Solving Systems of Nonlinear Equations Using a Globally Convergent Optimization Algorithm

Solving systems of nonlinear equations is a relatively complicated problem for which a number of different approaches have been presented. In this paper, a new algorithm is proposed for the solutions of systems of nonlinear equations. This algorithm uses a combination of the gradient and the Newton’s methods. A novel dynamic combinatory is developed to determine the contribution of the methods in the combination. Also, by using some parameters in the proposed algorithm, this contribution is adjusted. We use the gradient method due to its global convergence property, and the Newton’s method to speed up the convergence rate. We consider two different combinations. In the first one, a step length is determined only along the gradient direction. The second one is finding a step length along both the gradient and the Newton’s directions. The performance of the proposed algorithm in comparison to the Newton’s method, the gradient method and an existing combination method is explored on several well known test problems in solving systems of nonlinear equations. The numerical results provide evidence that the proposed combination algorithm is generally more robust and efficient than other mentioned methods on some important and difficult problems.