Journal article

A framework for generalising the Newton method and other iterative methods from Euclidean space to manifolds

JH Manton

Numerische Mathematik | Springer Berlin Heidelberg | Published : 2015


The Newton iteration is a popular method for minimising a cost function on Euclidean space. Various generalisations to cost functions defined on manifolds appear in the literature. In each case, the convergence rate of the generalised Newton iteration needed establishing from first principles. The present paper presents a framework for generalising iterative methods from Euclidean space to manifolds that ensures local convergence rates are preserved. It applies to any (memoryless) iterative method computing a coordinate independent property of a function (such as a zero or a local minimum). All possible Newton methods on manifolds are believed to come under this framework. Changes of coordin..

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Funding Acknowledgements

This work was funded in part by the Australian Research Council. Special thanks to Dr Jochen Trumpf for insightful and thought-provoking discussions during the preliminary stages of this paper, and to the two anonymous reviewers for excellent guidance on improving the presentation.