The Geometry of Algorithms with Orthogonality Constraints
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.
Keywords
Cite
@article{arxiv.physics/9806030,
title = {The Geometry of Algorithms with Orthogonality Constraints},
author = {Alan Edelman and T. A. Arias and Steven T. Smith},
journal= {arXiv preprint arXiv:physics/9806030},
year = {2009}
}
Comments
The condensed matter interest is as new methods for minimizing Kohn-Sham orbitals under the constraints of orthonormality and as "geometrically correct" generalizations and extensions of the analystically continued functional approach, Phys. Rev. Lett. 69, 1077 (1992). The problem of orthonormality constraints is quite general and the methods discussed are also applicable in a wide range of fields. To appear in SIAM Journal of Matrix Analysis and Applications, in press for sometime in August-October 1998; 52 pages, 8 figures