instead of : This is often done to ensure that the Wolfe conditions, or much simpler and efficient Armijo's condition, are satisfied at each step of the method. For step sizes other than 1, the method is often referred to as the relaxed or damped Newton's method. Convergence If is a strongly convex function with Lipschitz Hessian, then provided that is close enough to , the sequence generated by Newton's method will converge to the (necessarily unique) minimizer of quadratically fast. That is, Computing the Newton direction Finding the inverse of the Hessian in high dimensions to compute the Newton direction can be an expensive operation. In such cases, instead of directly inverting the Hessian, it is better to calculate the vector as the solution to the system of linear equations which may be solved by various factorizations or approximately (but to great accuracy) using iterative methods. Many of these methods are only applicable to certain types of equations, for example the Cholesky factorization and conjugate gradient will only work if is a positive definite matrix. While this may seem like a limitation, it is often a useful indicator of something gone wrong; for example if a minimization problem is being approached and is not positive definite, then the iterations are converging to a saddle point and not a minimum. On the other hand, if a constrained optimization is done (for example, with Lagrange multipliers), the problem may become one of saddle point finding, in which case the Hessian will be symmetric indefinite and the solution of will need to be done with a method that will work for such, such as the variant of Cholesky factorization or the conjugate residual method. There also exist various quasi-Newton methods, where an approximation for the Hessian (or its inverse directly) is built up from changes in the gradient. If the Hessian is close to a non-invertible matrix, the inverted Hessian can be numerically unstable and the solution may diverge. In this case, certain workarounds have been tried in the past, which have varied success with certain problems. One can, for example, modify the Hessian by adding a correction matrix so as to make positive definite. One approach is to diagonalize the Hessian and choose so that has the same eigenvectors as the Hessian, but with each negative eigenvalue replaced by . An approach exploited in the Levenberg–Marquardt algorithm (which uses an approximate