A Modefied Newton's Method

We propose a new modified Cholesky algorithm based on a symmetric indefinite factorization by using a new pivoting strategy。This algorithm computes a factorization of a only symmetric matrix , and keeps the negative eigenvalues and the nonsingularity of block diagonal matrix in the factorization such that the norm of the perturbation matrix can be small. Then we apply the modified Cholesky algorithm to Newton′s method, and get a modified Newton′s method for solving unconstrained optimization problems. When the Hessian matrix is indefinite,we consider negative curvature direction as search direction; when the Hessian matrix is positive definite, we consider Newton′s direction as search direction. Numerical experiments show that the modified Newton′s method has better practical performance.