scholarly journals Modified Ostrowski’s method with eighth-order convergence and high efficiency index

2010 ◽  
Vol 23 (5) ◽  
pp. 549-554 ◽  
Author(s):  
Xia Wang ◽  
Liping Liu
Keyword(s):  
High Efficiency ◽  
Efficiency Index ◽  
Order Convergence ◽  
Eighth Order ◽  
Ostrowski’S Method

scholarly journals On Some Iterative Methods with Memory and High Efficiency Index for Solving Nonlinear Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Tahereh Eftekhari
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
High Efficiency ◽  
Efficiency Index ◽  
Order Convergence ◽  
Eighth Order ◽  
Additional Function ◽  
Numerical Comparisons ◽  
With Memory

Based on iterative methods without memory of eighth-order convergence proposed by Thukral (2012), some iterative methods with memory and high efficiency index are presented. We show that the order of convergence is increased without any additional function evaluations. Numerical comparisons are made to show the performance of the presented methods.

scholarly journals Efficient Four-Parametric with-and-without-Memory Iterative Methods Possessing High Efficiency Indices

2018 ◽  
Vol 2018 ◽  
pp. 1-12 ◽  
Author(s):  
Alicia Cordero ◽  
Moin-ud-Din Junjua ◽  
Juan R. Torregrosa ◽  
Nusrat Yasmin ◽  
Fiza Zafar
Keyword(s):  
Iterative Methods ◽  
High Efficiency ◽  
Nonlinear Function ◽  
Efficiency Index ◽  
Simple Zero ◽  
Eighth Order ◽  
New Family ◽  
Derivative Free ◽  
With Memory ◽  
Theoretical Results

We construct a family of derivative-free optimal iterative methods without memory to approximate a simple zero of a nonlinear function. Error analysis demonstrates that the without-memory class has eighth-order convergence and is extendable to with-memory class. The extension of new family to the with-memory one is also presented which attains the convergence order 15.5156 and a very high efficiency index 15.51561/4≈1.9847. Some particular schemes of the with-memory family are also described. Numerical examples and some dynamical aspects of the new schemes are given to support theoretical results.

scholarly journals An Efficient Family of Root-Finding Methods with Optimal Eighth-Order Convergence

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
Rajni Sharma ◽  
Janak Raj Sharma
Keyword(s):  
Computational Cost ◽  
Efficiency Index ◽  
Optimal Order ◽  
Computational Results ◽  
Order Convergence ◽  
Eighth Order ◽  
Root Finding ◽  
First Derivative ◽  
The Family ◽  
Iteration Methods

We derive a family of eighth-order multipoint methods for the solution of nonlinear equations. In terms of computational cost, the family requires evaluations of only three functions and one first derivative per iteration. This implies that the efficiency index of the present methods is 1.682. Kung and Traub (1974) conjectured that multipoint iteration methods without memory based on n evaluations have optimal order . Thus, the family agrees with Kung-Traub conjecture for the case . Computational results demonstrate that the developed methods are efficient and robust as compared with many well-known methods.

scholarly journals Geometric Construction of Eighth-Order Optimal Families of Ostrowski’s Method

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Ramandeep Behl ◽  
S. S. Motsa
Keyword(s):  
Fourth Order ◽  
Weighted Average ◽  
Efficiency Index ◽  
Basins Of Attraction ◽  
Geometric Construction ◽  
The Other ◽  
Initial Value ◽  
Eighth Order ◽  
Numerical Examples ◽  
Ostrowski’S Method

Based on well-known fourth-order Ostrowski’s method, we proposed many new interesting optimal families of eighth-order multipoint methods without memory for obtaining simple roots. Its geometric construction consists in approximatingfn′atznin such a way that its average with the known tangent slopesfn′atxnandynis the same as the known weighted average of secant slopes and then we apply weight function approach. The adaptation of this strategy increases the convergence order of Ostrowski's method from four to eight and its efficiency index from 1.587 to 1.682. Finally, a number of numerical examples are also proposed to illustrate their accuracy by comparing them with the new existing optimal eighth-order methods available in the literature. It is found that they are very useful in high precision computations. Further, it is also noted that larger basins of attraction belong to our methods although the other methods are slow and have darker basins while some of the methods are too sensitive upon the choice of the initial value.

scholarly journals Basin of Attraction from Modification Variant of Chebyshev-Halley Methods

