Newton algorithm on Stiefel manifolds

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Abstract

We give an explicit construction of the Newton algorithm on orthogonal Stiefel manifolds. In order to do this we introduce a local frame appropriate for the computation of the Hessian matrix for a cost function defined on Stiefel manifolds. For a Brockett cost function defined on \(St^4_2\) we give a classification of the critical points and we present numerical simulations of the Newton algorithm.

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The Geometry of Algorithms with Orthogonality Constraints

Steven Smith, Alan Edelman, T. A. Arias (1998)

In this paper we develop new Newton and conjugate gradient algorithms on the Grassmann and Stiefel manifolds. These manifolds represent the constraints that arise in such areas as the symmetric eigenvalue problem, nonlinear eigenvalue problems, electronic structures computations, and signal processing. In addition to the new algorithms, we show how the geometrical framework gives penetrating new insights allowing us to create, understand, and compare algorithms. The theory proposed here provides a taxonomy for numerical linear algebra algorithms that provide a top level mathematical view of previously unrelated algorithms. It is our hope that developers of new algorithms and perturbation theories will benefit from the theory, methods, and examples in this paper.