THE GEOMETRY OF ALGORITHMS WITH ORTHOGONALITY CONSTRAINTS

Citation
A. Edelman et al., THE GEOMETRY OF ALGORITHMS WITH ORTHOGONALITY CONSTRAINTS, SIAM journal on matrix analysis and applications (Print), 20(2), 1998, pp. 303-353
Citations number
95
Categorie Soggetti
Mathematics,Mathematics
ISSN journal
08954798
Volume
20
Issue
2
Year of publication
1998
Pages
303 - 353
Database
ISI
SICI code
0895-4798(1998)20:2<303:TGOAWO>2.0.ZU;2-N
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 probl em, 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 her e provides a taxonomy for numerical linear algebra algorithms that pro vide a top level mathematical view of previously unrelated algorithms. It is our hope that developers of new algorithms and perturbation the ories will benefit from the theory, methods, and examples in this pape r.