• Ronny Ramlau, Lothar Reichel (Hg.)

ETNA - Electronic Transactions on Numerical Analysis

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Electronic Transactions on Numerical Analysis (ETNA) is an electronic journal for the publication of significant new developments in numerical analysis and scientific computing. Papers of the highest quality that deal with the analysis of algorithms for the solution of continuous models and numerical linear algebra are appropriate for ETNA, as are papers of similar quality that discuss implementation and performance of such algorithms. New algorithms for current or new computer architectures are appropriate provided that they are numerically sound. However, the focus of the publication should be on the algorithm rather than on the architecture. The journal is published by the Kent State University Library in conjunction with the Institute of Computational Mathematics at Kent State University, and in cooperation with the Johann Radon Institute for Computational and Applied Mathematics of the Austrian Academy of Sciences (RICAM). Reviews of all ETNA papers appear in Mathematical Reviews and Zentralblatt für Mathematik. Reference information for ETNA papers also appears in the expanded Science Citation Index. ETNA is registered with the Library of Congress and has ISSN 1068-9613.

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Verlag der Österreichischen Akademie der Wissenschaften
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ETNA - Electronic Transactions on Numerical Analysis



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Verlag der Österreichischen Akademie der Wissenschaften
Austrian Academy of Sciences Press
A-1011 Wien, Dr. Ignaz Seipel-Platz 2,
Tel. +43-1-515 81/DW 3420, Fax +43-1-515 81/DW 3400
https://verlag.oeaw.ac.at, e-mail: bestellung.verlag@oeaw.ac.at
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A single shooting method with approximate Fréchet derivative for computing geodesics on the Stiefel manifold

    Marco Sutti

ETNA - Electronic Transactions on Numerical Analysis, pp. 501-519, 2024/09/10

doi: 10.1553/etna_vol60s501

doi: 10.1553/etna_vol60s501


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doi:10.1553/etna_vol60s501



doi:10.1553/etna_vol60s501

Abstract

This paper shows how to use the shooting method, a classical numerical algorithm for solving boundary value problems, to compute the Riemannian distance on the Stiefel manifold $\mathrm{St}(n,p)$, the set of $ n \times p $ matrices with orthonormal columns. The proposed method is a shooting method in the sense of the classical shooting methods for solving boundary value problems; see, e.g., Stoer and Bulirsch, 1993.
The main feature is that we provide an approximate formula for the Fréchet derivative of the geodesic involved in our shooting method.
Numerical experiments demonstrate the algorithm\'s accuracy and performance. Comparisons with existing state-of-the-art algorithms for solving the same problem show that our method is competitive and even beats several algorithms in many cases.

Keywords: Stiefel manifold shooting methods endpoint geodesic problem Riemannian distance Newton's method Fréchet derivative

Published Online: 2024/09/10

Object Identifier: 0xc1aa5572 0x003f5dd2

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