Comparative study of eighth-order methods for finding simple roots of nonlinear equations

Chun, Changbum; Neta, Beny

doi:10.1007/s11075-016-0191-y

Cited by 34 publications

(28 citation statements)

References 48 publications

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“…In this section, we compared the performance of the fourth cases of the new optimal eighth-order methods (2) and (10) with some famous iterative methods. In case NS1 and NS3 we take = 0, 0.5 ,1, −1 but in case NS2 and NS4 we only take = 0 and we take 1 = 1 = 1 = 1 = 0 and for method NS4 we take = 3.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…It is clear that this variant requires to be of order eight to be an optimal iterative method, so we multiply the last step by two weight functions which they satisfy some conditions and without using more evaluations. These weight functions increased the order of this method from seven to eight in order to be an optimal iterative method, see [2].…”

Section: Introductionmentioning

confidence: 99%

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Optimal Families of Eighth-Order Convergent Iterative Methods for Solving Nonlinear Equations Based on Ostrowsk's Method

Alrasheedi¹,

Alsubiahi²

2018

ujcmj

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Solving nonlinear equations is one of the most important problems in numerical analysis, and has a wide range of application in various aspects, as well as many branches of science, engineering, physics, computing, astronomy, finance, ... Generally, it is difficult to find the exact root of the nonlinear equations, and so iterative methods become the efficient way to obtain approximate solutions. Recently Kou et. al. presented a class of new variants of Ostrowski's method with order of convergence equals seven (OSM7) for solving simple roots of nonlinear equations proposed in [7]. Ostrowski's method (OSM7) has efficiency index equals to √7 4

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Section: Numerical Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Optimal Families of Eighth-Order Convergent Iterative Methods for Solving Nonlinear Equations Based on Ostrowsk's Method

Alrasheedi¹,

Alsubiahi²

2018

ujcmj

View full text Add to dashboard Cite

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“…Iterative solutions and approximations for the calculation of the flow friction factor are implemented in software packages which are in common use in everyday engineering practice [88]. So in this paper, we analyzed selected iterative procedures in order to solve the Colebrook equation [93,94], and we found that up 2 to 3 iterations of the Halley and the Schröder method are suitable for the accuracy required by engineering practice, when the fixed initial starting point described in Section 2.2.2 is applied. On the other hand, using a threepoint iterative method with the same initial conditions, the required high accuracy can be reached after only 1 iteration (2 in the worst case) but using three internal steps [26][27][28].…”

Section: Discussionmentioning

confidence: 99%

Advanced Iterative Procedures for Solving the Implicit Colebrook Equation for Fluid Flow Friction

Praks

Brkić

2018

Advances in Civil Engineering

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The empirical Colebrook equation from 1939 is still accepted as an informal standard way to calculate the friction factor of turbulent flows (4000 < Re < 108) through pipes with roughness between negligible relative roughness (ε/D ⟶ 0) to very rough (up to ε/D = 0.05). The Colebrook equation includes the flow friction factor λ in an implicit logarithmic form, λ being a function of the Reynolds number Re and the relative roughness of inner pipe surface ε/D: λ = f(λ, Re, ε/D). To evaluate the error introduced by the many available explicit approximations to the Colebrook equation, λ ≈ f(Re, ε/D), it is necessary to determinate the value of the friction factor λ from the Colebrook equation as accurately as possible. The most accurate way to achieve that is by using some kind of the iterative method. The most used iterative approach is the simple fixed-point method, which requires up to 10 iterations to achieve a good level of accuracy. The simple fixed-point method does not require derivatives of the Colebrook function, while the most of the other presented methods in this paper do require. The methods based on the accelerated Householder’s approach (3rd order, 2nd order: Halley’s and Schröder’s method, and 1st order: Newton–Raphson) require few iterations less, while the three-point iterative methods require only 1 to 3 iterations to achieve the same level of accuracy. The paper also discusses strategies for finding the derivatives of the Colebrook function in symbolic form, for avoiding the use of the derivatives (secant method), and for choosing an optimal starting point for the iterative procedure. The Householder approach to the Colebrook’ equations expressed through the Lambert W-function is also analyzed. Finally, it is presented one approximation to the Colebrook equation with an error of no more than 0.0617%.

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“…The behavior of iterative methods has been examined from a global point of view by using ideas of dynamical systems, see for example [10,11,12,13,14,15,16,17]. Complex dynamics is the most used tool for the study of the global iterative methods, no only because of the good properties of the analytic functions in the complex domain, but also because they provide good pictorial representations in two dimensions.…”

Section: Dynamics Of Osada's Methodsmentioning

confidence: 99%

A note on “Convergence radius of Osada’s method under Hölder continuous condition”

Hueso

Martínez

Gupta

et al. 2018

Applied Mathematics and Computation

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In this paper we revise the proofs of the results obtained in "Convergence radius of Osada's method under Hölder continuous condition" [4], because the remainder of the Taylor's expansion used for the obtainment of the local convergence radius is not correct. So we perform the complete study in order to modify the equation for getting the local convergence radius, the uniqueness radius and the error bounds. Moreover a dynamical study for the third order Osada's method is also developed.

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Comparative study of eighth-order methods for finding simple roots of nonlinear equations

Cited by 34 publications

References 48 publications

Optimal Families of Eighth-Order Convergent Iterative Methods for Solving Nonlinear Equations Based on Ostrowsk's Method

Optimal Families of Eighth-Order Convergent Iterative Methods for Solving Nonlinear Equations Based on Ostrowsk's Method

Advanced Iterative Procedures for Solving the Implicit Colebrook Equation for Fluid Flow Friction

A note on “Convergence radius of Osada’s method under Hölder continuous condition”

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