Newton-type methods for solving nonlinear equations on quadratic matrix groups

In this paper we consider numerical methods for solving nonlinear equations on matrix Lie groups. Recently Owren and Welfert (Technical Report Numerics, No 3/1996, Norwegian University of Science and Technology, Trondheim, Norway, 1996) have proposed a method where the original nonlinear equation F(Y) = 0 is transformed into a nonlinear equation on the Lie algebra of the group, thus Newton-type methods may be applied which require the evaluation of exponentials of matrices. Here the previous transformation will be performed by the Cayley approximant of the exponential map. This approach has the advantage that no exponentials of matrices are needed. The numerical tests reported in the last section seem to show that our approach is less expensive and provides a larger convergence region than the method of Owren and Welfert.