A New Derivative-Free Quasi-Secant Algorithm For Solving Non-Linear Equations

Authors: F. Soleymani, M. Sharifi

Abstract:

Most of the nonlinear equation solvers do not converge always or they use the derivatives of the function to approximate the root of such equations. Here, we give a derivative-free algorithm that guarantees the convergence. The proposed two-step method, which is to some extent like the secant method, is accompanied with some numerical examples. The illustrative instances manifest that the rate of convergence in proposed algorithm is more than the quadratically iterative schemes.

Keywords: Non-linear equation, iterative methods, derivative-free, convergence.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1062098

References:

[1] K.E. Atkinson, An Introduction to Numerical Analysis, 2nd ed., John Wiley & Sons, Singapore, 1988.
[2] R.P. Brent, Algorithms for Minimization without Derivatives, Prentice Hall, Englewood Cliffs, NJ, 1973.
[3] C. Chun, Some fourth-order iterative methods for solving nonlinear equations, Applied Mathematics and Computation 195 (2008), 454-459.
[4] M. Grau and M. Noguera, A variant of Cauchy-s method with accelerated fifth-order convergence, Applied Mathematics Letters 17 (2004) 509-517.
[5] M. Grau-Snchez, Improvements of the efficiency of some three-step iterative like-Newton methods, Numerical Mathematics 107 (2007) 131-146.
[6] Z. Hui, L. De-Sheng and L. Yu-Zhong, A new method of secant-like for nonlinear equations, Communications for Nonlinear Sciences and Numerical Simulation 14 (2009) 2923-2927.
[7] J. Kou and X. Wang, Y. Li, Some eighth-order root-finding threestep methods, Communications for Nonlinear Sciences and Numerical Simulation, In Press (2009).
[8] J. Kou and X. Wang, Some improvements of Ostrowski-s method, Applied Mathematics Letters, In Press (2009).
[9] A.M. Ostrowski, Solution of equations in Euclidean and Banach space, Academic Press, New York 1973.
[10] T. Sauer, Numerical Analysis, Addison Wesley Publication, USA, 2005.
[11] B. I. Yun and M.S. Petkovi, A quadratically convergent iterative method for nonlinear equations, In Press. (2009).