The Geometry of Algorithms with Orthogonality Constraints

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 processi...

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Bibliographic Details
Published in:SIAM journal on matrix analysis and applications Vol. 20; no. 2; pp. 303 - 353
Main Authors: Edelman, Alan, Arias, Tomás A., Smith, Steven T.
Format: Journal Article
Language:English
Published: Philadelphia Society for Industrial and Applied Mathematics 1998
Subjects:
Algorithms
Applied mathematics
Eigenvalues
Eigenvectors
Geometry
Linear algebra
Optimization
Physics
Quadratic programming
Signal processing
ISSN:0895-4798, 1095-7162
Online Access:Get full text
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Summary: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.
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ISSN:0895-4798
1095-7162
DOI:10.1137/S0895479895290954