A super-fast tri-parametric iterative method with memory

Ullah, Malik Zaka; Kosari, Sajad; Soleymani, Fazlollah; Haghani, F. Khaksar; Al-Fhaid, A. S.

doi:10.1016/j.amc.2016.05.029

Cited by 6 publications

(6 citation statements)

References 13 publications

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“…In this section, the family of methods with memory (14) and (26) are tested for three experiments. Also, we compare the implementations with some kinds of methods with memory in the literature.…”

Section: Numerical Results and Comparisonsmentioning

confidence: 99%

“…This system has the solution p ≃ 5.7 and q ≃ 2.8, which shows a better convergence order for the adaptive method with memory (14). If we set n = 2, then p ≃ 5.95, so we have increased the convergence order near 50% without any new functional evaluations.…”

Section: Theorem 3 the Iterative Methods (22) Has Convergence Order 4mentioning

confidence: 92%

“…Now, we attempt to increase the convergence order of (14) by some modification on it. To this end, consider the following new modified version of (14) at which another parameter has been considered in the second step as well…”

Section: Construction Of Newton-type Adaptive Methods and Convergencementioning

confidence: 99%

“…Note that the notation of divided differences, such as f[·, ·], are used throughout this paper. In fact, in recent years, many developments have been done so as to improve the efficiency and reliability of iterative root-finding methods using the notion of with memorization; see, eg, other studies [12][13][14] to obtain a general perspective on such generalizations.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Developing two efficient adaptive Newton‐type methods with memory

Zadeh

Lotfi

Amirfakhrian

2018

Math Methods in App Sciences

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In this paper, we derive two general adaptive methods with memory in the class of Newton‐type methods by modifying and introducing one and two self accelerators over a variant of Ostrowski's method. The idea of introducing adaptive self‐accelerator (via all the old information for Newton‐type methods) is new and efficient in order to obtain a higher high efficiency index. Finally, we provide convergence analysis and numerical implementations to show the feasibility and applicability of the proposed methods.

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Section: Numerical Results and Comparisonsmentioning

confidence: 99%

Section: Theorem 3 the Iterative Methods (22) Has Convergence Order 4mentioning

confidence: 92%

Section: Construction Of Newton-type Adaptive Methods and Convergencementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Developing two efficient adaptive Newton‐type methods with memory

Zadeh

Lotfi

Amirfakhrian

2018

Math Methods in App Sciences

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show abstract

“…Over the past few years, many authors [5][6][7][8][9][10][11][12][13] have examined the use of memory to achieve high orders of convergence that surpass the corresponding Kung-Traub limit for methods without memory. These studies frequently make use of self-correcting parameters that use previously generated data in new ways to boost convergence.…”

Section: Introductionmentioning

confidence: 99%

A class of efficient high‐order iterative methods with memory for nonlinear equations and their dynamics

Howk

Hueso

Martínez

et al. 2018

Math Methods in App Sciences

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In this paper we obtain some theoretical results about iterative methods with memory for nonlinear equations. The class of algorithms we consider focus on incorporating memory without increasing the computational cost of the algorithm. This class uses for the predictor step of each iteration a quantity that has already been calculated in the previous iteration, typically the quantity governing the slope from the previous corrector step. In this way we do not introduce any extra computation, and more importantly, we avoid new function evaluations, allowing us to obtain high‐order iterative methods in a simple way. A specific class of methods of this type is introduced, and we prove the convergence order is 2n+2n−2 with n+1 function evaluations. An exhaustive efficiency study is performed to show the competitiveness of these methods. Finally, we test some specific examples and explore the effect that this predictor may have on the convergence set by setting a dynamical study.

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A new family of adaptive methods with memory for solving nonlinear equations

2018

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In this work, an adaptive method with memory is developed such that all previous information are applied. The importance of the proposed method can be seen because of the optimization in important effecting factors, i.e., least number of iterations steps, least number of functional evaluations, least value of absolute error, and maximum efficiency index in final as well as in individual step as compared with the other methods. Indeed, it is proved that this adaptive method with memory has efficiency index 2 and competes all the existing methods without and with memory in the literature. The order of convergence is obtained by using two self-accelerating parameters, which is increased from 2 to 4 without any new function evaluation. It means that, the order of convergence can be improved until 100%. Numerical examples and the comparison with existing methods are included to demonstrate exceptional convergence speed of the proposed method and confirm theoretical results.

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A super-fast tri-parametric iterative method with memory

Cited by 6 publications

References 13 publications

Developing two efficient adaptive Newton‐type methods with memory

Developing two efficient adaptive Newton‐type methods with memory

A class of efficient high‐order iterative methods with memory for nonlinear equations and their dynamics

A new family of adaptive methods with memory for solving nonlinear equations

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