Constructive Theory of Designing Optimal Eighth-Order Derivative-Free Methods for Solving Nonlinear Equations

Abstract

This paper stresses the theoretical nature of constructing the optimal derivative-free iterations. We give necessary and sufficient conditions for derivative-free three-point iterations with the eighth-order of convergence. We also establish the connection of derivative-free and derivative presence three-point iterations. The use of the sufficient convergence conditions allows us to design wide class of optimal derivative-free iterations. The proposed family of iterations includes not only existing methods but also new methods with a higher order of convergence.

On the development and extensions of some classes of optimal three-point iterations for solving nonlinear equations

“…The iteration ( 1), ( 20) can be considered as another variant of iterations given by Sharma et al in [12][13][14] and given by Zhanlav et al in [23]. (5) Let…”

“…In order to show the convergence behavior and to check the validity of theoretical results of the presented family (1) with parameters τn and α n , we make some numerical experiments. We also compare our methods with existing methods of same order in [13], [14] and [23] that denoted by (SAWN8) and (ZO8). Here all the computations are performed using the programming package MATHEMATICA with multiple-precision arithmetic and 1000 significant digits.…”

“…To generate basin attraction for complex polynomials using the methods, we take a grid of 400 × 400 points z 0 in the square [−3, 3] × [−3, 3] ⊂ C. We have used the method (1) for cubic polynomial p(z) = z 3 − 1 having three simple zeros. case iii 3 0.1077e-184 8.00 3 0.9069e-201 8.00 case iv, β = 0 3 0.1073e-184 8.00 3 0.1044e-227 8.00 SAWN8 [14] 3 0.5207e-153 8.00 3 0.1715e-172 8.00 ZO8 [23] 3 0.1036e-137 7.99 3 0.3317e-178 8.00…”

“…The optimal methods (1) distinguish each other only by choices of parameters τn and α n . It should be pointed out that to establish the convergence order of iterative methods often used either the error equation, see for example [1-5, 8-15, 17-19], or the nonlinear residuals [20][21][22][23]. An more detailed explanation of various aspects of convergence order based on error analysis, corrections and nonlinear residuals was given in the excellent surveys [6,7].…”

A General Optimal Iterative Scheme with Arbitrary Order of Convergence

“…When an iterative scheme reaches this upper bound, it is said to be an optimal method. Many optimal schemes have been designed in the last years with different order of convergence (see, e.g., [5][6][7][8][9][10] and the references therein).…”