Kantorovich's theorems for Newton's method for mappings and optimization problems on Lie groups

With the classical assumptions on f, a convergence criterion of Newton's method (independent of affine connections) to find zeros of a mapping f from a Lie group to its Lie algebra is established, and estimates of the convergence domains of Newton's method are obtained, which improve the corresponding results in Owren & Welfert (2000, BIT Numer. Math., 40, 121–145) and Wang & Li (2006, J. Zhejiang Univ. Sci. A, 8, 978–986). Applications to optimization are provided and the results due to Mahony (1996, Linear Algebra Appl., 248, 67–89) are extended and improved accordingly.