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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Vydáno v:	SIAM journal on matrix analysis and applications Ročník 20; číslo 2; s. 303 - 353
Hlavní autoři:	Edelman, Alan, Arias, Tomás A., Smith, Steven T.
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Philadelphia Society for Industrial and Applied Mathematics 1998
Témata:	Algorithms Applied mathematics Eigenvalues Eigenvectors Geometry Linear algebra Optimization Physics Quadratic programming Signal processing
ISSN:	0895-4798, 1095-7162
On-line přístup:	Získat plný text
Tagy:	Přidat tag Žádné tagy, Buďte první, kdo vytvoří štítek k tomuto záznamu!

Abstract	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.
AbstractList	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.
Author	Edelman, Alan Arias, Tomás A. Smith, Steven T.
Author_xml	– sequence: 1 givenname: Alan surname: Edelman fullname: Edelman, Alan – sequence: 2 givenname: Tomás A. surname: Arias fullname: Arias, Tomás A. – sequence: 3 givenname: Steven T. surname: Smith fullname: Smith, Steven T.
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ContentType	Journal Article
Copyright	[Copyright] © 1998 Society for Industrial and Applied Mathematics
Copyright_xml	– notice: [Copyright] © 1998 Society for Industrial and Applied Mathematics
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Snippet	In this paper we develop new Newton and conjugate gradient algorithms on the Grassmann and Stiefel manifolds. These manifolds represent the constraints that...
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SubjectTerms	Algorithms Applied mathematics Eigenvalues Eigenvectors Geometry Linear algebra Optimization Physics Quadratic programming Signal processing
Title	The Geometry of Algorithms with Orthogonality Constraints
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