×

Polynomial and rational approximations and the link between Schröder’s processes of the first and second kind. (English) Zbl 1474.65138

Summary: We show that Schröder’s processes of the first kind and of the second kind to obtain a simple root of a nonlinear equation are related by polynomial and rational approximations.

MSC:

65H05 Numerical computation of solutions to single equations
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Schröder, E., Ueber unendlich viele Algorithmen zur Auflösung der Gleichungen, Mathematische Annalen, 2, 2, 317-365 (1870) · JFM 02.0042.02 · doi:10.1007/BF01444024
[2] Schröder’s, E., On infinitely many algorithms for solving equations, TR-92-121, TR-2990 (1992), Institute for advanced Computer Studies, University of Maryland
[3] Kalantari, B.; Kalantari, I.; Zaare-Nahandi, R., A basic family of iteration functions for polynomial root finding and its characterizations, Journal of Computational and Applied Mathematics, 80, 2, 209-226 (1997) · Zbl 0874.65037 · doi:10.1016/S0377-0427(97)00014-9
[4] Kalantari, B., Polynomial Root-Finding and Polynomiography (2009), Singapore: World Scientific, Singapore · Zbl 1218.37003
[5] Petković, L.; Petković, M., The link between Schröder’s iteration methods of the first and second kind, Novi Sad Journal of Mathematics, 38, 3, 55-63 (2008) · Zbl 1224.65128
[6] Petković, M. S.; Petković, L. D.; Herceg, D., On Schröder’s families of root-finding methods, Journal of Computational and Applied Mathematics, 233, 8, 1755-1762 (2010) · Zbl 1184.65052 · doi:10.1016/j.cam.2009.09.012
[7] Kalantari, B., Polynomial root-finding methods whose basins of attraction approximate Voronoi diagram, Discrete & Computational Geometry, 46, 1, 187-203 (2011) · Zbl 1221.65111 · doi:10.1007/s00454-011-9330-3
[8] Dubeau, F., On comparisons of Chebyshev-Halley iteration functions based on their asymptotic constants, International Journal of Pure and Applied Mathematics, 58, 965-981 (2013)
[9] Traub, J. F., Iterative Methods for the Solution of Equations (1964), Englewood Cliffs, NJ, USA: Prentice Hall, Englewood Cliffs, NJ, USA · Zbl 0121.11204
[10] Householder, A. S., Principles of Numerical Analysis (1953), Columbus, NY, USA: McGraw-Hill, Columbus, NY, USA · Zbl 0051.34602
[11] Bodewig, E., On types of convergence and on the behavior of approximations in the neighborhood of a multiple root of an equation, Quarterly of Applied Mathematics, 7, 325-333 (1949) · Zbl 0034.37701
[12] Shub, M.; Smale, S., Computational complexity. On the geometry of polynomials and a theory of cost. I, Annales Scientifiques de l’École Normale Supérieure, 18, 1, 107-142 (1985) · Zbl 0603.65027
[13] Petković, M.; Herceg, D., On rediscovered iteration methods for solving equations, Journal of Computational and Applied Mathematics, 107, 2, 275-284 (1999) · Zbl 0940.65046 · doi:10.1016/S0377-0427(99)00105-3
[14] Kalantari, B.; Gerlach, J., Newton’s method and generation of a determinantal family of iteration functions, Journal of Computational and Applied Mathematics, 116, 1, 195-200 (2000) · Zbl 0982.65065 · doi:10.1016/S0377-0427(99)00361-1
[15] Gander, W., On Halley’s iteration method, The American Mathematical Monthly, 92, 2, 131-134 (1985) · Zbl 0574.65041 · doi:10.2307/2322644
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.