A QUADRATICALLY CONVERGENT ITERATIVE METHOD FOR NONLINEAR EQUATIONS Yun, Beong-In; Petkovic, Miodrag S.;
In this paper we propose a simple iterative method for finding a root of a nonlinear equation. It is shown that the new method, which does not require any derivatives, has a quadratic convergence order. In addition, one can find that a hybrid method combined with the non-iterative method can further improve the convergence rate. To show the efficiency of the presented method we give some numerical examples.
Solving nonlinear equations by a new derivative free iterative method, Applied Mathematics and Computation, 2011, 217, 12, 5768
M. Basto, V. Semiao, and F. L. Calheiros, A new iterative method to compute nonlinear equations, Appl. Math. Comput. 173 (2006), no. 1, 468-483.
A. Ben-Israel, Newton's method with modified functions, Recent developments in optimization theory and nonlinear analysis (Jerusalem, 1995), 39-50, Contemp. Math., 204, Amer. Math. Soc., Providence, RI, 1997.
J. Chen, New modified regula falsi method for nonlinear equations, Appl. Math. Comput. 184 (2007), no. 2, 965-971.
J. Chen and W. Li, On new exponential quadratically convergent iterative formulae, Appl. Math. Comput. 180 (2006), no. 1, 242-246.
H. H. H. Homeier, A modified Newton method for rootfinding with cubic convergence, J. Comput. Appl. Math. 157 (2003), no. 1, 227-230.
V. Kanwar, S. Singh, and S. Bakshi, Simple geometric constructions of quadratically and cubically convergent iterative functions to solve nonlinear equations, Numer. Algorithm 47 (2008), no. 1, 95-107.
M. A. Noor, F. Ahmad, and S. Javeed, Two-step iterative methods for nonlinear equations, Appl. Math. Comput. 181 (2006), no. 2, 1068-1075.
A. M. Ostrowski, Solution of Equations and Systems of Equations, Second edition. Pure and Applied Mathematics, Vol. 9 Academic Press, New York-London, 1966.
L. D. Petkovic and M. S. Petkovic, A note on some recent methods for solving nonlinear equations, Appl. Math. Comput. 185 (2007), no. 1, 368-374.
M. S. Petkovic and B. I. Yun, Sigmoid-like functions and root finding methods, Appl. Math. Comput. 204 (2008), no. 2, 784-793.
R. G. Voigt, Orders of convergence for iterative procedures, SIAM J. Numer. Anal. 8 (1971), 222-243.
X. Y. Wu, Z. H. Shen, and J. L. Xia, An improved regula falsi method with quadratic convergence of both diameter and point for enclosing simple zeros of nonlinear equations, Appl. Math. Comput. 144 (2003), no. 2-3, 381-388.
B. I. Yun, A non-iterative method for solving non-linear equations, Appl. Math. Comput. 198 (2008), no. 2, 691-699.
B. I. Yun and M. S. Petkovic, Iterative methods based on the signum function approach for solving nonlinear equations, Numer. Algorithm 52 (2009), no. 4, 649-662.