Local convergence of Newton’s method on the Heisenberg group

摘要

In the present paper, we study Newton’s method on the Heisenberg group for solving the equation f(x)=0, where f is a mapping from Heisenberg group to its Lie algebra. Under certain generalized Lipschitz condition, we obtain the convergence radius of Newton’s method and the estimation of the uniqueness ball of the zero point of f. Some applications to special cases including Kantorovich’s condition and γ-condition are provided. The determination of an approximate zero point of an analytic mapping is also presented. Concrete examples are given to illustrate applications of our results.