Comparison of Some Iterative Methods for Finding Simple Root of Nonlinear Equations
[1]
Mehdi Salimi, Department of Mathematics, Tuyserkan Branch, Islamic Azad University, Tuyserkan, Iran.
[2]
Davood Nezami Behrooz, Department of Mathematics, Tuyserkan Branch, Islamic Azad University, Tuyserkan, Iran.
With the advancement of digital computer, advanced computer arithmetic and symbolic computation, several scholars presented a good number of iterative methods without memory for approximating a simple root of nonlinear equations. A main tool for solving nonlinear problems is the approximation of simple roots x* of a nonlinear equation f(x*) equal to zero, with a scalar function f, from D in R to R which is defined on an open interval D. In this paper, some third and fourth order methods without memory to compute the approximate simple roots of nonlinear equations in one variable presented in recent years are compared. Per iteration, the methods need two evaluations of the function and one evaluation of its first derivation. The efficacy of the present methods is tested on a number of numerical examples. The efficiency index of the methods is equal to 1.442 for non-optimal methods and 1.587 for optimal methods. Consequently, this method possesses very high computational efficiency. The principal aim and motivation in constructing the iterative methods for solving nonlinear equations is to reach the highest attainable order of convergence with the minimum number of function and derivation evaluations for each iteration. The main purpose of present research is to adjust the third and fourth order methods, which are rediscovery of Traub's method [1], for solving nonlinear equations for simple zeros with the equal order of convergence and the efficiency index. In the simple words, the methods are tested on several test problems and compared together with the same convergence order methods.
Simple Root, Two-point Iterative Method, Kung and Traub Conjecture, Optimal Order of Convergence, Computational Efficacy
[1]
Kung, H. T., Traub, J. F., Optimal order of one-point and multipoint iteration, Journal Assoc. Computer Mathematics, 21, 634 - 651, (1974).
[2]
Ostrowski, A. M., Solution of Equations and Systems of Equations, Academic Pres, New York (1960).
[3]
Traub, J. F., Iterative Methods for the Solution of Equations, Prentice Hall, New York, (1964).
[4]
Weerakoon, S., Fernando, T. G. I., A variant of Newton's method with accelerated third-order convergence, Applied Mathematics Letters, 13, 87-93 (2000).
[5]
Cordero, A., Torregrosa, J. R., Variants of Newton's method using fifth-order quadrature formulas, Applied Mathematics and Computation, 190, 686-698 (2007).
[6]
Bi, W., Ren, H., Wu, Q., Three-step iterative methods with eighth-order convergence for solving nonlinear equations, Journal of Computational and Applied Mathematics, 225, 105-112 (2009).
[7]
Bi, W., Wu, Q., Ren, H., A new family of eighth-order iterative methods for solving nonlinear equations, Applied Mathematics and Computation, 214, 236-245 (2009).
[8]
Lotfi, T., Sharifi, S., Salimi, M., Siegmund, S., A new class of three-point methods with optimal convergence order eight and its dynamics, Numerical Algorithms, 68, 261-288 (2015).
[9]
Salimi, M., Lotfi, T., Sharifi, S., Siegmund, S., Optimal Newton-Secant like methods without memory for solving nonlinear equations with its dynamics, International Journal of Computer Mathematics, 94 (9), 1759-1777 (2017).
[10]
Sharifi, S., Ferrara, M., Salimi, M., Siegmund, S., New modification of Maheshwari method with optimal eighth order of convergence for solving nonlinear equations, Open Mathematics (formerly Central European Journal of Mathematics), 14, 443-451 (2016).
[11]
Sharifi, S., Salimi, M., Siegmund, S., Lotfi, T., A new class of optimal four-point methods with convergence order 16 for solving nonlinear equations, Mathematics and Computers in Simulation, 119, 69-90 (2016).
[12]
Sharifi, S., Siegmund, S., Salimi, M., Solving nonlinear equations by a derivative-free form of the King's family with memory, Calcolo, 53, 201-215 (2016).
[13]
Potra, F. A., Ptak, V., Nondiscrete introduction and iterative processes, in: Research Note in Mathematics, vol 103, Pitman, Boston, (1984).
[14]
Sharma, J. R., A composite third order Newton–Steffensen method for solving nonlinear equations, Applied Mathematics and Computation, 169, 242-246 (2005).
[15]
King, R. F., A family of fourth order methods for nonlinear equations, SIAM Journal on Numerical Analysis, 10, 876-879 (1973).
[16]
Maheshwari, A. K., A fourth-order iterative method for solving nonlinear equations, Applied Mathematics and Computation, 211, 383-391 (2009).
[17]
Kou, J., Li, Y., Wang, X., A composite fourth-order iterative method, Applied Mathematics and Computation, 184, 471-475 (2007).
[18]
Chun, C., A family of composite fourth-order iterative methods for solving nonlinear equations, Applied Mathematics and Computation, 187, 951-956 (2007).
