Methods Of Solving Nonlinear Equations

The solving of Nonlinear algebraic equation is a basic and important problem, because there are a lot of practical problems in engineering practice, economics, information security, and dynamics which are eventually converted to algebraic equations. The method of solving the nonlinear equations has long been important research in engineering applications and numerical calculation, The classical methods for solving nonlinear equations, including the Newton method, quasi-Newton method, gradient method, and other improvements, These classical methods in the solution process are faster local convergence, but the shortcomings are very obvious. For example:the selection of initial point must be very close to the exact solution, or will lead to the solution does not converge. However, in terms of theory and practical applications, the solution of nonlinear equations interval or convergence interval is difficult to determine, which makes the selecting of appropriate initial root is almost impossible, In addition, these methods in the solution process often requires the derivative of the equation, the derivative of nonlinear equations is not easy to obtain. So.The traditional method is difficult in the practical.In recent years, many bionic optimization algorithm have been used in the modern field, such as particle swarm optimization, genetic algorithms, simulated annealing, neural networks, fish school algorithm, etc., they all have strong global search ability of the optimal solution. Kennedy and Eberhart propose the Particle Swarm Optimization (PSO) algorithm which is inspired by artificial life research results by simulating birds foraging in the process of migration and clustering behavior of a random search algorithm based on swarm intelligence, the algorithm has manys advantage,such as warm intelligence, inherent parallelism, the simple iterative scheme,and the fast convergence to the optimal solution, but due to the lack of sophisticated search capabilities of the local area, the algorithm later in the search there will be convergence stagnation.The author of this paper summarizes the advantages and disadvantages of Newton method, quasi-Newton method and particle swarm optimization, Finally, the quasi-Newton-Particle Swarm (QNPSO)has been proposed which combines the strengths of both Newton’s method and particle swarm optimization, First, particle swarm optimization is used to large-scale searching that provide a good beginning for the quasi-Newton method,then the refine searching to find the accurate root of the equation. Not only has the global convergence of the particle swarm algorithm and a faster convergence rate, the solution of nonlinear equations can be obtained with greater probability, but also can overcome the slower convergence and even premature "phenomenon". The numerical results show that quasi-Newton-hybrid algorithm has the better ability to converges to the global optimal solution and the convergence rate is higher than the simple particle swarm optimization significantly which compared with the simple particle swarm optimization. During the searching which is controlled by the same number of iterations, the convergence rate has improved greatly. Numerical experiments show that this algorithm is stability, convergence and high efficiency.