Multiple roots of systems of equations by repulsion merit functions

In this paper we address the problem of computing multiple roots of a system of nonlinear equations through the global optimization of an appropriate merit function. The search procedure for a global min- imizer of the merit function is carried out by a metaheuristic, known as harmony search, which does not require any derivative information. The multiple roots of the system are sequentially determined along several ite- rations of a single run, where the merit function is accordingly modified by penalty terms that aim to create repulsion areas around previously computed minimizers. A repulsion algorithm based on a multiplicative kind penalty function is proposed. Preliminary numerical experiments with a benchmark set of problems show the effectiveness of the proposed method.Fundação para a Ciência e a Tecnologia (FCT

Remarks on Solving Methods of Nonlinear Equations

Abstract: In the field of mechanical engineering, many practical problems can be converted into nonlinear problems, such as the meshing problem of mechanical transmission. So the solution of nonlinear equations has important theoretical research and practical application significance. Whether the traditional Newton iteration method or the intelligent optimization algorithm after the popularization of computers, both them have been greatly enriched and developed through the continuous in-depth research of scholars at home and abroad, and a series of improved algorithms have emerged. This paper mainly reviews the research status of solving nonlinear equations from two aspects of traditional iterative method and intelligent optimization algorithm, systematically reviews the research achievements of domestic and foreign scholars, and puts forward prospects for future research directions