An Improvement of Ostrowski’s and King’s Techniques with Optimal Convergence Order Eight

Soleymani, Fazlollah; Sharifi, Mahdi; Mousavi, B. Somayeh

doi:10.1007/s10957-011-9929-9

Cited by 38 publications

(17 citation statements)

References 7 publications

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“…Theorem 4. Let the varying parameter λ k in the iterative Equation (17) be calculated by Equation (15). If an initial approximation x 0 is sufficiently close to a simple zero a of f (x), then the R-order of convergence of the n-point methods Equation (17) with memory is at least 2 n + 2 n−3 + 2 n−4…”

Section: Lemma 1 Letmentioning

confidence: 99%

“…Multipoint iterative methods can overcome theoretical limits of one-point methods concerning the convergence order and computational efficiency. In recent years, many multipoint iterative methods have been proposed for solving nonlinear equations, see [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. Wang and Liu in [4] developed the following eighth-order iterative method without memory by Hermite interpolation methods…”

Section: Introductionmentioning

confidence: 99%

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A Family of Newton Type Iterative Methods for Solving Nonlinear Equations

et al. 2015

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In this paper, a general family of n-point Newton type iterative methods for solving nonlinear equations is constructed by using direct Hermite interpolation. The order of convergence of the new n-point iterative methods without memory is 2 n requiring the evaluations of n functions and one first-order derivative in per full iteration, which implies that this family is optimal according to Kung and Traub's conjecture (1974). Its error equations and asymptotic convergence constants are obtained. The n-point iterative methods with memory are obtained by using a self-accelerating parameter, which achieve much faster convergence than the corresponding n-point methods without memory. The increase of convergence order is attained without any additional calculations so that the n-point Newton type iterative methods with memory possess a very high computational efficiency. Numerical examples are demonstrated to confirm theoretical results.

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Section: Lemma 1 Letmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Family of Newton Type Iterative Methods for Solving Nonlinear Equations

et al. 2015

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“…To improve the local order of convergence different techniques have been used getting iterative schemes with and without memory (see, for instance [5][6][7]) and to upgrade the efficiency index, many optimal methods without memory have been proposed; see, for example, [3,[8][9][10][11][12][13][14][15][16], and the references therein.…”

Section: Introductionmentioning

confidence: 99%

On improved three-step schemes with high efficiency index and their dynamics

et al. 2013

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This paper presents an improvement of the sixth-order method of Chun and Neta as a class of three-step iterations with optimal efficiency index, in the sense of Kung-Traub conjecture. Each member of the presented class reaches the highest possible order using four functional evaluations. Error analysis will be studied and numerical examples are also made to support the theoretical results. We then present results which describe the dynamics of the presented optimal methods for complex polynomials. The basins of attraction of the existing optimal methods and our methods are presented and compared to illustrate their performances.

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“…Multi-step quasi-linearization methods are constructed in [5,6], but both methods face low convergence-order. Many authors [7][8][9][10][11][12][13][14] has investigated the iterative methods for nonlinear equations with higher convergence-order. For the systems of nonlinear equation recent advancements are addressed in [15,[17][18][19][20][21][22][23][24][25][26][27].…”

mentioning

confidence: 99%