An initialization strategy for improving newton's method

Among the methods for solving a system of nonlinear equations, Newton's method is distinguished because of its significant advantage of converging quadratically. For this, a nonsingular Jacobian matrix and a good starting point are required, but they can rarely be available especially in application problems. On the other hand, Dimension Reducing method also of quadratic convergence works well even from initial points far away from the solution as well as in cases of singular or ill-conditioned Jacobian matrix. In this paper, DR and Newton methods are properly incorporated into a new algorithm of quadratic convergence to contribute to the important issue of initializing Newton's method. The quadratic convergence of the proposed method is proven and the numerical results on tested problems are promising.