The Newton Iteration on Lie Groups

We define the Newton iteration for solving the equation f(y) = 0, where f is a map from a Lie group to its corresponding Lie algebra. Two versions are presented, which are formulated independently of any metric on the Lie group. Both formulations reduce to the standard method in the Euclidean case, and are related to existing algorithms on certain Riemannian manifolds. In particular, we show that, under classical assumptions on f, the proposed method converges quadratically. We illustrate the techniques by solving a fixed-point problem arising from the numerical integration of a Lie-type initial value problem via implicit Euler.