On the construction of some tri-parametric iterative methods with memory

Lotfi, Taher; Soleymani, Fazlollah; Ghorbanzadeh, Mohammad; Assari, Paria

doi:10.1007/s11075-015-9976-7

Cited by 25 publications

(25 citation statements)

References 14 publications

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“…More recently, some authors have constructed iterative schemes with memory from optimal procedures of different orders, mainly four (see, e.g., [10][11][12]), eight ( [13][14][15], among others), or even general n-point schemes [16,17]. Some good reviews regarding the acceleration of convergence order by using memory are [18,19].…”

Section: Introductionmentioning

confidence: 99%

Dynamical Techniques for Analyzing Iterative Schemes with Memory

Choubey¹,

Cordero

Jaiswal

et al. 2018

Complexity

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We construct a new biparametric three-point method with memory to highly improve the computational efficiency of its original partner, without adding functional evaluations. In this way, through different estimations of self-accelerating parameters, we have modified an existing seventh-order method. The parameters have been defined by Hermite interpolating polynomial that allows the accelerating effect. In particular, the R-order of the proposed iterative method with memory is increased from seven to ten. A real multidimensional analysis of the stability of this method with memory is made, in order to study its dependence on the initial estimations. Taking into account that usually iterative methods with memory are more stable than their derivative-free partners and the obtained results in this study, the behavior of this scheme shows to be excellent, but for a small domain. Numerical examples and comparison are also provided, confirming the theoretical results.

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Section: Introductionmentioning

confidence: 99%

Dynamical Techniques for Analyzing Iterative Schemes with Memory

Choubey¹,

Cordero

Jaiswal

et al. 2018

Complexity

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“…In this section, numerical examples are taken from [10] to test the proposed with-memory root solver (3.3) (FWM) in comparison with the with-memory family of methods of Kung and Traub [8] and with-memory method of Lotfi et al [10] (1.4). Kung and Traub [8] presented the iterative method Φ r (f ) (r = −1, 0, .…”

Section: Numerical Resultsmentioning

confidence: 99%

“…Sometimes it is not possible to improve the convergence order and the efficiency index of without memory methods without additional functional evaluations based on free parameters [3]. Recently, several multi-step with-memory, specially two-steps and three-steps root solvers based on without memory derivative free methods have been developed, which can be seen in [2,4,5,9,10,12,15,19]. Traub [18], was the first who developed the first with-memory method by modifying the famous Steffensen's iterative scheme (1.1).…”

Section: Introductionmentioning

confidence: 99%

“…It is proved that the with-memory method (1.3) has R-order of convergence at least 7 and its index of efficiency is 1.913. Lotfi et al in [10], presented a new tri-parametric with-memory method based on without memory two-step variant of Steffensen's method: 5) where N 4 (t), N 5 (t) and N 6 (t) are fourth, fifth and sixth degree interpolating polynomials respectively which pass through best saved points. It is demonstrated that the R-order of convergence of (1.4) is at least 7.77200 and the efficiency index is 7.77200 ≈ 1.98082.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On the construction of three step derivative free four-parametric methods with accelerated order of convergence

Zafar¹,

Akram²,

Yasmin³

et al. 2016

J. Nonlinear Sci. Appl.

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In this paper, a general procedure to develop some four-parametric with-memory methods to find simple roots of nonlinear equations is proposed. The new methods are improved extensions of with derivative without memory iterative methods. We used four self-accelerating parameters to boost up the convergence order and computational efficiency of the proposed methods without using any additional function evaluations. Numerical examples are presented to support the theoretical results of the methods. We further investigate the dynamics of the methods in the complex plane.

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“…To state the matter differently, it is necessary to pursue methods of higher computational efficiency. Hence, in this work we try to extend the recent findings of Lotfi et al [7] so as to present a super high computational efficiency index. We need fast iterative algorithms to approximate the solution of nonlinear equations arising from the application of shooting methods to solve boundary value problems.…”

Section: Introductory Notesmentioning

confidence: 90%