For Solving Nonlinear The Pathological Equations Of A Modified Newton Method And Its Applications

It is well-known that when solving a system of nonlinear equations, the damped Newton method is globally and superlinearly/quadratically convergent. However, when it is applied to solve an ill-conditioned problem, the method may meet some numerical difficulty. This is caused by the large condition number of the Jacobian matrix. In the case where the condition number of the Jacobian matrix is very large, the Newton direction is likely to be orthogonal to the gradient of the general level function. As a consequent, the steplength generated by the line search might be very small, which affects the effect of the method. A practical way to overcome this difficulty is to use the so-called "natural level function" as the merit function instead of the general level function. Numerical experiments shows that this technique works well when the method is applied to solve many practical problems. However, so far, there is no theory to show the global convergence of the method.In this thesis, we make a minor modification to the line search process and propose a Newton method with the modified line searches. We show that the proposed method is globally and superlinearly/quadratically convergent for solving smooth nonlinear equations. We also extend the technique to a Newton method for solving a nonsmooth equation reformulation of the nonlinear complementarity problem. Under appropriate conditions, we obtain the global and superlinear/qudratic convergence of the method. The reported numerical results show that the proposed method performs well for the test problems.