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The Newton iteration on Lie groups. (English) Zbl 0957.65054
The authors are concerned with the numerical solution of a nonlinear equation on a manifold. They present two versions of Newton’s iterative method for solving \(f(y)= 0\), where \(f\) maps from a Lie group into its corresponding Lie algebra. Both versions reduce to the standard method in Euclidean coordinates. Local quadratic convergence is proved under suitable assumptions on \(f\).
The investigations presented has been mainly motivated by the use of implicit methods (such as the backward Euler method) for solving initial-value problems for ordinary differential equations on manifolds. The numerical example presented at the end of the paper comes from that field. Finally, some possible extension (e.g. the use of higher-order implicit methods for the time integration) and open problems are discussed.

MSC:
65J15 Numerical solutions to equations with nonlinear operators
22E30 Analysis on real and complex Lie groups
34C40 Ordinary differential equations and systems on manifolds
65L05 Numerical methods for initial value problems involving ordinary differential equations
47J25 Iterative procedures involving nonlinear operators
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