scholarly journals Potra-Pták Iterative Method with Memory

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Taher Lotfi ◽  
Stanford Shateyi ◽  
Sommayeh Hadadi
Keyword(s):  
Iterative Method ◽  
Free Parameter ◽  
Order Of Convergence ◽  
The Third ◽  
With Memory ◽  
Interpolatory Polynomial

The problem is to extend the method proposed by Soleymani et al. (2012) to a method with memory. Following this aim, a free parameter is calculated using Newton’s interpolatory polynomial of the third degree. So the R-order of convergence is increased from 4 to 6 without any new function evaluations. Numerically the extended method is examined along with comparison to some existing methods with the similar properties.

scholarly journals A New High-Order Jacobian-Free Iterative Method with Memory for Solving Nonlinear Systems

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2122
Author(s):  
Ramandeep Behl ◽  
Alicia Cordero ◽  
Juan R. Torregrosa ◽  
Sonia Bhalla
Keyword(s):  
Nonlinear Systems ◽  
Iterative Method ◽  
Order Of Convergence ◽  
Recent Literature ◽  
Basin Of Attraction ◽  
Qualitative Behavior ◽  
Order Convergence ◽  
Numerical Tests ◽  
Initial Scheme ◽  
With Memory

We used a Kurchatov-type accelerator to construct an iterative method with memory for solving nonlinear systems, with sixth-order convergence. It was developed from an initial scheme without memory, with order of convergence four. There exist few multidimensional schemes using more than one previous iterate in the very recent literature, mostly with low orders of convergence. The proposed scheme showed its efficiency and robustness in several numerical tests, where it was also compared with the existing procedures with high orders of convergence. These numerical tests included large nonlinear systems. In addition, we show that the proposed scheme has very stable qualitative behavior, by means of the analysis of an associated multidimensional, real rational function and also by means of a comparison of its basin of attraction with those of comparison methods.

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.

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.

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 Impact on Stability by the Use of Memory in Traub-Type Schemes

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 274
Author(s):  
Francisco I. Chicharro ◽  
Alicia Cordero ◽  
Neus Garrido ◽  
Juan R. Torregrosa
Keyword(s):  
Order Of Convergence ◽  
Nonlinear Problems ◽  
Higher Order ◽  
Lower Number ◽  
Dynamical Analysis ◽  
The Real ◽  
Original Scheme ◽  
Interpolation Polynomials ◽  
With Memory ◽  
Number Of Iterations

In this work, two Traub-type methods with memory are introduced using accelerating parameters. To obtain schemes with memory, after the inclusion of these parameters in Traub’s method, they have been designed using linear approximations or the Newton’s interpolation polynomials. In both cases, the parameters use information from the current and the previous iterations, so they define a method with memory. Moreover, they achieve higher order of convergence than Traub’s scheme without any additional functional evaluations. The real dynamical analysis verifies that the proposed methods with memory not only converge faster, but they are also more stable than the original scheme. The methods selected by means of this analysis can be applied for solving nonlinear problems with a wider set of initial estimations than their original partners. This fact also involves a lower number of iterations in the process.

An optimized Steffensen-type iterative method with memory associated with annuity calculation

2019 ◽  
Vol 134 (4) ◽  
Author(s):  
Fuad W. Khdhr ◽  
Fazlollah Soleymani ◽  
Rostam K. Saeed ◽  
Ali Akgül
Keyword(s):  
Iterative Method ◽  
With Memory

scholarly journals Memory in a New Variant of King’s Family for Solving Nonlinear Systems

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1251
Author(s):  
Munish Kansal ◽  
Alicia Cordero ◽  
Sonia Bhalla ◽  
Juan R. Torregrosa
Keyword(s):  
Nonlinear Systems ◽  
Convergence Analysis ◽  
Numerical Experiments ◽  
Order Of Convergence ◽  
Recent Literature ◽  
Divided Difference ◽  
Difference Operators ◽  
New Variant ◽  
Order Four ◽  
With Memory

In the recent literature, very few high-order Jacobian-free methods with memory for solving nonlinear systems appear. In this paper, we introduce a new variant of King’s family with order four to solve nonlinear systems along with its convergence analysis. The proposed family requires two divided difference operators and to compute only one inverse of a matrix per iteration. Furthermore, we have extended the proposed scheme up to the sixth-order of convergence with two additional functional evaluations. In addition, these schemes are further extended to methods with memory. We illustrate their applicability by performing numerical experiments on a wide variety of practical problems, even big-sized. It is observed that these methods produce approximations of greater accuracy and are more efficient in practice, compared with the existing methods.

scholarly journals A Family of Fourteenth-Order Convergent Iterative Methods for Solving Nonlinear Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Fiza Zafar ◽  
Gulshan Bibi
Keyword(s):  
Iterative Method ◽  
Iterative Methods ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Efficiency Index ◽  
Numerical Examples ◽  
New Class ◽  
First Derivative ◽  
New Methods ◽  
Specific Step

We present a family of fourteenth-order convergent iterative methods for solving nonlinear equations involving a specific step which when combined with any two-step iterative method raises the convergence order by n+10, if n is the order of convergence of the two-step iterative method. This new class include four evaluations of function and one evaluation of the first derivative per iteration. Therefore, the efficiency index of this family is 141/5 =1.695218203. Several numerical examples are given to show that the new methods of this family are comparable with the existing methods.

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