Optimal Steffensen-type methods with eighth order of convergence

Soleymani, Fazlollah; Vanani, S. Karimi

doi:10.1016/j.camwa.2011.10.047

Cited by 46 publications

(24 citation statements)

References 9 publications

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“…Three log-calls methods are the most powerful among the presented procedures. The simplest, but accurate, can be recommended for use: 3.3.1) Neta [47]; Equation (16), 3.3.2) Chun-Neta [48] based on Kung-Traub [41]; Equation (17), 3.3.3) Džunić-Petković-Petković method [57][58][59]; Equation (18), and 3.3.5) Jain [51] based on Steffensen method [52][53][54][55]; Equation (20). Those accurate, but with two-parameter function can be used, but for the Colebrook equation they seem to be too complex and their use should be limited to some rare very well elaborated cases and calculations: Sharma-Arora [56]; Equation (23), 3.3.9) Sharma-Sharma [61]; Equation (24), and 3.3.10) Sharma-Guha-Gupta method [62]; Equation (25).…”

Section: Summary -Discussion and Analysismentioning

confidence: 99%

Choosing the Optimal Multi-Point Iterative Method for the Colebrook Flow Friction Equation – Numerical Validation

Praks¹,

Brkić²

2018

Preprint

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Abstract:The Colebrook equation is implicitly given in respect to the unknown flow friction factor ; = ( , * , ) which cannot be expressed explicitly in exact way without simplifications and use of approximate calculus. Common approach to solve it is through the Newton-Raphson iterative procedure or through the fixed-point iterative procedure. Both requires in some case even eight iterations. On the other hand numerous more powerful iterative methods such as three-or twopoint methods, etc. are available. The purpose is to choose optimal iterative method in order to solve the implicit Colebrook equation for flow friction accurately using the least possible number of iterations. The methods are thoroughly tested and those which require the least possible number of iterations to reach the accurate solution are identified. The most powerful three-point methods require in worst case only two iterations to reach final solution. The recommended representatives are Sharma-Guha-Gupta, Sharma-Sharma, Sharma-Arora, Džunić-Petković-Petković; Bi-Ren-Wu, Chun-Neta based on Kung-Traub, Neta, and Jain method based on Steffensen scheme. The recommended iterative methods can reach the final accurate solution with the least possible number of iterations. The approach is hybrid between iterative procedure and one-step explicit approximations and can be used in engineering design for initial rough, but also for final fine calculations.

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Section: Summary -Discussion and Analysismentioning

confidence: 99%

Choosing the Optimal Multi-Point Iterative Method for the Colebrook Flow Friction Equation – Numerical Validation

Praks¹,

Brkić²

2018

Preprint

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“…• The paper SK [29] proposes two classes of three-step without memory iterations based on the well known second-order method of Steffensen. Per computing step, the methods from the developed classes reach the order of convergence eight using only four evaluations, while they are totally free from derivative evaluation.…”

Section: Arg3mentioning

confidence: 99%

A Steffensen type method of two steps in Banach spaces with applications

Amat

Busquier

Ezquerro

et al. 2016

Journal of Computational and Applied Mathematics

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Please cite this article as: S. Amat, S. Busquier, J.A. Ezquerro, M.A. Hernández-Verón, A Steffensen type method of two steps in Banach spaces with applications, Journal of Computational and Applied Mathematics (2015), http://dx. AbstractThis paper is devoted to the analysis of a Steffensen-type of two steps with order of convergence at least three. The main advantage of this method is that it does not need to evaluate any Fréchet derivative or any bilinear operator. The method includes extra parameters in the divided difference in order to ensure a good approximation to the first derivative in each iteration. We prove, using recurrence relations, a semilocal convergence result in Banach spaces and do a detailed study of the domain of parameters associated to this result. Finally, some numerical results, including differentiable and nondifferentiable operators, are presented. Special attention is paid in the approximation of solutions of boundary problems using the multiple shooting method and in the approximation of a nonlinear model related with image processing.

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“…The results are provided in Tables 1-3. For methods Eqs (17), (18) and (19), we have selected −0.5 for any parameter which is involved. Example 1.…”

Section: Numerical Computationsmentioning

confidence: 99%

“…The new method has iteration stabilities to the original iteration value and behave acceptable. For further reading on this topic, one may refer [17][18][19][20].…”

Section: Numerical Computationsmentioning

confidence: 99%

An efficient twelfth-order iterative method for finding all the solutions of nonlinear equations

Soleymani

2013

JCM

Self Cite

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In this study, the sixteenth-order iterative scheme of Li et al. [X. Li, C. Mu, J. Ma, C. Wang, Sixteenth-order method for nonlinear equations, Appl. Math. Com. 215 (2010), 3754-3758] is considered. We increase its efficiency index from 1.587 to 1.644, by reducing the number of evaluations from six to five per iteration. This goal is achieved by providing an approximation for the first-order derivative of the function in the fourth step. Error analysis will also be studied. In the sequel, some numerical instances are given to show the accuracy of the new obtained twelfth-order technique. Therein, another objective of this paper is achieved by proposing a hybrid method for finding all the real solutions of nonlinear equations.

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Optimal Steffensen-type methods with eighth order of convergence

Cited by 46 publications

References 9 publications

Choosing the Optimal Multi-Point Iterative Method for the Colebrook Flow Friction Equation – Numerical Validation

Choosing the Optimal Multi-Point Iterative Method for the Colebrook Flow Friction Equation – Numerical Validation

A Steffensen type method of two steps in Banach spaces with applications

An efficient twelfth-order iterative method for finding all the solutions of nonlinear equations

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Optimal Steffensen-type methods with eighth order of convergence

Cited by 46 publications

References 9 publications

Choosing the Optimal Multi-Point Iterative Method for the Colebrook Flow Friction Equation &ndash; Numerical Validation

Choosing the Optimal Multi-Point Iterative Method for the Colebrook Flow Friction Equation &ndash; Numerical Validation

A Steffensen type method of two steps in Banach spaces with applications

An efficient twelfth-order iterative method for finding all the solutions of nonlinear equations

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Product

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Choosing the Optimal Multi-Point Iterative Method for the Colebrook Flow Friction Equation – Numerical Validation

Choosing the Optimal Multi-Point Iterative Method for the Colebrook Flow Friction Equation – Numerical Validation