A simple and efficient method with high order convergence for solving systems of nonlinear equations

Xiao, Xiaoyong; Yin, Hongwei

doi:10.1016/j.camwa.2015.03.018

Cited by 14 publications

(11 citation statements)

References 29 publications

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“…Firstly, consider a third order method [24], (M 13 ), which does not use F (y k ) in its iteration function. Applying Theorem 1 for a = −1, we develop a fifth order method, denoted by M 15 :…”

Section: Extended Methods and Computational Efficiencymentioning

confidence: 99%

Increasing the order of convergence for iterative methods to solve nonlinear systems

Xiao

Yin

2015

Calcolo

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For solving nonlinear systems, we introduce a technique that improves the order of convergence of any given iterative method which uses the extended Newton iteration as a predictor. Based on a given iterative method of order p ≥ 2, a new method of order p + 2 is developed by introducing just one evaluation of the function. We obtain some new methods with higher order of convergence by applying this procedure to some known methods. Computational efficiency in the general form is discussed and comparisons are made between these new methods and the ones from which have been derived. We also perform several numerical tests to show the asymptotic behaviors which confirm the theoretical results.

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Section: Extended Methods and Computational Efficiencymentioning

confidence: 99%

Increasing the order of convergence for iterative methods to solve nonlinear systems

Xiao

Yin

2015

Calcolo

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“…so Equation (17) holds for n = 0 and y 0 ∈ U(u * , r). We need the estimate obtained using (a 2 ) and (20)…”

Section: Lemma 1 Suppose That Equationmentioning

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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“…, (2) which has a quadratic order of convergence. In order to achieve higher convergence order, a number of modified, multistep Newton's or Newton-type iterations have been developed in the literature; see [3,4,6,7,[9][10][11][12][15][16][17][18][19] and references cited therein. There is another important class of multistep methods based on Jarratt methods or Jarratt-type methods [20][21][22].…”

Section: Introductionmentioning

confidence: 99%

Local Convergence and Attraction Basins of Higher Order, Jarratt-Like Iterations

2019

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We studied the local convergence of a family of sixth order Jarratt-like methods in Banach space setting. The procedure so applied provides the radius of convergence and bounds on errors under the conditions based on the first Fréchet-derivative only. Such estimates are not proposed in the approaches using Taylor expansions of higher order derivatives which may be nonexistent or costly to compute. In this sense we can extend usage of the methods considered, since the methods can be applied to a wider class of functions. Numerical testing on examples show that the present results can be applied to the cases where earlier results are not applicable. Finally, the convergence domains are assessed by means of a geometrical approach; namely, the basins of attraction that allow us to find members of family with stable convergence behavior and with unstable behavior.

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A simple and efficient method with high order convergence for solving systems of nonlinear equations

Cited by 14 publications

References 29 publications

Increasing the order of convergence for iterative methods to solve nonlinear systems

Increasing the order of convergence for iterative methods to solve nonlinear systems

Local Convergence of an Efficient Multipoint Iterative Method in Banach Space

Local Convergence and Attraction Basins of Higher Order, Jarratt-Like Iterations

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