scholarly journals Higher-Order Iteration Schemes for Solving Nonlinear Systems of Equations

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 937 ◽  
Author(s):  
Hessah Faihan Alqahtani ◽  
Ramandeep Behl ◽  
Munish Kansal
Keyword(s):  
Nonlinear Systems ◽  
Convergence Analysis ◽  
Fourth Order ◽  
Numerical Experiments ◽  
Multidimensional Case ◽  
Systems Of Nonlinear Equations ◽  
Systems Of Equations ◽  
Nonlinear Systems Of Equations ◽  
Validity And Reliability ◽  
Mild Conditions

We present a three-step family of iterative methods to solve systems of nonlinear equations. This family is a generalization of the well-known fourth-order King’s family to the multidimensional case. The convergence analysis of the methods is provided under mild conditions. The analytical discussion of the work is upheld by performing numerical experiments on some application oriented problems. Finally, numerical results demonstrate the validity and reliability of the suggested methods.

scholarly journals On a Bi-Parametric Family of Fourth Order Composite Newton–Jarratt Methods for Nonlinear Systems

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 492 ◽  
Author(s):  
Janak Raj Sharma ◽  
Deepak Kumar ◽  
Ioannis K. Argyros ◽  
Ángel Alberto Magreñán
Keyword(s):  
Nonlinear Systems ◽  
Boundary Value Problem ◽  
Fourth Order ◽  
Order Of Convergence ◽  
Systems Of Nonlinear Equations ◽  
Systems Of Equations ◽  
Nonlinear Systems Of Equations ◽  
Numerical Experimentation ◽  
Theoretical Results ◽  
Two Parameter

We present a new two-parameter family of fourth-order iterative methods for solving systems of nonlinear equations. The scheme is composed of two Newton–Jarratt steps and requires the evaluation of one function and two first derivatives in each iteration. Convergence including the order of convergence, the radius of convergence, and error bounds is presented. Theoretical results are verified through numerical experimentation. Stability of the proposed class is analyzed and presented by means of using new dynamics tool, namely, the convergence plane. Performance is exhibited by implementing the methods on nonlinear systems of equations, including those resulting from the discretization of the boundary value problem. In addition, numerical comparisons are made with the existing techniques of the same order. Results show the better performance of the proposed techniques than the existing ones.

scholarly journals A nonmonotone inexact Newton algorithm for nonlinear systems of equations

1995 ◽  
Vol 36 (4) ◽  
pp. 460-492 ◽  
Author(s):  
Yi Xiao ◽  
Eric King-Wah Chu
Keyword(s):  
Hopf Bifurcation ◽  
Nonlinear Systems ◽  
Numerical Experiments ◽  
Standard Test ◽  
Test Problems ◽  
Systems Of Equations ◽  
Nonlinear Systems Of Equations ◽  
Special Cases ◽  
Nonmonotone Techniques ◽  
Newton's Methods

AbstractIn this paper, an inexact Newton's method for nonlinear systems of equations is proposed. The method applies nonmonotone techniques and Newton's as well as inexact Newton's methods can be viewed as special cases of this new method. The method converges globally and quadratically. Some numerical experiments are reported for both standard test problems and an application in the computation of Hopf bifurcation points.

scholarly journals Generalized High-Order Classes for Solving Nonlinear Systems and Their Applications

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1194 ◽  
Author(s):  
Francisco I. Chicharro ◽  
Alicia Cordero ◽  
Neus Garrido ◽  
Juan R. Torregrosa
Keyword(s):  
Nonlinear Systems ◽  
Iterative Method ◽  
Fourth Order ◽  
Order Of Convergence ◽  
Divided Difference ◽  
High Order ◽  
Weight Functions ◽  
Systems Of Equations ◽  
Nonlinear Systems Of Equations ◽  
New Methods

