DOI QR코드

DOI QR Code

A QUADRATICALLY CONVERGENT ITERATIVE METHOD FOR NONLINEAR EQUATIONS

  • Yun, Beong-In (Department of Informatics and Statistics Kunsan National University) ;
  • Petkovic, Miodrag S. (Department of Mathematics Faculty of Electronic Engineering University of Nis)
  • Received : 2009.11.20
  • Published : 2011.05.01

Abstract

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.

References

  1. 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. https://doi.org/10.1016/j.amc.2005.04.045
  2. 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. https://doi.org/10.1090/conm/204/02621
  3. J. Chen, New modified regula falsi method for nonlinear equations, Appl. Math. Comput. 184 (2007), no. 2, 965-971. https://doi.org/10.1016/j.amc.2006.05.203
  4. J. Chen and W. Li, On new exponential quadratically convergent iterative formulae, Appl. Math. Comput. 180 (2006), no. 1, 242-246. https://doi.org/10.1016/j.amc.2005.11.143
  5. H. H. H. Homeier, A modified Newton method for rootfinding with cubic convergence, J. Comput. Appl. Math. 157 (2003), no. 1, 227-230. https://doi.org/10.1016/S0377-0427(03)00391-1
  6. 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. https://doi.org/10.1007/s11075-007-9149-4
  7. M. A. Noor, F. Ahmad, and S. Javeed, Two-step iterative methods for nonlinear equations, Appl. Math. Comput. 181 (2006), no. 2, 1068-1075. https://doi.org/10.1016/j.amc.2006.01.065
  8. A. M. Ostrowski, Solution of Equations and Systems of Equations, Second edition. Pure and Applied Mathematics, Vol. 9 Academic Press, New York-London, 1966.
  9. 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. https://doi.org/10.1016/j.amc.2006.06.118
  10. M. S. Petkovic and B. I. Yun, Sigmoid-like functions and root finding methods, Appl. Math. Comput. 204 (2008), no. 2, 784-793. https://doi.org/10.1016/j.amc.2008.07.017
  11. R. G. Voigt, Orders of convergence for iterative procedures, SIAM J. Numer. Anal. 8 (1971), 222-243. https://doi.org/10.1137/0708023
  12. 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. https://doi.org/10.1016/S0096-3003(02)00414-9
  13. B. I. Yun, A non-iterative method for solving non-linear equations, Appl. Math. Comput. 198 (2008), no. 2, 691-699. https://doi.org/10.1016/j.amc.2007.09.006
  14. 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. https://doi.org/10.1007/s11075-009-9305-0

Cited by

  1. Solving nonlinear equations by a new derivative free iterative method vol.217, pp.12, 2011, https://doi.org/10.1016/j.amc.2010.12.055