An eighth-order family of optimal multiple root finders and its dynamics

Behl, Ramandeep; Cordero, Alicia; Motsa, S. S.; Torregrosa, Juan R.

doi:10.1007/s11075-017-0361-6

Cited by 34 publications

(27 citation statements)

References 16 publications

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“…Now, we want to compare our methods with other existing robust schemes of the same order on the basis of the difference between two consecutive iterations, the residual errors in the function, the computational order of convergence ρ, and asymptotic error constant η. We have chosen eighth-order iterative methods for multiple zeros given by Behl et al [19,23]. We take the following particular case (Equation (27)) for (a 1 = 1, a 2 = −2, G 02 = 2m) of the family by Behl et al [19] and denote it by BM1 as follows:…”

Section: Numerical Experimentsmentioning

confidence: 99%

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An Efficient Family of Optimal Eighth-Order Multiple Root Finders

2018

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Finding a repeated zero for a nonlinear equation f (x) = 0, f : I ⊆ R → R has always been of much interest and attention due to its wide applications in many fields of science and engineering. Modified Newton's method is usually applied to solve this kind of problems. Keeping in view that very few optimal higher-order convergent methods exist for multiple roots, we present a new family of optimal eighth-order convergent iterative methods for multiple roots with known multiplicity involving a multivariate weight function. The numerical performance of the proposed methods is analyzed extensively along with the basins of attractions. Real life models from life science, engineering, and physics are considered for the sake of comparison. The numerical experiments and dynamical analysis show that our proposed methods are efficient for determining multiple roots of nonlinear equations.

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Section: Numerical Experimentsmentioning

confidence: 99%

“…We have chosen eighth-order iterative methods for multiple zeros given by Behl et al [19,23]. We take the following particular case (Equation (27)) for (a 1 = 1, a 2 = −2, G 02 = 2m) of the family by Behl et al [19] and denote it by BM1 as follows:…”

Section: Numerical Experimentsmentioning

confidence: 99%

An Efficient Family of Optimal Eighth-Order Multiple Root Finders

2018

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“…have been proposed and analyzed in literature, see [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24]. Such methods require the evaluation of derivative.…”

Section: Introductionmentioning

confidence: 99%

Development of Optimal Eighth Order Derivative-Free Methods for Multiple Roots of Nonlinear Equations

2019

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A number of higher order iterative methods with derivative evaluations are developed in literature for computing multiple zeros. However, higher order methods without derivative for multiple zeros are difficult to obtain and hence such methods are rare in literature. Motivated by this fact, we present a family of eighth order derivative-free methods for computing multiple zeros. Per iteration the methods require only four function evaluations, therefore, these are optimal in the sense of Kung-Traub conjecture. Stability of the proposed class is demonstrated by means of using a graphical tool, namely, basins of attraction. Boundaries of the basins are fractal like shapes through which basins are symmetric. Applicability of the methods is demonstrated on different nonlinear functions which illustrates the efficient convergence behavior. Comparison of the numerical results shows that the new derivative-free methods are good competitors to the existing optimal eighth-order techniques which require derivative evaluations.

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“…There is a plethora of problems from diverse disciplines, such as mathematics [1][2][3][4][5][6][7][8][9][10][11][12][13], optimization [3][4][5][6][7][8], mathematical programming [7,8], chemistry [7], biology [1,2,12], physics [9,13], economics [8], statistics [13], engineering [1,2,[9][10][11][12][13] and other disciplines, that can be reduced to finding a solution x * of the equation:…”

Section: Introductionmentioning

confidence: 99%

“…The order of convergence p ∈ N depends explicitly on the first p − 1 derivatives of the functions appearing in the method. Moreover, the computational cost increases in general especially when the convergence order increases, since successive derivatives must be computed [1][2][3][4][5][6][7][8][9][10][11][12][13].…”

Section: Introductionmentioning

confidence: 99%

Unified Semi-Local Convergence for k—Step Iterative Methods with Flexible and Frozen Linear Operator

Argyros

George

2018

Mathematics

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The aim of this article is to present a unified semi-local convergence analysis for a k-step iterative method containing the inverse of a flexible and frozen linear operator for Banach space valued operators. Special choices of the linear operator reduce the method to the Newton-type, Newton’s, or Stirling’s, or Steffensen’s, or other methods. The analysis is based on center, as well as Lipschitz conditions and our idea of the restricted convergence region. This idea defines an at least as small region containing the iterates as before and consequently also a tighter convergence analysis.

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An eighth-order family of optimal multiple root finders and its dynamics

Cited by 34 publications

References 16 publications

An Efficient Family of Optimal Eighth-Order Multiple Root Finders

An Efficient Family of Optimal Eighth-Order Multiple Root Finders

Development of Optimal Eighth Order Derivative-Free Methods for Multiple Roots of Nonlinear Equations

Unified Semi-Local Convergence for k—Step Iterative Methods with Flexible and Frozen Linear Operator

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