Stability and applicability of iterative methods with memory

Chicharro, Francisco Israel; Cordero, Alicia; Garrido, Neus; Torregrosa, Juan R.

doi:10.1007/s10910-018-0952-z

Cited by 23 publications

(27 citation statements)

References 25 publications

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“…Numerical tests examples from [32][33][34] are provided in Tables 7-13. In Tables 7, 9, 11, and 12, we present the numerical results for simultaneous determination of all roots, while Tables 8, 10, and 13 represent for single root finding methods.…”

Section: Numerical Resultsmentioning

confidence: 99%

On Dynamics of Iterative Techniques for Nonlinear Equation with Applications in Engineering

Shams

Mir

Rafiq

et al. 2020

Mathematical Problems in Engineering

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In this article, we construct an optimal family of iterative methods for finding the single root and then extend this family for determining all the distinct as well as multiple roots of single-variable nonlinear equations simultaneously. Convergence analysis is presented for both the cases to show that the optimal order of convergence is 4 in the case of single root finding methods and 6 for simultaneous determination of all distinct as well as multiple roots of a nonlinear equation. The computational cost, basins of attraction, efficiency, log of residual, and numerical test examples show that the newly constructed methods are more efficient as compared to the existing methods in the literature.

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Section: Numerical Resultsmentioning

confidence: 99%

On Dynamics of Iterative Techniques for Nonlinear Equation with Applications in Engineering

Shams

Mir

Rafiq

et al. 2020

Mathematical Problems in Engineering

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“…In this section, we analyze the performance of three different schemes with memoryour proposed scheme (22) and two known similar schemes, which are defined in (5), taken from Chicharro et al [18]-on quadratic polynomials that we denote by AM 5 , which use Kurchatov's divided difference in order to introduce memory. We also analyze scheme…”

Section: A Qualitative Study Of Iterative Methods With Memory: New and Knownmentioning

confidence: 99%

“…where u (j) = x (j) − 2x (j) − x (j−1) , x (j−1) ; F −1 F(x (j) ). In the following result, we analyze the convergence order of scheme (22) with memory, denoted by PM 6 .…”

Section: Extension To a Higher-order Scheme With Memorymentioning

confidence: 99%

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

et al. 2021

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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.

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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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Stability and applicability of iterative methods with memory

Cited by 23 publications

References 25 publications

On Dynamics of Iterative Techniques for Nonlinear Equation with Applications in Engineering

On Dynamics of Iterative Techniques for Nonlinear Equation with Applications in Engineering

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

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

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