A study on the local convergence and the dynamics of Chebyshev–Halley–type methods free from second derivative

Argyros, Ioannis K.; Magreñán, Á. Alberto

doi:10.1007/s11075-015-9981-x

Cited by 57 publications

(32 citation statements)

References 25 publications

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“…The semilocal convergence analysis here is different from the local convergence studied in references [2,3]. The local convergence requires the assumptions around a solution, whereas the semilocal convergence needs the conditions around an initial point.…”

Section: Introductionmentioning

confidence: 95%

“…The second-order Newton's method [10] is widely applied for solving this equation. Recently, third-order Chebyshev-Halley methods and some of their variants have been developed [1][2][3][4][5][6][7][8][9]. In reference [8], Gutiérrez and Hernández studied the convergence of super-Halley method given by…”

Section: Introductionmentioning

confidence: 99%

“…The local convergence requires the assumptions around a solution, whereas the semilocal convergence needs the conditions around an initial point. In references [2,3], to establish the local convergence for Chebyshev-Halley-type methods, two of the required assumptions are as listed below.…”

Section: Introductionmentioning

confidence: 99%

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Analysis of semilocal convergence for ameliorated super-Halley methods with less computation for inversion

Wang¹,

Kou²

2016

LMS J. Comput. Math.

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In this paper, the semilocal convergence for ameliorated super-Halley methods in Banach spaces is considered. Different from the results in [J. M. Gutiérrez and M. A. Hernández, Comput. Math. Appl. 36 (1998) 1–8], these ameliorated methods do not need to compute a second derivative, the computation for inversion is reduced and the $R$-order is also heightened. Under a weaker condition, an existence–uniqueness theorem for the solution is proved.

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Section: Introductionmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Analysis of semilocal convergence for ameliorated super-Halley methods with less computation for inversion

Wang¹,

Kou²

2016

LMS J. Comput. Math.

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“…Nowadays, with the use of computers, and symbolic and numerical software we can draw those basins and we can understand how difficult it is to draw them without computers. In recent times, many authors have analyzed the complex dynamics of iteration functions in their work, see [17,[25][26][27][28] and references therein.…”

Section: Introductionmentioning

confidence: 99%

On a Bi-Parametric Family of Fourth Order Composite Newton–Jarratt Methods for Nonlinear Systems

et al. 2019

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We present a new two-parameter family of fourth-order iterative methods for solving systems of nonlinear equations. The scheme is composed of two Newton–Jarratt steps and requires the evaluation of one function and two first derivatives in each iteration. Convergence including the order of convergence, the radius of convergence, and error bounds is presented. Theoretical results are verified through numerical experimentation. Stability of the proposed class is analyzed and presented by means of using new dynamics tool, namely, the convergence plane. Performance is exhibited by implementing the methods on nonlinear systems of equations, including those resulting from the discretization of the boundary value problem. In addition, numerical comparisons are made with the existing techniques of the same order. Results show the better performance of the proposed techniques than the existing ones.

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“…Another way to analyze the behavior of a numerical method is to study it from a dynamical point of view, i.e., to consider the iterative method as a discrete dynamical system and to study its stability. This is a line of work that has proven to be especially fruitful in recent years (see, for example, the papers [1], [2], [6], [9], [11] and, more recently, [8], [12], [13] and [14]).…”

mentioning

confidence: 99%

Dynamical Analysis to Explain the Numerical Anomalies in the Family of Ermakov-Kalitlin Type Methods

Cordero

Torregrosa

Vindel

2019

Mathematical Modelling and Analysis

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In this paper, we study the dynamics of an iterative method based on the Ermakov-Kalitkin class of iterative schemes for solving nonlinear equations. As it was proven in ”A new family of iterative methods widening areas of convergence, Appl. Math. Comput.”, this family has the property of getting good estimations of the solution when Newton’s method fails. Moreover, the set of converging starting points for several non-polynomial test functions was plotted and they showed to be wider in the case of proposed methods than in Newton’s case, for small values of the parameter. Now, we make a complex dynamical analysis of this parametric class in order to justify the stability properties of this family.

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A study on the local convergence and the dynamics of Chebyshev–Halley–type methods free from second derivative

Cited by 57 publications

References 25 publications

Analysis of semilocal convergence for ameliorated super-Halley methods with less computation for inversion

Analysis of semilocal convergence for ameliorated super-Halley methods with less computation for inversion

On a Bi-Parametric Family of Fourth Order Composite Newton–Jarratt Methods for Nonlinear Systems

Dynamical Analysis to Explain the Numerical Anomalies in the Family of Ermakov-Kalitlin Type Methods

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