scholarly journals Two Bi-Accelerator Improved with Memory Schemes for Solving Nonlinear Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
J. P. Jaiswal
Keyword(s):  
Theoretical Result ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Order Convergence ◽  
Derivative Free ◽  
Derivative Free Methods ◽  
Memory Schemes ◽  
With Memory ◽  
Interpolatory Polynomial

The present paper is devoted to the improvement of theR-order convergence of with memory derivative free methods presented by Lotfi et al. (2014) without doing any new evaluation. To achieve this aim one more self-accelerating parameter is inserted, which is calculated with the help of Newton’s interpolatory polynomial. First theoretically it is proved that theR-order of convergence of the proposed schemes is increased from 6 to 7 and 12 to 14, respectively, without adding any extra evaluation. Smooth as well as nonsmooth examples are discussed to confirm theoretical result and superiority of the proposed schemes.

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 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 Derivative-Free Methods with High Order of Convergence and Its Application to Nonsmooth Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Alicia Cordero ◽  
José L. Hueso ◽  
Eulalia Martínez ◽  
Juan R. Torregrosa
Keyword(s):  
Fourth Order ◽  
Order Of Convergence ◽  
Nonsmooth Equations ◽  
Order Convergence ◽  
Linear Combinations ◽  
Numerical Examples ◽  
Derivative Free ◽  
Derivative Free Methods ◽  
Seventh Order ◽  
The Given

A family of derivative-free methods of seventh-order convergence for solving nonlinear equations is suggested. In the proposed methods, several linear combinations of divided differences are used in order to get a good estimation of the derivative of the given function at the different steps of the iteration. The efficiency indices of the members of this family are equal to 1.6266. Also, numerical examples are used to show the performance of the presented methods, on smooth and nonsmooth equations, and to compare with other derivative-free methods, including some optimal fourth-order ones, in the sense of Kung-Traub’s conjecture.

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 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.

On Some Highly Efficient Derivative Free Methods With and Without Memory for Solving Nonlinear Equations

2015 ◽  
Vol 12 (01) ◽  
pp. 1350093 ◽  
Author(s):  
J. R. Sharma ◽  
Puneet Gupta
Keyword(s):  
Nonlinear Equations ◽  
Free Parameter ◽  
Order Of Convergence ◽  
Discrete Math ◽  
Rational Approximations ◽  
Computational Results ◽  
Additional Function ◽  
Derivative Free ◽  
Point Scheme ◽  
With Memory

We present derivative free multipoint methods of optimal eighth and sixteenth order convergence for solving nonlinear equations. The schemes are based on derivative free two-point methods proposed by Petković et al. [Petković, M. S., Džunić, J. and Petković, L. D. [2011] "A family of two-point methods with memory for solving nonlinear equations," Appl. Anal. Discrete Math.5, 298–317], which further developed by using rational approximations. Extending the work further, we explore four-point methods with memory with increasing order of convergence from the basic four-point scheme without memory. The order is increased from 16 of the basic method to 20, 22, 23, 23.662, and 24 by suitable variation of a free parameter in each iterative step. This increase in the convergence order is achieved without any additional function evaluations and therefore, the methods with memory possess better computational efficiency than the methods without memory. Numerical examples are presented and the performance is compared with the existing optimal three and four-point methods. Computational results and comparison with the existing methods confirm efficient and robust character of present methods.

Optimal Derivative-Free Methods for Solving Nonlinear Equations

2011 ◽  
Author(s):  
Alicia Cordero ◽  
José L. Hueso ◽  
Eulalia Martínez ◽  
Juan R. Torregrosa ◽  
Theodore E. Simos ◽  
...  
Keyword(s):  
Nonlinear Equations ◽  
Derivative Free ◽  
Derivative Free Methods
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