scholarly journals Improving the Computational Efficiency of a Variant of Steffensen’s Method for Nonlinear Equations

Mathematics ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 306 ◽  
Author(s):  
Fuad Khdhr ◽  
Rostam Saeed ◽  
Fazlollah Soleymani
Keyword(s):  
Nonlinear Equations ◽  
Computational Efficiency ◽  
High Efficiency ◽  
Efficiency Index ◽  
Numerical Investigations ◽  
Steffensen’S Method ◽  
Acceleration Technique ◽  
Derivative Free ◽  
Interpolation Polynomials ◽  
With Memory

Steffensen-type methods with memory were originally designed to solve nonlinear equations without the use of additional functional evaluations per computing step. In this paper, a variant of Steffensen’s method is proposed which is derivative-free and with memory. In fact, using an acceleration technique via interpolation polynomials of appropriate degrees, the computational efficiency index of this scheme is improved. It is discussed that the new scheme is quite fast and has a high efficiency index. Finally, numerical investigations are brought forward to uphold the theoretical discussions.

scholarly journals On Some Iterative Methods with Memory and High Efficiency Index for Solving Nonlinear Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Tahereh Eftekhari
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
High Efficiency ◽  
Efficiency Index ◽  
Order Convergence ◽  
Eighth Order ◽  
Additional Function ◽  
Numerical Comparisons ◽  
With Memory

Based on iterative methods without memory of eighth-order convergence proposed by Thukral (2012), some iterative methods with memory and high efficiency index are presented. We show that the order of convergence is increased without any additional function evaluations. Numerical comparisons are made to show the performance of the presented methods.

scholarly journals Efficient Four-Parametric with-and-without-Memory Iterative Methods Possessing High Efficiency Indices

2018 ◽  
Vol 2018 ◽  
pp. 1-12 ◽  
Author(s):  
Alicia Cordero ◽  
Moin-ud-Din Junjua ◽  
Juan R. Torregrosa ◽  
Nusrat Yasmin ◽  
Fiza Zafar
Keyword(s):  
Iterative Methods ◽  
High Efficiency ◽  
Nonlinear Function ◽  
Efficiency Index ◽  
Simple Zero ◽  
Eighth Order ◽  
New Family ◽  
Derivative Free ◽  
With Memory ◽  
Theoretical Results

We construct a family of derivative-free optimal iterative methods without memory to approximate a simple zero of a nonlinear function. Error analysis demonstrates that the without-memory class has eighth-order convergence and is extendable to with-memory class. The extension of new family to the with-memory one is also presented which attains the convergence order 15.5156 and a very high efficiency index 15.51561/4≈1.9847. Some particular schemes of the with-memory family are also described. Numerical examples and some dynamical aspects of the new schemes are given to support theoretical results.

scholarly journals Efficient Iterative Methods with and without Memory Possessing High Efficiency Indices

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
T. Lotfi ◽  
F. Soleymani ◽  
Z. Noori ◽  
A. Kılıçman ◽  
F. Khaksar Haghani
Keyword(s):  
Nonlinear Equation ◽  
High Efficiency ◽  
Efficiency Index ◽  
Simple Zero ◽  
Numerical Examples ◽  
Derivative Free ◽  
Derivative Free Methods ◽  
With Memory ◽  
Theoretical Results ◽  
High Computational Efficiency

Two families of derivative-free methods without memory for approximating a simple zero of a nonlinear equation are presented. The proposed schemes have an accelerator parameter with the property that it can increase the convergence rate without any new functional evaluations. In this way, we construct a method with memory that increases considerably efficiency index from81/4≈1.681to121/4≈1.861. Numerical examples and comparison with the existing methods are included to confirm theoretical results and high computational efficiency.

scholarly journals Solving Nondifferentiable Nonlinear Equations by New Steffensen-Type Iterative Methods with Memory

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
J. P. Jaiswal
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Efficiency Index ◽  
The Self ◽  
Function Evaluation ◽  
Eighth Order ◽  
Derivative Free ◽  
With Memory

It is attempted to present two derivative-free Steffensen-type methods with memory for solving nonlinear equations. By making use of a suitable self-accelerator parameter in the existing optimal fourth- and eighth-order without memory methods, the order of convergence has been increased without any extra function evaluation. Therefore, its efficiency index is also increased, which is the main contribution of this paper. The self-accelerator parameters are estimated using Newton’s interpolation. To show applicability of the proposed methods, some numerical illustrations are presented.

