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

Sharma, Janak Raj; Argyros, Ioannis K.; Kumar, Sunil

doi:10.3390/math6110260

Cited by 7 publications

(8 citation statements)

References 17 publications

(25 reference statements)

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“…A compilation of many available multi-point methods and their properties are discussed in the texts 1,3,4,6,7,8,10 . Furthermore, numerous higher order Newton-like multi-step methods were constructed at the expense of auxiliary calculation of functions, derivatives, and modification in the points of iteration as found in 9,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25 . For example, Frontini et al 11 have devised a modification of the Newton method of order three to solve equations in -dimensional spaces.…”

Section: Introductionmentioning

confidence: 99%

Iterative methods of order four and five for solving non-linear system : A study on local convergence

John¹,

Jayaraman²

2022

Preprint

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In this paper, we develop the local convergence analysis of Newton-like fourth and fifth order iterative methods for solving a system of non-linear equations. Earlier studies as in Petkovic (Multipoint methods for solving nonlinear equations, Elsevier, Amsterdam,2013), Traub (Iterative methods for the solution of equations, AMS Chelsea Publishing, Providence, 1982) and Kalyanasundaram et al (International Journal of Applied and Computational Mathematics 3 3:2213-2230, 2017) shows that the local convergence was proved using Taylor series expansion which involved the computation of derivatives of order higher than one. For the fourth and fifth order iterative methods under consideration in this paper, it is required that the functions should be at least five and six times differentiable respectively so that the method is applicable to find the solution. This restricts the applicability of the method and also the cost in finding the solution increases as it involves the computation of higher order derivatives. The local convergence analysis derived in this paper uses Lipschitz and ω-continuity conditions which involves only first derivative (present in the method) to prove the convergence. Moreover, the present study also provides details about the radii of domain of convergence and also estimates on error bounds. Therefore, it is evident that the present study enhances the applicability of the methods under consideration. The obtained results have been verified with suitable numerical illustrations.

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Section: Introductionmentioning

confidence: 99%

Iterative methods of order four and five for solving non-linear system : A study on local convergence

John¹,

Jayaraman²

2022

Preprint

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“…where F : Ω −→ B 2 is Fréchet differentiable operator. Notice that a plethora of applications from Mathematics, Science and Engineering are reduced to a form as (1.1) by utilizing Mathematical modeling [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. The solution x * is sought in closed form, but this can be achieved only in some cases.…”

Section: Introductionmentioning

confidence: 99%

Extended local convergence for Newton-type solver under weak conditions

Argyros¹,

George²,

Senapati³

2021

Stud. Univ. Babes-Bolyai Math.

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We present the local convergence of a Newton-type solver for equations involving Banach space valued operators. The eighth order of convergence was shown earlier in the special case of the k-dimensional Euclidean space, using hypotheses up to the eighth derivative although these derivatives do not appear in the method. We show convergence using only the rst derivative. This way we extend the applicability of the methods. Numerical examples are used to show the convergence conditions. Finally, the basins of attraction of the method, on some test problems are presented.

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“…for Banach space valued mappings with F : Ω ⊂ B → B, where F is differentiable in the sense of Fréchet [1,2]. For a good survey of literature on local and semilocal convergence criteria of iterative methods see [3][4][5][6][7][8][9][10][11][12][13].…”

Section: Introductionmentioning

confidence: 99%

“…In quest of efficient higher order method, a number of improved, multipoint Newton's or Newton-like iterative schemes have been proposed in literature; see, for example [3,5,[8][9][10][12][13][14][15][16][17][18][19] and references cited therein.…”

Section: Introductionmentioning

confidence: 99%

Local Convergence of an Efficient Multipoint Iterative Method in Banach Space

2020

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We discuss the local convergence of a derivative-free eighth order method in a Banach space setting. The present study provides the radius of convergence and bounds on errors under the hypothesis based on the first Fréchet-derivative only. The approaches of using Taylor expansions, containing higher order derivatives, do not provide such estimates since the derivatives may be nonexistent or costly to compute. By using only first derivative, the method can be applied to a wider class of functions and hence its applications are expanded. Numerical experiments show that the present results are applicable to the cases wherein previous results cannot be applied.

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

Cited by 7 publications

References 17 publications

Iterative methods of order four and five for solving non-linear system : A study on local convergence

Iterative methods of order four and five for solving non-linear system : A study on local convergence

Extended local convergence for Newton-type solver under weak conditions

Local Convergence of an Efficient Multipoint Iterative Method in Banach Space

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