Newton's method has proved to be a very efficient method for solving strictly convex unconstrained minimization problems. For the nonconvex case, various modified Newton methods have been proposed. In this paper, a new modified Newton method is presented. The method is a linesearch method, utilizing the Cholesky factorization of a positive-definite portion of the Hessian matrix. The search direction is defined as a linear combination of a descent direction and a direction of negative curvature. Theoretical properties of the method are established and its behaviour is studied when applied to a set of test problems. 27 refs., 6 figs., 4 tabs.