Dynamical Techniques for Analyzing Iterative Schemes with Memory

Choubey, Neha; Cordero, Alicia; Jaiswal, J. P.; Torregrosa, Juan R.

doi:10.1155/2018/1232341

Cited by 8 publications

(6 citation statements)

References 31 publications

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“…In this section, we provide numerical examples to elucidate the efficiency and effectiveness of the newly formulated three-step with-memory methods NWM9 (7) and NWM10 (47). These methods are compared with some existing well-known three-step methods, SA8, NAJJ10, XT10 and NJ10, as presented in references [5,6,10,11], respectively, using the test functions mentioned in Table 1. The aforementioned iterative methods are now listed below.…”

Section: Numerical Results and Discussionmentioning

confidence: 99%

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An Efficient Bi-Parametric With-Memory Iterative Method for Solving Nonlinear Equations

Sharma,

Mittal,

Jaiswal

et al. 2023

AppliedMath

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New three-step with-memory iterative methods for solving nonlinear equations are presented. We have enhanced the convergence order of an existing eighth-order memory-less iterative method by transforming it into a with-memory method. Enhanced acceleration of the convergence order is achieved by introducing two self-accelerating parameters computed using the Hermite interpolating polynomial. The corresponding R-order of convergence of the proposed uni- and bi-parametric with-memory methods is increased from 8 to 9 and 10, respectively. This increase in convergence order is accomplished without requiring additional function evaluations, making the with-memory method computationally efficient. The efficiency of our with-memory methods NWM9 and NWM10 increases from 1.6818 to 1.7320 and 1.7783, respectively. Numeric testing confirms the theoretical findings and emphasizes the superior efficacy of suggested methods when compared to some well-known methods in the existing literature.

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Section: Numerical Results and Discussionmentioning

confidence: 99%

“…In 2018, Choubey et al [10] proposed a tenth-order with-memory iterative method (NAJJ10) using two self-accelerating parameters, which is defined as:…”

Section: Numerical Results and Discussionmentioning

confidence: 99%

An Efficient Bi-Parametric With-Memory Iterative Method for Solving Nonlinear Equations

Sharma,

Mittal,

Jaiswal

et al. 2023

AppliedMath

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“…The proposed family includes memory using self-accelerating parameters and holds a similar expression than the original scheme. It is based on the techniques presented in Chicharro et al 26 and Choubey et al 27 among others. The following result, which can be found in Ortega and Reinhboldt, 28 is useful to analyze the order of convergence of a method with memory.…”

Section: Iterative Methods With Memorymentioning

confidence: 99%

On the choice of the best members of the Kim family and the improvement of its convergence

Chicharro

Cordero

Garrido

et al. 2019

Math Methods in App Sciences

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The best members of the Kim family, in terms of stability, are obtained by using complex dynamics. From this elements, parametric iterative methods with memory are designed. A dynamical analysis of the methods with memory is presented in order to obtain information about the stability of them. Numerical experiments are shown for confirming the theoretical results. KEYWORDSbasin of attraction, iterative methods with memory, low-dimensional dynamical systems, nonlinear algebraic or transcendental equations, parameter plane, stability MSC CLASSIFICATION 65H05 INTRODUCTIONTo find a solution of the nonlinear equation f(x) = 0 is a common problem that appears frequently in different fields of science and engineering. These nonlinear problems require iterative methods to be solved. In the last decades, there has been an extensive literature focused on the generation of iterative procedures to find a solution . We can see a good overview on classical and recent results in Amat and Busquier 1 or Petković et al. 2 The iterative methods are fixed-point schemes that, starting from one or more initial estimations, obtain a new value that approaches our solution asIf we use only the last iteration (p = 0), we have an iterative method without memory and, if we use more than one previous iterations (p > 0), the iterative scheme is with memory.Related with the order of convergence of iterative methods without memory, the Kung-Traub conjecture 3 establishes that the order of a scheme without memory is always lower than 2 d−1 , where d is the number of functional evaluations per iteration. When this bound is reached, the iterative method is called optimal. One technique for improving the order of convergence without increasing the number of functional evaluations is to introduce memory in the scheme, that is, the new iterate depend on more than one previous iterates.

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“…In recent years, the development of with-memory iterative methods has garnered considerable interest among researchers. For a deeper understanding, one can refer to [23][24][25][26][27][28][29][30][31][32][33][34] and the references cited therein.…”

Section: Introductionmentioning

confidence: 99%

A New Adaptive Eleventh-Order Memory Algorithm for Solving Nonlinear Equations

Panday,

Mittal,

Stoenoiu

et al. 2024

Mathematics

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In this article, we introduce a novel three-step iterative algorithm with memory for finding the roots of nonlinear equations. The convergence order of an established eighth-order iterative method is elevated by transforming it into a with-memory variant. The improvement in the convergence order is achieved by introducing two self-accelerating parameters, calculated using the Hermite interpolating polynomial. As a result, the R-order of convergence for the proposed bi-parametric with-memory iterative algorithm is enhanced from 8 to 10.5208. Notably, this enhancement in the convergence order is accomplished without the need for extra function evaluations. Moreover, the efficiency index of the newly proposed with-memory iterative algorithm improves from 1.5157 to 1.6011. Extensive numerical testing across various problems confirms the usefulness and superior performance of the presented algorithm relative to some well-known existing algorithms.

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Dynamical Techniques for Analyzing Iterative Schemes with Memory

Cited by 8 publications

References 31 publications

An Efficient Bi-Parametric With-Memory Iterative Method for Solving Nonlinear Equations

An Efficient Bi-Parametric With-Memory Iterative Method for Solving Nonlinear Equations

On the choice of the best members of the Kim family and the improvement of its convergence

A New Adaptive Eleventh-Order Memory Algorithm for Solving Nonlinear Equations

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