A new family of fourth-order methods for multiple roots of nonlinear equations

Liu, Baoqing; Zhou, Xiaojian

doi:10.15388/na.18.2.14018

Cited by 40 publications

(28 citation statements)

References 15 publications

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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%

“…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 8 Soleymani and Babajee, 9 Liu and Zhou, 10 and Zhou et al 11 in (2013), Thukral 12 in (2014), Behl et al 13 Out of them, iterative functions proposed by Li et al 5 (expect two of them are optimal), Neta and Johnson, 1 and Neta 3 are nonoptimal schemes of fourth order. On the other hand, rest of them are optimal according to classical Kung-Traub conjecture.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Introductionmentioning

confidence: 99%

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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“…have been proposed in the literature; see, for example, [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] and the references therein. The techniques based on Newton's or the Newton-like method require the evaluations of derivatives of first order.…”

Section: Introductionmentioning

confidence: 99%

An Efficient Class of Traub-Steffensen-Like Seventh Order Multiple-Root Solvers with Applications

2019

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Many higher order multiple-root solvers that require derivative evaluations are available in literature. Contrary to this, higher order multiple-root solvers without derivatives are difficult to obtain, and therefore, such techniques are yet to be achieved. Motivated by this fact, we focus on developing a new family of higher order derivative-free solvers for computing multiple zeros by using a simple approach. The stability of the techniques is checked through complex geometry shown by drawing basins of attraction. Applicability is demonstrated on practical problems, which illustrates the efficient convergence behavior. Moreover, the comparison of numerical results shows that the proposed derivative-free techniques are good competitors of the existing techniques that require derivative evaluations in the iteration.

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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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A new family of fourth-order methods for multiple roots of nonlinear equations

Cited by 40 publications

References 15 publications

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

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

An Efficient Class of Traub-Steffensen-Like Seventh Order Multiple-Root Solvers with Applications

An Efficient Family of Optimal Eighth-Order Multiple Root Finders

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