scholarly journals An Efficient Derivative Free One-Point Method with Memory for Solving Nonlinear Equations

Mathematics ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 604 ◽  
Author(s):  
Janak Raj Sharma ◽  
Sunil Kumar ◽  
Clemente Cesarano
Keyword(s):  
Nonlinear Equations ◽  
Efficiency Index ◽  
Point Method ◽  
Function Evaluation ◽  
Systems Of Nonlinear Equations ◽  
Academic Problems ◽  
Similar Nature ◽  
Newton's Iteration ◽  
Derivative Free ◽  
With Memory

We propose a derivative free one-point method with memory of order 1.84 for solving nonlinear equations. The formula requires only one function evaluation and, therefore, the efficiency index is also 1.84. The methodology is carried out by approximating the derivative in Newton’s iteration using a rational linear function. Unlike the existing methods of a similar nature, the scheme of the new method is easy to remember and can also be implemented for systems of nonlinear equations. The applicability of the method is demonstrated on some practical as well as academic problems of a scalar and multi-dimensional nature. In addition, to check the efficacy of the new technique, a comparison of its performance with the existing techniques of the same order is also provided.

scholarly journals On a New Three-Step Class of Methods and Its Acceleration for Nonlinear Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
T. Lotfi ◽  
K. Mahdiani ◽  
Z. Noori ◽  
F. Khaksar Haghani ◽  
S. Shateyi
Keyword(s):  
Nonlinear Equation ◽  
Convergence Rate ◽  
Nonlinear Equations ◽  
Efficiency Index ◽  
Simple Zero ◽  
Convergence Order ◽  
Underlying Theory ◽  
Derivative Free ◽  
Derivative Free Methods ◽  
With Memory

A class of derivative-free methods without memory for approximating a simple zero of a nonlinear equation is presented. The proposed class uses four function evaluations per iteration with convergence order eight. Therefore, it is an optimal three-step scheme without memory based on Kung-Traub conjecture. Moreover, the proposed class has an accelerator parameter with the property that it can increase the convergence rate from eight to twelve without any new functional evaluations. Thus, we construct a with memory method that increases considerably efficiency index from81/4≈1.681to121/4≈1.861. Illustrations are also included to support the underlying theory.

scholarly journals On a Derivative-Free Variant of King’s Family with Memory

2015 ◽  
Vol 2015 ◽  
pp. 1-5
Author(s):  
M. Sharifi ◽  
S. Karimi Vanani ◽  
F. Khaksar Haghani ◽  
M. Arab ◽  
S. Shateyi
Keyword(s):  
Nonlinear Equations ◽  
Numerical Experiments ◽  
Order Of Convergence ◽  
Derivative Free ◽  
With Memory

The aim of this paper is to construct a method with memory according to King’s family of methods without memory for nonlinear equations. It is proved that the proposed method possesses higherR-order of convergence using the same number of functional evaluations as King’s family. Numerical experiments are given to illustrate the performance of the constructed scheme.

scholarly journals A family of two-point methods with memory for solving nonlinear equations

2011 ◽  
Vol 5 (2) ◽  
pp. 298-317 ◽  
Author(s):  
Miodrag Petkovic ◽  
Jovana Dzunic ◽  
Ljiljana Petkovic
Keyword(s):  
Nonlinear Equations ◽  
Free Parameter ◽  
Convergence Order ◽  
Optimal Order ◽  
Additional Function ◽  
Numerical Examples ◽  
Derivative Free ◽  
With Memory ◽  
Theoretical Results ◽  
High Computational Efficiency

An efficient family of two-point derivative free methods with memory for solving nonlinear equations is presented. It is proved that the convergence order of the proposed family is increased from 4 to at least 2 + ?6 ? 4.45, 5, 1/2 (5 + ?33) ? 5.37 and 6, depending on the accelerating technique. The increase of convergence order is attained using a suitable accelerating technique by varying a free parameter in each iteration. The improvement of convergence rate is achieved without any additional function evaluations meaning that the proposed methods with memory are very efficient. Moreover, the presented methods are more efficient than all existing methods known in literature in the class of two-point methods and three-point methods of optimal order eight. Numerical examples and the comparison with the existing two-point methods are included to confirm theoretical results and high computational efficiency. 2010 Mathematics Subject Classification. 65H05

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