scholarly journals Ball Comparison for Some Efficient Fourth Order Iterative Methods Under Weak Conditions

Mathematics ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 89
Author(s):  
Ioannis Argyros ◽  
Ramandeep Behl
Keyword(s):  
Banach Space ◽  
Iterative Methods ◽  
Nonlinear Equations ◽  
Fourth Order ◽  
Numerical Experiments ◽  
Order Derivative ◽  
First Derivative

We provide a ball comparison between some 4-order methods to solve nonlinear equations involving Banach space valued operators. We only use hypotheses on the first derivative, as compared to the earlier works where they considered conditions reaching up to 5-order derivative, although these derivatives do not appear in the methods. Hence, we expand the applicability of them. Numerical experiments are used to compare the radii of convergence of these methods.

scholarly journals A Two-Parameter Family of Fourth-Order Iterative Methods with Optimal Convergence for Multiple Zeros

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Young Ik Kim ◽  
Young Hee Geum
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Fourth Order ◽  
Numerical Experiments ◽  
Harmonic Mean ◽  
Optimal Order ◽  
Multiple Roots ◽  
Multiple Zeros ◽  
Asymptotic Error ◽  
Two Parameter

We develop a family of fourth-order iterative methods using the weighted harmonic mean of two derivative functions to compute approximate multiple roots of nonlinear equations. They are proved to be optimally convergent in the sense of Kung-Traub’s optimal order. Numerical experiments for various test equations confirm well the validity of convergence and asymptotic error constants for the developed methods.

scholarly journals A new family of fourth-order methods for multiple roots of nonlinear equations

2013 ◽  
Vol 18 (2) ◽  
pp. 143-152 ◽  
Author(s):  
Baoqing Liu ◽  
Xiaojian Zhou
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Fourth Order ◽  
Newton’S Method ◽  
Newton's Method ◽  
Optimal Order ◽  
Multiple Roots ◽  
First Derivative ◽  
New Family ◽  
Modified Newton’S Method

Recently, some optimal fourth-order iterative methods for multiple roots of nonlinear equations are presented when the multiplicity m of the root is known. Different from these optimal iterative methods known already, this paper presents a new family of iterative methods using the modified Newton’s method as its first step. The new family, requiring one evaluation of the function and two evaluations of its first derivative, is of optimal order. Numerical examples are given to suggest that the new family can be competitive with other fourth-order methods and the modified Newton’s method for multiple roots.

scholarly journals Local Convergence for Multi-Step High Order Solvers under Weak Conditions

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 179
Author(s):  
Ramandeep Behl ◽  
Ioannis K. Argyros
Keyword(s):  
Banach Space ◽  
Nonlinear Equations ◽  
Local Convergence ◽  
Nonlinear Problems ◽  
Fréchet Derivative ◽  
High Order ◽  
Order Derivative ◽  
First Order ◽  
Lipschitz Constants ◽  
Convergence Study

Our aim in this article is to suggest an extended local convergence study for a class of multi-step solvers for nonlinear equations valued in a Banach space. In comparison to previous studies, where they adopt hypotheses up to 7th Fŕechet-derivative, we restrict the hypotheses to only first-order derivative of considered operators and Lipschitz constants. Hence, we enlarge the suitability region of these solvers along with computable radii of convergence. In the end of this study, we choose a variety of numerical problems which illustrate that our works are applicable but not earlier to solve nonlinear problems.

scholarly journals A New Higher-Order Iterative Scheme for the Solutions of Nonlinear Systems

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 271 ◽  
Author(s):  
Ramandeep Behl ◽  
Ioannis K. Argyros
Keyword(s):  
Mathematical Modeling ◽  
Banach Space ◽  
Nonlinear Equations ◽  
Numerical Experiments ◽  
Iterative Scheme ◽  
Real Life ◽  
System Of Nonlinear Equations ◽  
Life Problems ◽  
Lipschitz Constants ◽  
Better Than

