Local convergence of Newton’s method on the Heisenberg group

In the present paper, we study Newton’s method on the Heisenberg group for solving the equation f(x)=0f(x)=0, where ff 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 ff. Some applications to special cases including Kantorovich’s condition and γγ-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.