Finding the solution of nonlinear equations by a class of optimal methods

Sharifi, Mahdi; Babajee, Diyashvir Kreetee Rajiv; Soleymani, Fazlollah

doi:10.1016/j.camwa.2011.11.040

Cited by 74 publications

(53 citation statements)

References 9 publications

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“…Thus, it is suitable to develop a fourth-order method from the third-order method to improve the efficiency. For the scalar case, we can suggest the following quartical iteration, which is in fact a Jarratt-type iterative method including three functional evaluations to reach the highest possible order four [13] y…”

Section: Description Of a New Methodsmentioning

confidence: 99%

A Quartically Convergent Jarratt-Type Method for Nonlinear System of Equations

Ghorbanzadeh

Soleymani

2015

Algorithms

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Section: Description Of a New Methodsmentioning

confidence: 99%

A Quartically Convergent Jarratt-Type Method for Nonlinear System of Equations

Ghorbanzadeh

Soleymani

2015

Algorithms

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“…With the advancement of computer algebra, many researchers like Chun [8], Chun and Ham [9], Cordero et al [10], Sharma and Ghua [11], Kanwar et al [12], Sharifi et al [13], Soleymani et al [14] and Behl et al [15], among others, proposed various optimal schemes or families of methods of order four. But Ostrowski', Jarratt' and King's methods are between the most efficient fourth-order methods known to date.…”

Section: Introductionmentioning

confidence: 99%

Construction of fourth-order optimal families of iterative methods and their dynamics

Behl

Cordero

Motsa

et al. 2015

Applied Mathematics and Computation

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In this paper, we propose a general class of fourth-order optimal multi-point methods without memory for obtaining simple roots. This class requires only three functional evaluations (viz. two evaluations of function f (x n ), f (y n ) and one of its first-order derivative f (x n )) per iteration. Further, we show that the well-known Ostrowski's method and King's family of fourth-order procedures are special cases of our proposed schemes. One of the new particular subclasses is a biparametric family of iterative methods. By using complex dynamics tools, its stability is analyzed, showing stable members of the family. Further on, one of the parameters is fixed and the stability of the resulting class is studied. On the other hand, the accuracy and validity of new schemes is tested by a number of numerical examples by comparing them with recent and classical optimal fourth-order methods available in the literature. It is found that they are very useful in high precision computations.

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“…This method is a three-point method requiring 1 function and 3 first derivative evaluations and has an efficiency index of 3 1/4 = 1.316 which is lower than 2 1/2 = 1.414 of the 1-point Newton method. Recently, the order of many variants of Newton's method have been improved using the same number of functional evaluations by means of weight functions (see [2][3][4][5][6][7] and the references therein).…”

Section: Introductionmentioning

confidence: 99%

Some Improvements to a Third Order Variant of Newton’s Method from Simpson’s Rule

Babajee¹

2015

Algorithms

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In this paper, we present three improvements to a three-point third order variant of Newton's method derived from the Simpson rule. The first one is a fifth order method using the same number of functional evaluations as the third order method, the second one is a four-point 10th order method and the last one is a five-point 20th order method. In terms of computational point of view, our methods require four evaluations (one function and three first derivatives) to get fifth order, five evaluations (two functions and three derivatives) to get 10th order and six evaluations (three functions and three derivatives) to get 20th order. Hence, these methods have efficiency indexes of 1.495, 1.585 and 1.648, respectively which are better than the efficiency index of 1.316 of the third order method. We test the methods through some numerical experiments which show that the 20th order method is very efficient.

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Finding the solution of nonlinear equations by a class of optimal methods

Cited by 74 publications

References 9 publications

A Quartically Convergent Jarratt-Type Method for Nonlinear System of Equations

A Quartically Convergent Jarratt-Type Method for Nonlinear System of Equations

Construction of fourth-order optimal families of iterative methods and their dynamics

Some Improvements to a Third Order Variant of Newton’s Method from Simpson’s Rule

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