Many real-life problems can be reduced to scalar and vectorial nonlinear equations by using mathematical modeling. In this paper, we introduce a new iterative family of the sixth-order for a system of nonlinear equations. In addition, we present analyses of their convergences, as well as the computable radii for the guaranteed convergence of them for Banach space valued operators and error bounds based on the Lipschitz constants. Moreover, we show the applicability of them to some real-life problems, such as kinematic syntheses, Bratu’s, Fisher’s, boundary value, and Hammerstein integral problems. We finally wind up on the ground of achieved numerical experiments, where they perform better than other competing schemes.

scholarly journals Three New Optimal Fourth-Order Iterative Methods to Solve Nonlinear Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Gustavo Fernández-Torres ◽  
Juan Vásquez-Aquino
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Fourth Order ◽  
Newton’S Method ◽  
Newton's Method ◽  
Classical Method ◽  
Multiple Roots ◽  
Order Convergence ◽  
Numerical Examples ◽  
Optimal Methods

We present new modifications to Newton's method for solving nonlinear equations. The analysis of convergence shows that these methods have fourth-order convergence. Each of the three methods uses three functional evaluations. Thus, according to Kung-Traub's conjecture, these are optimal methods. With the previous ideas, we extend the analysis to functions with multiple roots. Several numerical examples are given to illustrate that the presented methods have better performance compared with Newton's classical method and other methods of fourth-order convergence recently published.

scholarly journals Some New Iterative Methods for Nonlinear Equations

2010 ◽  
Vol 2010 ◽  
pp. 1-12 ◽  
Author(s):  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor ◽  
Eisa Al-Said ◽  
Muhammad Waseem
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Fourth Order ◽  
System Of Equations ◽  
Decomposition Technique ◽  
Methods Comparison ◽  
Numerical Examples ◽  
New Methods ◽  
And Performance

We suggest and analyze some new iterative methods for solving the nonlinear equationsf(x)=0using the decomposition technique coupled with the system of equations. We prove that new methods have convergence of fourth order. Several numerical examples are given to illustrate the efficiency and performance of the new methods. Comparison with other similar methods is given.

scholarly journals A Family of Fourteenth-Order Convergent Iterative Methods for Solving Nonlinear Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Fiza Zafar ◽  
Gulshan Bibi
Keyword(s):  
Iterative Method ◽  
Iterative Methods ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Efficiency Index ◽  
Numerical Examples ◽  
New Class ◽  
First Derivative ◽  
New Methods ◽  
Specific Step

We present a family of fourteenth-order convergent iterative methods for solving nonlinear equations involving a specific step which when combined with any two-step iterative method raises the convergence order by n+10, if n is the order of convergence of the two-step iterative method. This new class include four evaluations of function and one evaluation of the first derivative per iteration. Therefore, the efficiency index of this family is 141/5 =1.695218203. Several numerical examples are given to show that the new methods of this family are comparable with the existing methods.

scholarly journals On a Family of High-Order Iterative Methods under Kantorovich Conditions and Some Applications

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
S. Amat ◽  
C. Bermúdez ◽  
S. Busquier ◽  
M. J. Legaz ◽  
S. Plaza
Keyword(s):  
Banach Spaces ◽  
Integral Equations ◽  
Iterative Methods ◽  
Nonlinear Equations ◽  
Numerical Experiments ◽  
High Order ◽  
High Order Iterative Methods

This paper is devoted to the study of a class of high-order iterative methods for nonlinear equations on Banach spaces. An analysis of the convergence under Kantorovich-type conditions is proposed. Some numerical experiments, where the analyzed methods present better behavior than some classical schemes, are presented. These applications include the approximation of some quadratic and integral equations.

scholarly journals A New Biparametric Family of Two-Point Optimal Fourth-Order Multiple-Root Finders

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Young Ik Kim ◽  
Young Hee Geum
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Fourth Order ◽  
Order Of Convergence ◽  
Multiple Root ◽  
Multiple Roots ◽  
Asymptotic Error ◽  
Error Constant ◽  
Asymptotic Error Constant

We construct a biparametric family of fourth-order iterative methods to compute multiple roots of nonlinear equations. This method is verified to be optimally convergent. Various nonlinear equations confirm our proposed method with order of convergence of four and show that the computed asymptotic error constant agrees with the theoretical one.

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