über die Divergenzpunkte des Newtonschen Verfahrens zur Bestimmung von Wurzeln algebraischer Gleichungen, I, Publ. Math. Debrecen, Vol. 3, pp. 109–118; (1956).II, Vol. 4, pp. 384–397.
Benzinger, H. E., Burns, S., and Palmore, J. (1987). Complex analytic dynamics and Newton's method.Phys. Lett. A,119, 441–446.
Khinchin, A. Ya. (1964).Continued Fractions, University of Chicago Press, Chicago, p. 45.
Palmore, J. (1993a). A relation between Newton's method and successive approximations for quadratic irrationals. In Hirsch, M., Marsden, J., and Shub, M. (eds.),From Topology to Computation, Springer, New York, pp. 254–259.
Palmore, J. (1993b). Shadowing by computable orbits of continued fraction convergents for algebraic numbers.Complex Variables (in press).
Singer, D. (1978). Stable orbits and bifurcations of maps of the interval.SIAM J. Appl. Math. 35, 260–267.
Wong, S. (1984). Newton's method and symbolic dynamics.Proc. Am. Math. Soc. 91(2), 245–253.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Palmore, J. Newton's method and Schwarzian derivatives. J Dyn Diff Equat 6, 507–511 (1994). https://doi.org/10.1007/BF02218860
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02218860