New family of seventh-order methods for nonlinear equations

Bi, Weihong; Ren, Hang; Wu, Qingbiao

doi:10.1016/j.amc.2008.04.048

Cited by 25 publications

(12 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The numerical results show that the methods associated with a multiprecision arithmetic floating point are very useful, because these methods yield a clear reduction in number of iterations. Finally, we conclude that the methods presented in this paper are preferable to other recognized efficient methods, namely Newton's method, Ostrowski's method, sixth order methods [4,6,12], seventh order methods [1,8] etc.…”

Section: Discussionmentioning

confidence: 85%

“…Kou et al [8] presented a family of variants of Ostrowski's method with seventh order convergence requiring three f and one f evaluations. With the same number of evaluations, Bi et al [1] developed a seventh order family of modified King's methods. Bi et al [2] also presented an eighth order family of modified King's methods requiring four evaluations which agrees with the Kung-Traub conjecture.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A new family of modified Ostrowski’s methods with accelerated eighth order convergence

Sharma

Sharma²

2009

Numer Algor

100

View full text Add to dashboard Cite

Based on Ostrowski's fourth order method, we derive a family of eighth order methods for the solution of nonlinear equations. In terms of computational cost the family requires three evaluations of the function and one evaluation of first derivative. Therefore, the efficiency index of the present methods is 1.682 which is better than the efficiency index 1.587 of Ostrowski's method. Kung and Traub conjectured that multipoint iteration methods without memory based on n evaluations have optimal order 2 n−1 . Thus, the family agrees with Kung-Traub conjecture for the case n = 4. The efficacy of the present methods is tested on a number of numerical examples. It is observed that our methods are competitive with other similar robust methods and very effective in high precision computations.

show abstract

Section: Discussionmentioning

confidence: 85%

Section: Introductionmentioning

confidence: 99%

A new family of modified Ostrowski’s methods with accelerated eighth order convergence

Sharma

Sharma²

2009

Numer Algor

100

View full text Add to dashboard Cite

show abstract

“…(See [25][26][27][28] for the detail discussions of Equations (11)- (13) respectively.) Substituting the approximations of F (x n ), F (y n ) and F (z n ) given by Equations (11)- (13) in Equation (10), we establish the following new iterative method:…”

Section: Iterative Methods With Fifth-order Convergence For Solving Mumentioning

confidence: 99%

Fifth-Order Iterative Method for Solving Multiple Roots of the Highest Multiplicity of Nonlinear Equation

Liang

et al. 2015

Algorithms

View full text Add to dashboard Cite

show abstract

“…(See [20,21] and [22] for the detail discussions of (11) and (12), respectively.) Substituting the approximations of f (x n ) and f (z n ) given by (11) and (12) in (10) we establish the following new iterative method:…”

Section: The Methods and Convergence Analysismentioning

confidence: 99%

Fifth-order iterative method for finding multiple roots of nonlinear equations

et al. 2010

Numer Algor

View full text Add to dashboard Cite

This paper presents a fifth-order iterative method as a new modification of Newton's method for finding multiple roots of nonlinear equations with unknown multiplicity m. Its convergence order is analyzed and proved. Moreover, several numerical examples demonstrate that the proposed iterative method is superior to the existing methods. Keywords Nonlinear equation · Multiple roots · Newton-like method · High-order convergence · Iterative methodsWe consider the iterative method to solve multiple roots x * of a nonlinear equation f 0 (x) = 0, where f 0 : [a, b] ⊂ R → R is a nonlinear differential function on [a, b]. Schröder [1] has modified Newton's method to obtain a multiple root when the multiplicity m is known

show abstract

New family of seventh-order methods for nonlinear equations

Cited by 25 publications

References 5 publications

A new family of modified Ostrowski’s methods with accelerated eighth order convergence

A new family of modified Ostrowski’s methods with accelerated eighth order convergence

Fifth-Order Iterative Method for Solving Multiple Roots of the Highest Multiplicity of Nonlinear Equation

Fifth-order iterative method for finding multiple roots of nonlinear equations

Contact Info

Product

Resources

About