Increasing the interval of convergence for a generalized Newton’s method of solving nonlinear equations
DOI:
https://doi.org/10.26089/NumMet.v17r102Keywords:
iterative processes, Newton’s method, logarithmic derivative, continuous functions defined on a segment, higher order methods, interval of convergence, transcendental equationsAbstract
An approach to the construction of an extended interval of convergence for a previously proposed generalization of Newton’s method to solve nonlinear equations of one variable. This approach is based on the boundedness of a continuous function defined on a segment. It is proved that, for the search for the real roots of a real-valued polynomial with complex roots, the proposed approach provides iterations with nonlocal convergence. This result is generalized to the case transcendental equations.
References
A. N. Gromov, “An Approach for Constructing One-Point Iterative Methods for Solving Nonlinear Equations of One Variable,” Vychisl. Metody Programm. 16, 298-306 (2015).
T. Zhanlav and O. Chuluunbaatar, “Convergence of a Continuous Analog of Newton’s Method for Solving Nonlinear Equations,” Vychisl. Metody Programm. 10, 402-407 (2009).
F. Zafar and N. A. Mir, “A Generalized Family of Quadrature Based Iterative Methods,” General Math. 18 (4), 43-51 (2010).
M. Baghmisheh, Y. Mahmoudi, and M. Jahangirirad, “A New Modification of Newton’s Method by Gauss Integration Formula,” Life Sci. J. 10, 288-291 (2013).
H. H. Omran, “Modified Third Order Iterative Method for Solving Nonlinear Equations,” J. AI-Nahrain Univ. 16 (3), 239-245 (2013).
S. D. Conte and C. W. De Boor, Elementary Numerical Analysis: An Algorithmic Approach (McGraw-Hill, New York, 1980).
A. I. Markushevich, The Theory of Analytic Functions (Nauka, Moscow, 1967; Chelsea, New York, 1977).
