Higher order derivative-free iterative methods with and without memory for systems of nonlinear equations

2017 ◽  
Vol 314 ◽  
pp. 199-211 ◽  
Author(s):  
F. Ahmad ◽  
F. Soleymani ◽  
F. Khaksar Haghani ◽  
S. Serra-Capizzano
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Higher Order ◽  
Order Derivative ◽  
Systems Of Nonlinear Equations ◽  
Derivative Free ◽  
Higher Order Derivative

scholarly journals An Efficient Class of Traub–Steffensen-Type Methods for Computing Multiple Zeros

Axioms ◽  
2019 ◽  
Vol 8 (2) ◽  
pp. 65 ◽  
Author(s):  
Deepak Kumar ◽  
Janak Raj Sharma ◽  
Clemente Cesarano
Keyword(s):  
Iterative Methods ◽  
Basins Of Attraction ◽  
Higher Order ◽  
Order Derivative ◽  
Third Order ◽  
Higher Order Methods ◽  
Multiple Zeros ◽  
Derivative Free ◽  
Graphical Tool ◽  
Numerical Experimentation

Numerous higher-order methods with derivative evaluations are accessible in the literature for computing multiple zeros. However, higher-order methods without derivatives are very rare for multiple zeros. Encouraged by this fact, we present a family of third-order derivative-free iterative methods for multiple zeros that require only evaluations of three functions per iteration. Convergence of the proposed class is demonstrated by means of using a graphical tool, namely basins of attraction. Applicability of the methods is demonstrated through numerical experimentation on different functions that illustrates the efficient behavior. Comparison of numerical results shows that the presented iterative methods are good competitors to the existing techniques.

scholarly journals Higher-Order Derivative-Free Iterative Methods for Solving Nonlinear Equations and Their Basins of Attraction

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1052 ◽  
Author(s):  
Jian Li ◽  
Xiaomeng Wang ◽  
Kalyanasundaram Madhu
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Complex Plane ◽  
Dynamical Behavior ◽  
Efficiency Index ◽  
Basins Of Attraction ◽  
Type Method ◽  
New Methods ◽  
Derivative Free ◽  
Higher Order Derivative

Based on the Steffensen-type method, we develop fourth-, eighth-, and sixteenth-order algorithms for solving one-variable equations. The new methods are fourth-, eighth-, and sixteenth-order converging and require at each iteration three, four, and five function evaluations, respectively. Therefore, all these algorithms are optimal in the sense of Kung–Traub conjecture; the new schemes have an efficiency index of 1.587, 1.682, and 1.741, respectively. We have given convergence analyses of the proposed methods and also given comparisons with already established known schemes having the same convergence order, demonstrating the efficiency of the present techniques numerically. We also studied basins of attraction to demonstrate their dynamical behavior in the complex plane.

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