The Research Based On Particle Swarm Optimization For Solving Systems Of Nonlinear Equations

Solving nonlinear equations problem is one of the most basic problems in modern mathematics. A long period of time, many scholars are done a lot of research in the theory of nonlinear equations and numerical calculations, but solving nonlinear equations is still a problem. Particularly for high nonlinear equations derived from practical engineering problems, the efficient and reliable algorithms are always absent. Traditional numerical algorithms have fast convergence property. In general, these algorithms have local convergence, that is to say, their convergence is related to the initial point, so they often fail to solve some non-linear equations because of the local convergence. The validity of algorithms is low. In recent years, people try to introduce some algorithm with large scale convergence to solve non-linear equations, and have some new breakthroughs.Particle Swarm Optimization (PSO) is based on swarm intelligence theory. The algorithm can provide efficient solutions for optimization problems through intelligence generated from complex activities such as cooperation and competition among individuals in the biologic colony. PSO has been widely applied in function optimization and shows great potential in practice. As an effective global optimization method, compared with the conventional optimization method, PSO is simple in concept, few in parameters and easy in implementation. So the algorithm is used to solve complex, large-scale, non-linear, non-differentiable optimization problems.This paper firstly introduces the traditional method of solving nonlinear equations and the basic concepts of particle swarm optimization, and explains the major work.Secondly, we change the problem of solving nonlinear equations into the problem of function optimization; then we choose the PSO algorithm to solve optimization problem. In the process of solving nonlinear equations, the equations may exist multiple solutions in a certain region. So, in order to obtain all the real solution of the equations, the paper uses function"stretching"technique in PSO, which can obtain all the real solution.Thirdly, this paper analyzes the characteristics of the optimization problem of solving equations, which can be seen as the anti-optimization problem. Therefore we propose the PSO as object, fitness as controlled variable and inertia weight as a control variable. And then a closed-loop control system is constituted. The traditional controller is introduced in the system. Modified algorithm can improve the convergence speed and global search capability. Fourthly, the PSO algorithm itself is nonlinear, random, and the traditional controller is linear law. So we use single neuron controller instead of the traditional controller. Meanwhile we chose typical equations. Compared with the former method, the results showed that the adaptive control PSO algorithm has the stronger capacity of converging to the global optimum and the faster speed of convergence. In the same number of iterations of the search process, the success rate is high.Fifthly, in the chapter, we analysis the features of algorithm in search process, and find that premature convergence always is closely related with the convergence of individual stocks, the rapid decline of diversity. Thus in order to avoid premature convergence, we hope that the algorithm in the search process can maintain an appropriate diversity. The chapter uses the information of fitness to adjust the diversity. The algorithm not only can improve the diversity of the swarm, but also adjust the capability of exploitation and exploration.