A class of four parametric with‐ and without‐memory root finding methods

Zafar, Fiza; Cordero, Alicia; Torregrosa, Juan R.; Rafi, Aneeqa

doi:10.1002/cmm4.1024

Cited by 5 publications

(4 citation statements)

References 9 publications

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“…over methods (18) and (19), respectively. Notably, both methods MM b 4 and MM b 8 show no divergent points, as observed from Tables 5 and 6, respectively.…”

Section: Discussionmentioning

confidence: 99%

“…In this section, we examine the performance and the computational efficiency of the newly developed with and without-memory methods discussed in Sections 2 and 3 and compare with some methods of similar nature available in the literature. In particular, we have considered for the comparison the following derivative-free three-parametric methods: FZM 4 (4.1) [15], VTM 4 (28) [16], and SM 4 (4.1) [17], and the following four-parametric methods: AJM 8 [13], ZM 8 ( ZR1 from [18]), and ACM 8 (M1 from [19]).…”

Section: Numerical Experimentsmentioning

confidence: 99%

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Efficient Families of Multi-Point Iterative Methods and Their Self-Acceleration with Memory for Solving Nonlinear Equations

et al. 2023

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In this paper, we have constructed new families of derivative-free three- and four-parametric methods with and without memory for finding the roots of nonlinear equations. Error analysis verifies that the without-memory methods are optimal as per Kung–Traub’s conjecture, with orders of convergence of 4 and 8, respectively. To further enhance their convergence capabilities, the with-memory methods incorporate accelerating parameters, elevating their convergence orders to 7.5311 and 15.5156, respectively, without introducing extra function evaluations. As such, they exhibit exceptional efficiency indices of 1.9601 and 1.9847, respectively, nearing the maximum efficiency index of 2. The convergence domains are also analysed using the basins of attraction, which exhibit symmetrical patterns and shed light on the fascinating interplay between symmetry, dynamic behaviour, the number of diverging points, and efficient root-finding methods for nonlinear equations. Numerical experiments and comparison with existing methods are carried out on some nonlinear functions, including real-world chemical engineering problems, to demonstrate the effectiveness of the new proposed methods and confirm the theoretical results. Notably, our numerical experiments reveal that the proposed methods outperform their existing counterparts, offering superior precision in computation.

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“…over methods (18) and (19), respectively. Notably, both methods MM b 4 and MM b 8 show no divergent points, as observed from Tables 5 and 6, respectively.…”

Section: Discussionmentioning

confidence: 99%

Section: Numerical Experimentsmentioning

confidence: 99%

Efficient Families of Multi-Point Iterative Methods and Their Self-Acceleration with Memory for Solving Nonlinear Equations

et al. 2023

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“…The iterative scheme with memory (3) has a convergence order of 2.41. Several researchers have developed root-finding methods using memory based on existing optimal methods without memory; see, e.g., [2,[6][7][8][9][10][11][12][13][14][15][16].…”

Section: Introductionmentioning

confidence: 99%

Extension of King’s Iterative Scheme by Means of Memory for Nonlinear Equations

et al. 2023

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We developed a new family of optimal eighth-order derivative-free iterative methods for finding simple roots of nonlinear equations based on King’s scheme and Lagrange interpolation. By incorporating four self-accelerating parameters and a weight function in a single variable, we extend the proposed family to an efficient iterative scheme with memory. Without performing additional functional evaluations, the order of convergence is boosted from 8 to 15.51560, and the efficiency index is raised from 1.6817 to 1.9847. To compare the performance of the proposed and existing schemes, some real-world problems are selected, such as the eigenvalue problem, continuous stirred-tank reactor problem, and energy distribution for Planck’s radiation. The stability and regions of convergence of the proposed iterative schemes are investigated through graphical tools, such as 2D symmetric basins of attractions for the case of memory-based schemes and 3D stereographic projections in the case of schemes without memory. The stability analysis demonstrates that our newly developed schemes have wider symmetric regions of convergence than the existing schemes in their respective domains.

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“…There is an extensive literature of methods with memory that include derivatives in their iterative expressions (see, for example, Petković et al 2 and the references therein) and other ones that are derivative free, such as the papers of Petković and Dzunić [6][7][8] or by other authors, [9][10][11][12][13] all of them by using similar techniques.…”

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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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A class of four parametric with‐ and without‐memory root finding methods

Cited by 5 publications

References 9 publications

Efficient Families of Multi-Point Iterative Methods and Their Self-Acceleration with Memory for Solving Nonlinear Equations

Efficient Families of Multi-Point Iterative Methods and Their Self-Acceleration with Memory for Solving Nonlinear Equations

Extension of King’s Iterative Scheme by Means of Memory for Nonlinear Equations

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

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