Memory in a New Variant of King’s Family for Solving Nonlinear Systems

Kansal, Munish; Cordero, Alicia; Bhalla, Sonia; Torregrosa, Juan R.

doi:10.3390/math8081251

Cited by 4 publications

(10 citation statements)

References 23 publications

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“…But this derivative is not bounded. Thus, the findings in [13] do not imply that lim n→∞ x n = 1. However, the sequence {x n } is convergent to 1.…”

Section: Introductionmentioning

confidence: 89%

“…We notice that equation h(t) = 0 is solvable by t * = 1 ∈ Ω. We notice that the results in [13] require the existence and boundedness of the seventh derivative F (7) about the solution. But this derivative is not bounded.…”

Section: Introductionmentioning

confidence: 98%

“…where B = B n = [w n , x n ; F] and w n = x n + F(x n ), is often employed to provide a sequence of iterates converging quadratically to x * . Many iterative approaches have been developed to improve efficiency and order convergence (see [9][10][11][12][13]). An approach established in [13] is given for x 0 ∈ Ω as…”

Section: Introductionmentioning

confidence: 99%

“…Many iterative approaches have been developed to improve efficiency and order convergence (see [9][10][11][12][13]). An approach established in [13] is given for x 0 ∈ Ω as…”

Section: Introductionmentioning

confidence: 99%

“…Note that the inversion of the same linear operator and a function evaluation are required per step. The local analysis (LA) of the method is shown to be of the order of six in [13] for a ̸ = b under conditions of F (1) , F (2) , . .…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Three-Step Derivative-Free Method of Order Six

Kumar,

Sharma,

Argyros

et al. 2023

Foundations

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Derivative-free iterative methods are useful to approximate the numerical solutions when the given function lacks explicit derivative information or when the derivatives are too expensive to compute. Exploring the convergence properties of such methods is crucial in their development. The convergence behavior of such approaches and determining their practical applicability require conducting local as well as semi-local convergence analysis. In this study, we explore the convergence properties of a sixth-order derivative-free method. Previous local convergence studies assumed the existence of derivatives of high order even when the method itself was not utilizing any derivatives. These assumptions imposed limitations on its applicability. In this paper, we extend the local analysis by providing estimates for the error bounds of the method. Consequently, its applicability expands across a broader range of problems. Moreover, the more important and challenging semi-local convergence not investigated in earlier studies is also developed. Additionally, we survey recent advancements in this field. The outcomes presented in this paper can be proved valuable to practitioners and researchers engaged in the development and analysis of derivative-free numerical algorithms. Numerical tests illuminate and validate further the theoretical results.

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“…But this derivative is not bounded. Thus, the findings in [13] do not imply that lim n→∞ x n = 1. However, the sequence {x n } is convergent to 1.…”

Section: Introductionmentioning

confidence: 89%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

“…Many iterative approaches have been developed to improve efficiency and order convergence (see [9][10][11][12][13]). An approach established in [13] is given for x 0 ∈ Ω as…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Three-Step Derivative-Free Method of Order Six

Kumar,

Sharma,

Argyros

et al. 2023

Foundations

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show abstract

An adaptive Steffensen-like families for solving nonlinear systems using frozen divided differences

Kansal,

Rani

2023

J Math Chem

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The Enhanced Fixed Point Method: An Extremely Simple Procedure to Accelerate the Convergence of the Fixed Point Method to Solve Nonlinear Algebraic Equations

et al. 2022

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This work proposes the Enhanced Fixed Point Method (EFPM) as a straightforward modification to the problem of finding an exact or approximate solution for a linear or nonlinear algebraic equation. The proposal consists of providing a versatile method that is easy to employ and systematic. Therefore, it is expected that this work contributes to breaking the paradigm that an effective modification for a known method has to be necessarily long and complicated. As a matter of fact, the method expresses an algebraic equation in terms of the same equation but multiplied for an adequate factor, which most of the times is just a simple numeric factor. The main idea is modifying the original equation, slightly changing it for others in such a way that both have the same solution. Next, the modified equation is expressed as a fixed point problem and the proposed parameters are employed to accelerate the convergence of the fixed point problem for the original equation. Since the Newton method results from a possible fixed point problem of an algebraic equation, we will see that it is relatively easy to get modified versions of the Newton method with orders of convergence major than two. We will see in this work the convenience of this procedure.

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Memory in a New Variant of King’s Family for Solving Nonlinear Systems

Cited by 4 publications

References 23 publications

Three-Step Derivative-Free Method of Order Six

Three-Step Derivative-Free Method of Order Six

An adaptive Steffensen-like families for solving nonlinear systems using frozen divided differences

The Enhanced Fixed Point Method: An Extremely Simple Procedure to Accelerate the Convergence of the Fixed Point Method to Solve Nonlinear Algebraic Equations

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