On the local convergence of \(m\)-step Newton methods and \(j\)-Frechet differentiable operators with applications on a vector supercomputer (Q2782916)

For solving nonlinear operator equations \(F(x)= 0\) in a Banach space setting, the \(m\)-step Newton method is studied. This modification of Newton's method uses several times the time evaluation of the Fréchet-derivative \(F'\).NEWLINENEWLINENEWLINEThe author proves sufficient conditions for local convergence that are based upon higher-order Fréchet-derivatives of \(F\). The starting value for the iteration can then be taken from a larger set as it is the case when only the first Fréchet-derivative fulfills some Lipschitz condition.NEWLINENEWLINENEWLINEThe only example given (\(F(x)= e^x- \alpha\); \(\alpha> 0\)) is rather simple and can also be found in a few other papers of the author [cf. e.g. Comput. Math. Appl. 39, No. 1-2, 69-75 (2000; Zbl 0976.65054)].NEWLINENEWLINENEWLINEThe applications on a vector supercomputer announced in the title cannot be found in the paper itself.