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Kantorovich’s theorems for Newton’s method for mappings and optimization problems on Lie groups. (English) Zbl 1215.65106
With the standard conditions on a mapping, the authors develop convergence criteria for Newton’s method in a Lie environment. This work extends the previous results of B. Owren and B. Welfert [BIT 40, No. 1, 121–145 (2000; Zbl 0957.65054)] and J.-H. Wang and C. Li [J. Zhejiang Univ. Sci. A, 8, 978–986 (2006)]. The main results develop a L-Lipschitz condition under which a sequence generated by Newton’s method is both well-defined and convergent. These results are independent of the metrics defined on the groups. Furthermore, they provide a number of illustrations indicating the advantage of their technique over earlier works.
65J15Equations with nonlinear operators (numerical methods)
22E30Analysis on real and complex Lie groups