algorithm newton-raphson for finding roots of ( inputan initial guess and stopping error outputthe approximate root of ( while ( ) and budget is not exhausted do solve the linear system (xd ( - while ( ) ( - ak ( ) do / xx+ad return we can adapt root-finding newton-like method in order to minimize differentiable function rn we simply try to locate zero of the gradient of when is functionthe function rn rn is continuousand so the root of leads to the search direction dt - - ( (xt )where ht is the hessian matrix at xt (the matrix of jacobi of the gradient is the hessianwhen the learning rate at is equal to the update xt - (xt can alternatively be derived by assuming that (xis approximately quadratic and convex in the neighborhood of xt that is (xf (xt ( xt ) (xt ( xt )ht ( xt ) ( and then minimizing the right-hand side of ( with respect to the following algorithm uses an armijo inexact line search for minimization and guards against the possibility that the hessian may not be positive definite (that isits cholesky decomposition does not existalgorithm newton-raphson for minimizing (xinputan initial guess xstopping error line search parameter ( outputan approximate minimizer of ( in (the identity matrix while ( ) and budget is not exhausted do compute the hessian at if then /cholesky is successful update to be the cholesky factor satisfying llh else do not update the lower triangular - - ( (computed by forward substitutiond - (computed by backward substitutiona- while ( adf (xa - ( ) do -axx xx+ad return