Simply constructed family of a Ostrowski’s method with optimal order of convergence

Kanwar, V.; Behl, Ramandeep; Sharma, Kapil K.

doi:10.1016/j.camwa.2011.09.039

Cited by 21 publications

(17 citation statements)

References 8 publications

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“…Therefore, the convergence order and computational efficiency of the one-point iterative methods are lower than multipoint iterative methods. In recent years, many multipoint iterative methods have been proposed for solving nonlinear equations that improve local convergence order of the classical Newton method; see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. In 1973, King [2,3] had proposed an optimal family of fourth-order multipoint methods requiring three functional evaluations per full iteration, which is given by = − ( ) ( ) ,…”

Section: International Journal Of Mathematics and Mathematical Sciencesmentioning

confidence: 99%

“…For = 0, one can easily get the well-known Ostrowski's method [2,4,5]. However, all these multipoint methods are the variants of Newton's method and the iteration can be aborted due to the overflow or leads to divergence, if the derivative of the function at an iterative point is singular or almost singular, which restrict their applications in practical.…”

Section: International Journal Of Mathematics and Mathematical Sciencesmentioning

confidence: 99%

“…For 1 = 0, scheme (23) reduces the well-known Ostrowski's method [4,5]. In order to overcome this problem, some attempts have been made by Kanwar and Tomar [6] to develop an optimal family of Ostrowski's method.…”

Section: Order Of Convergencementioning

confidence: 99%

See 2 more Smart Citations

New Highly Efficient Families of Higher-Order Methods for Simple Roots, Permittingf′(xn)=0

Behl

Kanwar

2014

International Journal of Mathematics and Mathematical Sciences

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Construction of higher-order optimal and globally convergent methods for computing simple roots of nonlinear equations is an earliest and challenging problem in numerical analysis. Therefore, the aim of this paper is to present optimal and globally convergent families of King's method and Ostrowski's method having biquadratic and eight-order convergence, respectively, permitting ( ) = 0 in the vicinity of the required root. Fourth-order King's family and Ostrowski's method can be seen as special cases of our proposed scheme. All the methods considered here are found to be more effective to the similar robust methods available in the literature. In their dynamical study, it has been observed that the proposed methods have equal or better stability and robustness as compared to the other methods.

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Section: International Journal Of Mathematics and Mathematical Sciencesmentioning

confidence: 99%

Section: International Journal Of Mathematics and Mathematical Sciencesmentioning

confidence: 99%

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New Highly Efficient Families of Higher-Order Methods for Simple Roots, Permittingf′(xn)=0

Behl

Kanwar

2014

International Journal of Mathematics and Mathematical Sciences

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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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“…Therefore, it is only possible to obtain approximate solutions and one has to be satisfied with approximate solutions up to any specified degree of accuracy, by relying on numerical methods which are based on iterative procedures. Therefore, researchers worldwide resort to an iterative method and they have proposed a plethora of iterative methods [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. While, using these iterative methods researchers face the problems of slow convergence, non-convergence, divergence, inefficiency or failure (for detail please see Traub [15] and Petkovic et al [13]).…”

Section: Introductionmentioning

confidence: 99%

Local Convergence of an Efﬁcient High Convergence Order Method Using Hypothesis Only on the First Derivative

2015

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We present a local convergence analysis of an eighth order three step method in order to approximate a locally unique solution of nonlinear equation in a Banach space setting. In an earlier study by Sharma and Arora (2015), the order of convergence was shown using Taylor series expansions and hypotheses up to the fourth order derivative or even higher of the function involved which restrict the applicability of the proposed scheme. However, only first order derivative appears in the proposed scheme. In order to overcome this problem, we proposed the hypotheses up to only the first order derivative. In this way, we not only expand the applicability of the methods but also propose convergence domain. Finally, where earlier studies cannot be applied, a variety of concrete numerical examples are proposed to obtain the solutions of nonlinear equations. Our study does not exhibit this type of problem/restriction.

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Simply constructed family of a Ostrowski’s method with optimal order of convergence

Cited by 21 publications

References 8 publications

New Highly Efficient Families of Higher-Order Methods for Simple Roots, Permittingf′(xn)=0

New Highly Efficient Families of Higher-Order Methods for Simple Roots, Permittingf′(xn)=0

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

Local Convergence of an Efﬁcient High Convergence Order Method Using Hypothesis Only on the First Derivative

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