The Geometry of Algorithms with Orthogonality Constraints

Details The Geometry of Algorithms with Orthogonality Constraints The Geometry of Algorithms with Orthogonality Constraints

1998-06-19

arxiv.org/abs/physics/9806030v1

Physics

Computational Physics

The condensed matter interest is as new methods for minimizing Kohn-Sham orbitals under the constraints of orthonormality and

Scientific paper

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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