dorsal/arxiv
View SchemaThe Geometry of Algorithms with Orthogonality Constraints
| Authors | Alan Edelman, T. A. Arias, Steven T. Smith |
|---|---|
| Categories | |
| ArXiv ID | physics/9806030 |
| URL | https://arxiv.org/abs/physics/9806030 |
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.
{
"annotation_id": "471fa28c-96a2-42c8-b1ff-af12de88f0ef",
"date_created": "2026-03-02T18:01:21.293000Z",
"date_modified": "2026-03-02T18:01:21.293000Z",
"file_hash": "5e61cdd769470dee6cdab488a62dd7b0bc31c6adea632342905c8df3aae2e68a",
"private": false,
"record": {
"abstract": "In this paper we develop new Newton and conjugate gradient algorithms on the\nGrassmann and Stiefel manifolds. These manifolds represent the constraints that\narise in such areas as the symmetric eigenvalue problem, nonlinear eigenvalue\nproblems, electronic structures computations, and signal processing. In\naddition to the new algorithms, we show how the geometrical framework gives\npenetrating new insights allowing us to create, understand, and compare\nalgorithms. The theory proposed here provides a taxonomy for numerical linear\nalgebra algorithms that provide a top level mathematical view of previously\nunrelated algorithms. It is our hope that developers of new algorithms and\nperturbation theories will benefit from the theory, methods, and examples in\nthis paper.",
"arxiv_id": "physics/9806030",
"authors": [
"Alan Edelman",
"T. A. Arias",
"Steven T. Smith"
],
"categories": [
"physics.comp-ph",
"cond-mat",
"math.NA",
"physics.chem-ph"
],
"title": "The Geometry of Algorithms with Orthogonality Constraints",
"url": "https://arxiv.org/abs/physics/9806030"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "08351414-78fb-4a72-8aed-591f5a33107e",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}