2019 ◽  
Vol 8 (6S3) ◽  
pp. 119-124
Keyword(s):  
Nonlinear Equation ◽  
Convergence Analysis ◽  
Hermite Interpolation ◽  
Basin Of Attraction ◽  
Efficiency Index ◽  
Sixth Order ◽  
Order Convergence ◽  
Eighth Order ◽  
Basin Of Attractions

The development of Chebyshev-Halley Method for solving nonlinear equation is presented in this paper. Varian of Chebyshev-Halley method by Xiaojian (2008) was modified using Hermite Interpolation. The convergence analysis shows that these methods have sixth-order convergence for   0 and   1 eighth-order convergence for   1 2 . The methods are classified by the order and efficiency index. Here, we considered other criteria, the basin of attractions which are presented for several examples.The development of Chebyshev-Halley Method for solving nonlinear equation is presented in this paper. Varian of Chebyshev-Halley method by Xiaojian (2008) was modified using Hermite Interpolation. The convergence analysis shows that these methods have sixth-order convergence for   0 and   1 eighth-order convergence for   1 2 . The methods are classified by the order and efficiency index. Here, we considered other criteria, the basin of attractions which are presented for several examples.

scholarly journals New modification of Maheshwari’s method with optimal eighth order convergence for solving nonlinear equations

2016 ◽  
Vol 14 (1) ◽  
pp. 443-451 ◽  
Author(s):  
Somayeh Sharifi ◽  
Massimiliano Ferrara ◽  
Mehdi Salimi ◽  
Stefan Siegmund
Keyword(s):  
Weight Function ◽  
Nonlinear Equations ◽  
Numerical Experiments ◽  
Efficiency Index ◽  
The Other ◽  
Order Convergence ◽  
Eighth Order ◽  
First Derivative ◽  
Attraction Basins

AbstractIn this paper, we present a family of three-point with eight-order convergence methods for finding the simple roots of nonlinear equations by suitable approximations and weight function based on Maheshwari’s method. Per iteration this method requires three evaluations of the function and one evaluation of its first derivative. These class of methods have the efficiency index equal to ${8^{{\textstyle{1 \over 4}}}} \approx 1.682$. We describe the analysis of the proposed methods along with numerical experiments including comparison with the existing methods. Moreover, the attraction basins of the proposed methods are shown with some comparisons to the other existing methods.

scholarly journals A Higher Order Chebyshev-Halley-Type Family of Iterative Methods for Multiple Roots

Mathematics ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 339
Author(s):  
Ramandeep Behl ◽  
Eulalia Martínez ◽  
Fabricio Cevallos ◽  
Diego Alarcón
Keyword(s):  
Iterative Methods ◽  
Computational Cost ◽  
Real Life ◽  
Efficiency Index ◽  
Maximum Efficiency ◽  
Multiple Roots ◽  
Order Convergence ◽  
Eighth Order ◽  
Optimal Method ◽  
Computational Mathematics

The aim of this paper is to introduce new high order iterative methods for multiple roots of the nonlinear scalar equation; this is a demanding task in the area of computational mathematics and numerical analysis. Specifically, we present a new Chebyshev–Halley-type iteration function having at least sixth-order convergence and eighth-order convergence for a particular value in the case of multiple roots. With regard to computational cost, each member of our scheme needs four functional evaluations each step. Therefore, the maximum efficiency index of our scheme is 1.6818 for α = 2 , which corresponds to an optimal method in the sense of Kung and Traub’s conjecture. We obtain the theoretical convergence order by using Taylor developments. Finally, we consider some real-life situations for establishing some numerical experiments to corroborate the theoretical results.

scholarly journals A New Optimal Eighth-Order Ostrowski-Type Family of Iterative Methods for Solving Nonlinear Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Taher Lotfi ◽  
Tahereh Eftekhari
Keyword(s):  
Weight Function ◽  
Iterative Methods ◽  
Nonlinear Equations ◽  
Efficiency Index ◽  
Eighth Order ◽  
New Class ◽  
First Derivative ◽  
New Methods ◽  
New Family ◽  
Ostrowski’S Method

Based on Ostrowski's method, a new family of eighth-order iterative methods for solving nonlinear equations by using weight function methods is presented. Per iteration the new methods require three evaluations of the function and one evaluation of its first derivative. Therefore, this family of methods has the efficiency index which equals 1.682. Kung and Traub conjectured that a multipoint iteration without memory based on n evaluations could achieve optimal convergence order 2n−1. Thus, we provide a new class which agrees with the conjecture of Kung-Traub for n=4. Numerical comparisons are made to show the performance of the presented methods.

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