Family of optimal two-step fourth order iterative method and its extension for solving nonlinear equations

Oghovese, Ogbereyivwe; Veronica, Ojo-Orobosa

doi:10.1080/09720502.2021.1884393

Cited by 2 publications

(2 citation statements)

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“…Next we establish the convergence of the families of means-based IM given in (2.12) for different types of mean used in M T . The Taylor's expansion technique also used in literature [15,16,17,13] and some reference therein, was used in proving the convergence of the method. First, the Proposition 2.1 is considered.…”

Section: Methods Formulationmentioning

confidence: 99%

“…In order to improve the CO and EI of (1.2), many authors have used different techniques. A good review on some of these techniques such as the geometric, functional, composition, sampling, Adomain decomposition, rational function and weight function techniques can be found in [1,15,16,17] and reference therein. The Weerakoon and Fernando (W-F) method in [25] is one modified form of the NM (1.2), put forward by replacing the function f (x) in (1.2) with the arithmetic mean of the functions f (x) and f (w).…”

Section: Introductionmentioning

confidence: 99%

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Families of Means-Based Modified Newtons Method for Solving Nonlinear Model

2021

Punjab Univ. j. math.

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In literature, the arithmetic mean of the two functions in the denominator of the second step of order three Weerakoon and Fernando (2000) iterative method have been replaced with other different means. However, these actions have not improve its order of convergence. To improve the order of convergence of these modified methods, a generic family of iterative methods that involve two weight functions and a generic consequential function for replacement of means is proposed. The analysis of convergence carried out on the families of methods, shows that they are of fourth order convergence and requires evaluation of three functions per iteration cycle. Further, the flexibility of the weight functions enables the re-discovery of some existing and construction of new families of iterative methods. Some concrete members of the family of methods are applied to solve some nonlinear equations and real life problems that are modeled into nonlinear equations

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Section: Methods Formulationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Families of Means-Based Modified Newtons Method for Solving Nonlinear Model

2021

Punjab Univ. j. math.

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Extension of an Eighth-Order Iterative Technique to Address Non-Linear Problems

Ramos,

Argyros,

Behl

et al. 2024

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The convergence order of an iterative method used to solve equations is usually determined by using Taylor series expansions, which in turn require high-order derivatives, which are not necessarily present in the method. Therefore, such convergence analysis cannot guarantee the theoretical convergence of the method to a solution if these derivatives do not exist. However, the method can converge. This indicates that the most sufficient convergence conditions required by the Taylor approach can be replaced by weaker ones. Other drawbacks exist, such as information on the isolation of simple solutions or the number of iterations that must be performed to achieve the desired error tolerance. This paper positively addresses all these issues by considering a technique that uses only the operators on the method and Ω-generalized continuity to control the derivative. Moreover, both local and semi-local convergence analyses are presented for Banach space-valued operators. The technique can be used to extend the applicability of other methods along the same lines. A large number of concrete examples are shown in which the convergence conditions are fulfilled.

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Family of optimal two-step fourth order iterative method and its extension for solving nonlinear equations

Cited by 2 publications

References 11 publications

Families of Means-Based Modified Newtons Method for Solving Nonlinear Model

Families of Means-Based Modified Newtons Method for Solving Nonlinear Model

Extension of an Eighth-Order Iterative Technique to Address Non-Linear Problems

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