A uniparametric family of three-step eighth-order multipoint iterative methods for simple roots

Based on iterative methods without memory of eighth-order convergence proposed by Thukral (2012), some iterative methods with memory and high efficiency index are presented. We show that the order of convergence is increased without any additional function evaluations. Numerical comparisons are made to show the performance of the presented methods.

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Multipoint Iterative Methods for Solving Certain Equations

The Computer Journal ◽

10.1093/comjnl/8.4.398 ◽

1966 ◽

Vol 8 (4) ◽

pp. 398-400 ◽

Cited By ~ 31

Author(s):

P. Jarratt

Keyword(s):

Iterative Methods ◽

Multipoint Iterative Methods

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Efficient Four-Parametric with-and-without-Memory Iterative Methods Possessing High Efficiency Indices

Mathematical Problems in Engineering ◽

10.1155/2018/8093673 ◽

2018 ◽

Vol 2018 ◽

pp. 1-12 ◽

Cited By ~ 4

Author(s):

Alicia Cordero ◽

Moin-ud-Din Junjua ◽

Juan R. Torregrosa ◽

Nusrat Yasmin ◽

Fiza Zafar

Keyword(s):

Iterative Methods ◽

High Efficiency ◽

Nonlinear Function ◽

Efficiency Index ◽

Simple Zero ◽

Eighth Order ◽

New Family ◽

Derivative Free ◽

With Memory ◽

Theoretical Results

We construct a family of derivative-free optimal iterative methods without memory to approximate a simple zero of a nonlinear function. Error analysis demonstrates that the without-memory class has eighth-order convergence and is extendable to with-memory class. The extension of new family to the with-memory one is also presented which attains the convergence order 15.5156 and a very high efficiency index 15.51561/4≈1.9847. Some particular schemes of the with-memory family are also described. Numerical examples and some dynamical aspects of the new schemes are given to support theoretical results.

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Ball Convergence of an Efficient Eighth Order Iterative Method Under Weak Conditions

Mathematics ◽

10.3390/math6110260 ◽

2018 ◽

Vol 6 (11) ◽

pp. 260 ◽

Cited By ~ 3

Author(s):

Janak Sharma ◽

Ioannis Argyros ◽

Sunil Kumar

Keyword(s):

Iterative Method ◽

Iterative Methods ◽

Higher Order ◽

High Order ◽

Convergence Order ◽

Eighth Order ◽

First Derivative ◽

Ball Convergence ◽

Derivatives Of

The convergence order of numerous iterative methods is obtained using derivatives of a higher order, although these derivatives are not involved in the methods. Therefore, these methods cannot be used to solve equations with functions that do not have such high-order derivatives, since their convergence is not guaranteed. The convergence in this paper is shown, relying only on the first derivative. That is how we expand the applicability of some popular methods.

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Some Real-Life Applications of a Newly Constructed Derivative Free Iterative Scheme

Symmetry ◽

10.3390/sym11020239 ◽

2019 ◽

Vol 11 (2) ◽

pp. 239 ◽

Cited By ~ 11

Author(s):

Ramandeep Behl ◽

M. Salimi ◽

M. Ferrara ◽

S. Sharifi ◽

Samaher Alharbi

Keyword(s):

Iterative Methods ◽

Chemical Engineering ◽

Flame Temperature ◽

Real Life ◽

Chemical Reactor ◽

Basins Of Attraction ◽

Eighth Order ◽

Present Scheme ◽

New Family ◽

Derivative Free

In this study, we present a new higher-order scheme without memory for simple zeros which has two major advantages. The first one is that each member of our scheme is derivative free and the second one is that the present scheme is capable of producing many new optimal family of eighth-order methods from every 4-order optimal derivative free scheme (available in the literature) whose first substep employs a Steffensen or a Steffensen-like method. In addition, the theoretical and computational properties of the present scheme are fully investigated along with the main theorem, which demonstrates the convergence order and asymptotic error constant. Moreover, the effectiveness of our scheme is tested on several real-life problems like Van der Waal’s, fractional transformation in a chemical reactor, chemical engineering, adiabatic flame temperature, etc. In comparison with the existing robust techniques, the iterative methods in the new family perform better in the considered test examples. The study of dynamics on the proposed iterative methods also confirms this fact via basins of attraction applied to a number of test functions.

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