An optimal fourth-order family of methods for multiple roots and its dynamics

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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“…Example 5. Next, we assume a standard nonlinear test function from Behl et al [17] which is defined by…”

Section: Examplementioning

confidence: 99%

“…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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“…Construction of higher‐order iterative methods for multiple roots having prior knowledge of multiplicity ( m >1) is one of the most important and challenging task in computational mathematics. No doubts, we have a small number of fourth‐order iterative methods for multiple roots, which were proposed by Neta and Johnson in (2008), Li et al in (2009), Neta, Sharma and Sharma, and Li et al in (2010), Zhou et al in (2011), Sharifi et al in (2012), Soleymani et al, Soleymani and Babajee, Liu and Zhou, and Zhou et al in (2013), Thukral in (2014), Behl et al and Hueso et al in (2015), Behl et al in (2016), and Zafar et al in (2018).…”

Section: Introductionmentioning

confidence: 99%

A new two‐point scheme for multiple roots of nonlinear equations

Behl

Zafar

Junjua

2020

Math Methods in App Sciences

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In most of the earlier research for multiple zeros, in order to obtain a new iteration function from the existing scheme, the usual practice is to make no change at the first substep. In this paper, we explore the idea that what are the advantages if the flexibility of choice is also given at the first substep. Therefore, we present a new two‐point sixth‐order scheme for multiple roots (m>1). The main advantages of our scheme over the existing schemes are flexibility at both substeps, simple body structure, smaller residual error, smaller error difference between two consecutive iterations, and smaller asymptotic error constant. The development of the scheme is based on midpoint formula and weight functions of two variables. We compare our methods with the existing methods of the same order with real‐life applications as well as standard test problems. From the numerical results, we find that our methods can be considered as better alternates for the existing methods of the same order. Finally, dynamical study of the proposed schemes is presented that confirms the theoretical results.

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“…In the last few decades, many researchers have worked to develop iterative methods for finding multiple roots with greater efficiency and higher order of convergence. Among them, Li et al [3] in 2009, Sharma and Sharma [4] and Li et al [5] in 2010, Zhou et al [6] in 2011, Sharifi et al [7] in 2012, Soleymani et al [8], Soleymani and Babajee [9], Liu and Zhou [10] and Zhou et al [11] in 2013, Thukral [12] in 2014, Behl et al [13] and Hueso et al [14] in 2015, and Behl et al [15] in 2016 presented optimal fourth-order methods for multiple zeros. Additionally, Li et al [5] (among other optimal methods) and Neta [16] presented non-optimal fourth-order iterative methods.…”

Section: Introductionmentioning

confidence: 99%

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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An optimal fourth-order family of methods for multiple roots and its dynamics

Cited by 50 publications

References 17 publications

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

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

A new two‐point scheme for multiple roots of nonlinear equations

An Efficient Family of Optimal Eighth-Order Multiple Root Finders

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