scholarly journals On the local convergence of the Modified Newton method

2019 ◽  
Vol 57 (1) ◽  
pp. 13-22
Author(s):  
Ştefan Măruşter
Keyword(s):  
Newton Method ◽  
Local Convergence ◽  
Convergence Order ◽  
Convergence Radius ◽  
First Derivative ◽  
Modified Newton Method

Abstract The aim of this paper is to investigate the local convergence of the Modified Newton method, i.e. the classical Newton method in which the first derivative is re-evaluated periodically after m steps. The convergence order is shown to be m + 1. A new algorithm is proposed for the estimation the convergence radius of the method. We propose also a threshold for the number of steps after which is recommended to re-evaluate the first derivative in the Modified Newton method.

scholarly journals Local convergence radius for the Mann-type iteration

2015 ◽  
Vol 53 (2) ◽  
pp. 109-120 ◽  
Author(s):  
Ştefan Măruşter
Keyword(s):  
Newton Method ◽  
Local Convergence ◽  
Numerical Experiments ◽  
Dimensional Space ◽  
Finite Dimensional Space ◽  
Convergence Radius ◽  
Finite Dimensional

Abstract A procedure to estimate the local convergence radius for a Mann-type iteration is given in the setting of a finite dimensional space. In particular we obtain the estimation of radius for classical Newton method. Numerical experiments are presented showing the efficiency of the proposed procedure in comparison with other known methods. In some cases our procedure gives the maximum local convergence radius.

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

Algorithms ◽  
2015 ◽  
Vol 8 (4) ◽  
pp. 1076-1087 ◽  
Author(s):  
Ioannis Argyros ◽  
Ramandeep Behl ◽  
S.S. Motsa
Keyword(s):  
Local Convergence ◽  
Convergence Order ◽  
Order Method ◽  
First Derivative

scholarly journals New Families of Third-Order Iterative Methods for Finding Multiple Roots

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
R. F. Lin ◽  
H. M. Ren ◽  
Z. Šmarda ◽  
Q. B. Wu ◽  
Y. Khan ◽  
...  
Keyword(s):  
Iterative Methods ◽  
Newton Method ◽  
Multiple Roots ◽  
Third Order ◽  
Cubic Convergence ◽  
Numerical Examples ◽  
Mild Conditions ◽  
First Derivative ◽  
Modified Newton Method ◽  
Iteration Methods

Two families of third-order iterative methods for finding multiple roots of nonlinear equations are developed in this paper. Mild conditions are given to assure the cubic convergence of two iteration schemes (I) and (II). The presented families include many third-order methods for finding multiple roots, such as the known Dong's methods and Neta's method. Some new concrete iterative methods are provided. Each member of the two families requires two evaluations of the function and one of its first derivative per iteration. All these methods require the knowledge of the multiplicity. The obtained methods are also compared in their performance with various other iteration methods via numerical examples, and it is observed that these have better performance than the modified Newton method, and demonstrate at least equal performance to iterative methods of the same order.

scholarly journals Modified Newton method to determine multiple zeros of nonlinear equations

2016 ◽  
Vol 11 (10) ◽  
pp. 5774-5780
Author(s):  
Rajinder Thukral
Keyword(s):  
Iterative Method ◽  
Nonlinear Equations ◽  
Newton Method ◽  
New Method ◽  
Convergence Order ◽  
Multiple Zeros ◽  
Numerical Comparisons ◽  
Modified Newton Method ◽  
Iteration Methods ◽  
Better Than

New one-point iterative method for solving nonlinear equations is constructed.  It is proved that the new method has the convergence order of three. Per iteration the new method requires two evaluations of the function.  Kung and Traub conjectured that the multipoint iteration methods, without memory based on n evaluations, could achieve maximum convergence order2n-1  but, the new method produces convergence order of three, which is better than expected maximum convergence order of two.  Hence, we demonstrate that the conjecture fails for a particular set of nonlinear equations. Numerical comparisons are included to demonstrate exceptional convergence speed of the proposed method using only a few function evaluations.

scholarly journals Local Convergence Analysis of an Efficient Fourth Order Weighted-Newton Method under Weak Conditions

2018 ◽  
Vol 56 (1) ◽  
pp. 23-34
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Convergence Analysis ◽  
Newton Method ◽  
Fourth Order ◽  
Local Convergence ◽  
Systems Of Nonlinear Equations ◽  
Order Method ◽  
First Derivative ◽  
Fourth Order Method ◽  
Order Four ◽  
Local Convergence Analysis

Abstract Local convergence analysis of a fourth order method considered by Sharma et. al in [19] for solving systems of nonlinear equations. Using conditions on derivatives upto the order five, they proved that the method is of order four. In this study using conditions only on the first derivative, we prove the convergence of the method in [19]. This way we extended the applicability of the method. Numerical example which do not satisfy earlier conditions but satisfy our conditions are presented in this study.

scholarly journals A New Newton Method with Memory for Solving Nonlinear Equations

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 108 ◽  
Author(s):  
Xiaofeng Wang ◽  
Yuxi Tao
Keyword(s):  
Convergence Rate ◽  
Nonlinear Equations ◽  
Newton Method ◽  
Numerical Experiments ◽  
The Self ◽  
Convergence Order ◽  
Additional Function ◽  
Modified Newton Method ◽  
Invariant Parameter ◽  
With Memory

A new Newton method with memory is proposed by using a variable self-accelerating parameter. Firstly, a modified Newton method without memory with invariant parameter is constructed for solving nonlinear equations. Substituting the invariant parameter of Newton method without memory by a variable self-accelerating parameter, we obtain a novel Newton method with memory. The convergence order of the new Newton method with memory is 1 + 2 . The acceleration of the convergence rate is attained without any additional function evaluations. The main innovation is that the self-accelerating parameter is constructed by a simple way. Numerical experiments show the presented method has faster convergence speed than existing methods.

scholarly journals Local convergence comparison between two novel sixth order methods for solving equations

2019 ◽  
Vol 18 (1) ◽  
pp. 5-19
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Banach Space ◽  
Convergence Analysis ◽  
Local Convergence ◽  
Sixth Order ◽  
Convergence Order ◽  
Convergence Criteria ◽  
Numerical Examples ◽  
First Derivative ◽  
Solving Equations ◽  
Local Convergence Analysis

Abstract The aim of this article is to provide the local convergence analysis of two novel competing sixth convergence order methods for solving equations involving Banach space valued operators. Earlier studies have used hypotheses reaching up to the sixth derivative but only the first derivative appears in these methods. These hypotheses limit the applicability of the methods. That is why we are motivated to present convergence analysis based only on the first derivative. Numerical examples where the convergence criteria are tested are provided. It turns out that in these examples the criteria in the earlier works are not satisfied, so these results cannot be used to solve equations but our results can be used.

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