On the global convergence of path-following methods to determine all solutions to a system of nonlinear equations

1987 ◽  
Vol 39 (2) ◽  
pp. 181-188 ◽  
Author(s):  
Immo Diener
Keyword(s):  
Global Convergence ◽  
Nonlinear Equations ◽  
Path Following ◽  
System Of Nonlinear Equations ◽  
All Solutions ◽  
Path Following Methods

scholarly journals An Improved Line Search Filter Method for the System of Nonlinear Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Zhong Jin ◽  
Yuqing Wang
Keyword(s):  
Global Convergence ◽  
Objective Function ◽  
Nonlinear Equations ◽  
Numerical Experiments ◽  
Line Search ◽  
Equality Constraints ◽  
Filter Method ◽  
System Of Nonlinear Equations ◽  
Search Filter

An improved line search filter algorithm for the system of nonlinear equations is presented. We divide the equations into two groups, one contains the equations that are treated as equality constraints and the square of other equations is regarded as objective function. Two groups of equations are updated at every iteration in the works by Nie (2004, 2006, and 2006), by Nie et al. (2008), and by Gu (2011), while we just update them at the iterations when it is needed indeed. As a consequence, the scale of the calculation is decreased in a certain degree. Under some suitable conditions the global convergence can be induced. In the end, numerical experiments show that the method in this paper is effective.

scholarly journals Solution of nonlinear equations and computation of multiple solutions of a simple reaction-diffusion equation

2000 ◽  
Vol 42 (1) ◽  
pp. 55-64
Author(s):  
Adrian Swift ◽  
Easwaran Balakrishnan
Keyword(s):  
Steady State ◽  
Diffusion Equation ◽  
Nonlinear Equations ◽  
Multiple Solutions ◽  
Path Following ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Research Involvement ◽  
Path Following Methods ◽  
Nonlinear Equation Systems

AbstractThe first part of this paper starts with a brief discussion of some methods for solution of nonlinear equations which have interested the first author over the last twenty years or so. In the second part we discuss a recent research involvement, the success of which relies heavily on the numerical solution of nonlinear equation systems. We briefly describe path-following methods and then present an application to a simple steady-state reaction-diffusion equation arising in combustion theory. Results for some regular geometric shapes are shown and compared with those from an approximate method.

scholarly journals An Adaptive Nonmonotone Line Search Technique for Solving Systems of Nonlinear Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Masoud Hatamian ◽  
Mahmoud Paripour ◽  
Farajollah Mohammadi Yaghoobi ◽  
Nasrin Karamikabir
Keyword(s):  
Global Convergence ◽  
Nonlinear Equations ◽  
Line Search ◽  
Nonmonotone Line Search ◽  
Systems Of Nonlinear Equations ◽  
System Of Nonlinear Equations ◽  
Search Technique ◽  
Numerical Examples ◽  
Line Search Technique ◽  
Novel Algorithm

In this article, a new nonmonotone line search technique is proposed for solving a system of nonlinear equations. We attempt to answer this question how to control the degree of the nonmonotonicity of line search rules in order to reach a more efficient algorithm? Therefore, we present a novel algorithm that can avoid the increase of unsuccessful iterations. For this purpose, we show the robust behavior of the proposed algorithm by solving a few numerical examples. Under some suitable assumptions, the global convergence of our strategy is proved.

Finding All Solutions of Systems of Nonlinear Equations Using Spiral Dynamics Inspired Optimization with Clustering

2015 ◽  
Vol 19 (5) ◽  
pp. 697-707 ◽  
Author(s):  
Kuntjoro Adji Sidarto ◽  
◽  
Adhe Kania
Keyword(s):  
Nonlinear Equations ◽  
Systems Of Nonlinear Equations ◽  
System Of Nonlinear Equations ◽  
Advantages And Disadvantages ◽  
Root Finding ◽  
Spiral Dynamics ◽  
All Solutions ◽  
Computational Sciences ◽  
System Equations ◽  
Novel Method

Nowadays the root finding problem for nonlinear system equations is still one of the difficult problems in computational sciences. Many attempts using deterministic and meta-heuristic methods have been done with their advantages and disadvantages, but many of them have fail to converge to all possible roots. In this paper, a novel method of locating and finding all of the real roots from the system of nonlinear equations is proposed mainly using the spiral dynamics inspired optimization by Tamura and Yasuda [1]. The method is improved by the usage of the Sobol sequence of points for generating initial candidates of roots which are uniformly distributed than of pseudo-random generated points. Using clustering technique, the method localizes all potential roots so the optimization is conducted in those points simultaneously. A set of problems as the benchmarks from the literature is given. Having only a single run for each problem, the proposed method has successfully found all possible roots within a bounded domain.

Solving System of Nonlinear Equations using Genetic Algorithm

2019 ◽  
Vol 10 (4) ◽  
pp. 877-886 ◽  
Author(s):  
Chhavi Mangla ◽  
Musheer Ahmad ◽  
Moin Uddin
Keyword(s):  
Genetic Algorithm ◽  
Nonlinear Equations ◽  
System Of Nonlinear Equations

New fourth-order convergent algorithm for analysis of trusses with material and geometric nonlinearities

2021 ◽  
pp. 030932472110005
Author(s):  
Luiz Antonio Farani de Souza ◽  
Douglas Fernandes dos Santos ◽  
Rodrigo Yukio Mizote Kawamoto ◽  
Leandro Vanalli
Keyword(s):  
Fourth Order ◽  
Nonlinear Behavior ◽  
Path Following ◽  
System Of Nonlinear Equations ◽  
The Finite Element Method ◽  
Step Method ◽  
Geometric Nonlinearities ◽  
Standard Procedures ◽  
Elastoplastic Constitutive Model ◽  
Convergent Algorithm

This paper presents a new algorithm to solve the system of nonlinear equations that describes the static equilibrium of trusses with material and geometric nonlinearities, adapting a three-step method with fourth-order convergence found in the literature. The co-rotational formulation of the Finite Element Method is used in the discretization of structures. The nonlinear behavior of the material is characterized by an elastoplastic constitutive model. The equilibrium paths with limit points of load and displacement are obtained using the linearized Arc-Length path-following technique. The numerical results obtained with the free program Scilab show that the new algorithm converges faster than standard procedures and modified Newton-Raphson, since the approximate solution of the problem is obtained with a smaller number of accumulated iterations and less CPU time. The equilibrium paths show that the structures exhibit a completely different behavior when the material nonlinearity is considered in the analysis with large displacements.

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