Improved Newton iteration for nonlinear matrix equations on quadratic Lie groups

In this paper we consider Newton iteration methods for solving nonlinear equations on matrix Lie groups. Recently, Owren and Welfert have proposed a method where the original nonlinear equation is transformed into a nonlinear equation on the Lie algebra of the group via the exponential map, thus Newton iteration methods may be applied. Based on this we suggest two improved variants of Newton iteration algorithm. One is that the exponential map would be approximated by Cayley map and give a Cayley version Newton iteration method for solving nonlinear equations on quadratic Lie groups, then we show that, the proposed method converges quadratically; Another is a variant of Newton type method with accelerated convergence and the numerical tests reported seem to support that it converges with cubically.