Inexact Newton methods on a vector supercomputer

We consider modifications of Newtonʹs method for solving a nonlinear system F(x) = 0 where F:Rn→Rn. Our modifications allow the repeated use of the same Jacobian F′(x) and they require the linear Newtonʹs equation F′(x)Δx = −F(x) to be solved only approximately. The use of these methods is particularly appropriate when an exact solution of Newtonʹs equation is difficult to obtain and/or when evaluating and preparing the Jacobian for the computation is costly. We establish the local convergence of our methods under rather simple and natural conditions on the exactness of the approximate solutions to Newtonʹs equation and prove theoretical results on the convergence order. We then give numerical examples on a vector supercomputer which show that these methods may perform serval times faster than the standard Newton method.