New third-order method for solving systems of nonlinear equations

2008 ◽  
Vol 50 (3) ◽  
pp. 271-282 ◽  
Author(s):  
Wang Haijun
Keyword(s):  
Nonlinear Equations ◽  
Systems Of Nonlinear Equations ◽  
Order Method ◽  
Third Order

scholarly journals Modified Potra-Pták Multi-step Schemes with Accelerated Order of Convergence for Solving Systems of Nonlinear Equations

2018 ◽  
Vol 24 (1) ◽  
pp. 3
Author(s):  
Himani Arora ◽  
Juan Torregrosa ◽  
Alicia Cordero
Keyword(s):  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Sixth Order ◽  
Systems Of Nonlinear Equations ◽  
Order Method ◽  
Third Order ◽  
Boundary Problems ◽  
Newton Step ◽  
The Family ◽  
Efficiency And Reliability

In this study, an iterative scheme of sixth order of convergence for solving systems of nonlinear equations is presented. The scheme is composed of three steps, of which the first two steps are that of third order Potra-Pták method and last is weighted-Newton step. Furthermore, we generalize our work to derive a family of multi-step iterative methods with order of convergence 3 r + 6 , r = 0 , 1 , 2 , … . The sixth order method is the special case of this multi-step scheme for r = 0 . The family gives a four-step ninth order method for r = 1 . As much higher order methods are not used in practice, so we study sixth and ninth order methods in detail. Numerical examples are included to confirm theoretical results and to compare the methods with some existing ones. Different numerical tests, containing academical functions and systems resulting from the discretization of boundary problems, are introduced to show the efficiency and reliability of the proposed methods.

scholarly journals On Some Efficient Techniques for Solving Systems of Nonlinear Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Janak Raj Sharma ◽  
Puneet Gupta
Keyword(s):  
Nonlinear Equations ◽  
Computational Efficiency ◽  
Newton Iteration ◽  
Systems Of Nonlinear Equations ◽  
Order Method ◽  
Third Order ◽  
Newton Step ◽  
The Third ◽  
Large Systems ◽  
Theoretical Results

We present iterative methods of convergence order three, five, and six for solving systems of nonlinear equations. Third-order method is composed of two steps, namely, Newton iteration as the first step and weighted-Newton iteration as the second step. Fifth and sixth-order methods are composed of three steps of which the first two steps are same as that of the third-order method whereas the third is again a weighted-Newton step. Computational efficiency in its general form is discussed and a comparison between the efficiencies of proposed techniques with existing ones is made. The performance is tested through numerical examples. Moreover, theoretical results concerning order of convergence and computational efficiency are verified in the examples. It is shown that the present methods have an edge over similar existing methods, particularly when applied to large systems of equations.

scholarly journals Third-Order Newton-Type Methods Combined with Vector Extrapolation for Solving Nonlinear Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Wen Zhou ◽  
Jisheng Kou
Keyword(s):  
Nonlinear Systems ◽  
Nonlinear Equations ◽  
Semilocal Convergence ◽  
Numerical Results ◽  
The Other ◽  
Systems Of Nonlinear Equations ◽  
Order Method ◽  
Third Order ◽  
Composite Method ◽  
Vector Extrapolation

We present a third-order method for solving the systems of nonlinear equations. This method is a Newton-type scheme with the vector extrapolation. We establish the local and semilocal convergence of this method. Numerical results show that the composite method is more robust and efficient than a number of Newton-type methods with the other vector extrapolations.

A Class of Higher-Order Newton-Like Methods for Systems of Nonlinear Equations

2021 ◽  
pp. 2150059
Author(s):  
Janak Raj Sharma ◽  
Sunil Kumar ◽  
Ioannis K. Argyros
Keyword(s):  
Nonlinear Equations ◽  
Inverse Operator ◽  
General Setting ◽  
Basins Of Attraction ◽  
Function Evaluation ◽  
Systems Of Nonlinear Equations ◽  
Third Order ◽  
Additional Function ◽  
Numerical Experimentation ◽  
The Cost

In this paper, a class of efficient iterative methods with increasing order of convergence for solving systems of nonlinear equations is developed and analyzed. The methodology uses well-known third-order Potra–Pták iteration in the first step and Newton-like iterations in the subsequent steps. Novelty of the methods is the increase in convergence order by an amount three per step at the cost of only one additional function evaluation. In addition, the algorithm uses a single inverse operator in each iteration, which makes it computationally more efficient and attractive. Local convergence is studied in the more general setting of a Banach space under suitable assumptions. Theoretical results of convergence and computational efficiency are verified through numerical experimentation. Comparison of numerical results indicates that the developed algorithms outperform the other similar algorithms available in the literature, particularly when applied to solve the large systems of equations. The basins of attraction of some of the existing methods along with the proposed method are given to exhibit their performance.

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