A generalized high-order class for approximating the solution of nonlinear systems of equations is introduced. First, from a fourth-order iterative family for solving nonlinear equations, we propose an extension to nonlinear systems of equations holding the same order of convergence but replacing the Jacobian by a divided difference in the weight functions for systems. The proposed GH family of methods is designed from this fourth-order family using both the composition and the weight functions technique. The resulting family has order of convergence 9. The performance of a particular iterative method of both families is analyzed for solving different test systems and also for the Fisher’s problem, showing the good performance of the new methods.

scholarly journals ON A CLASS OF EFFICIENT HIGHER ORDER NEWTON-LIKE METHODS

2019 ◽  
Vol 24 (1) ◽  
pp. 105-126 ◽  
Author(s):  
Janak Raj Sharma ◽  
Deepak Kumar
Keyword(s):  
Nonlinear Systems ◽  
Local Convergence ◽  
Convergence Order ◽  
Systems Of Nonlinear Equations ◽  
Systems Of Equations ◽  
Nonlinear Systems Of Equations ◽  
Uniqueness Results ◽  
Numerical Experimentation ◽  
Local Convergence Analysis ◽  
Theoretical Results

Based on a two-step Newton-like scheme, we propose a three-step scheme of convergence order p+2 (p >=3) for solving systems of nonlinear equations. Furthermore, on the basis of this scheme a generalized k+2-step scheme with increasing convergence order p+2k is presented. Local convergence analysis including radius of convergence and uniqueness results of the methods is presented. Computational efficiency in the general form is discussed. Theoretical results are verified through numerical experimentation. Finally, the performance is demonstrated by the application of the methods on some nonlinear systems of equations.

scholarly journals Advanced Interior Optimization Methods with Electronics Applications

2021 ◽  
pp. 97-110
Author(s):  
Francisco Casesnoves
Keyword(s):  
Nonlinear Systems ◽  
Numerical Optimization ◽  
Global Minimum ◽  
Optimization Methods ◽  
Optimization Method ◽  
Systems Of Nonlinear Equations ◽  
Systems Of Equations ◽  
Nonlinear Systems Of Equations ◽  
Further Development

<p>In a previous contribution, the mathematical-computational base of Interior Optimization Method was demonstrated. Electronics applications were performed with numerical optimization data and graphical proofs. In this evoluted-improved paper a series of electronics applications of Interior Optimization in superconductors BCS algorithms/theory are shown. In addition, mathematical developments of Interior Optimization Methods related to systems of Nonlinear Equations are proven. The nonlinear multiobjective optimization problem constitutes a difficult task to find/determine a global minimum, approximated-global minimum, or a convenient local minimum whith/without constraints. Nonlinear systems of equations principles set the base in the previous article for further development of Interior Optimization and Interior-Graphical Optimization [Casesnoves, 2016-7]. From Graphical Optimization 3D optimization stages [Casesnoves, 2016-7], the demonstration that solution of nonlinear systems of equations is not unique in general emerges. Software-engineering and computational simulations are shown with electronics superconductors [several elements, Type 1 superconductors] and electronics physics applications. Extensions to similar applications for materials-tribology models and Biomedical Tribology are explained.</p>

scholarly journals Fast intersection methods for the solution of some nonlinear systems of equations

2004 ◽  
Vol 2004 (2) ◽  
pp. 127-136
Author(s):  
Bernard Beauzamy
Keyword(s):  
Nonlinear Systems ◽  
Euclidean Space ◽  
Nonlinear Equations ◽  
Systems Of Nonlinear Equations ◽  
Fast Method ◽  
Systems Of Equations ◽  
Nonlinear Systems Of Equations ◽  
Nonlinear Operators

We give a fast method to solve numerically some systems of nonlinear equations. This method applies basically to all systems which can be put in the formU∘V(X)=Y, whereUandVare two possibly nonlinear operators. It uses a modification of Newton's algorithm, in the sense that one projects alternatively onto two subsets. But, here, these subsets are not subspaces any more, but manifolds in a Euclidean space.

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