A Fast and Stable Square-Root-Free Unitary Core-Chasing Algorithm
Abstract
Symmetric and unitary matrices are among the most important classes of structured matrices admitting efficient and stable eigenvalue algorithms. Notable examples include structure-preserving variants of the QR algorithm and their square-root-free counterparts. For unitary matrices, these algorithms were historically derived from recurrence relations for orthogonal polynomials on the unit circle. A matrix-based derivation using core chasing was later given in [3]; this family of algorithms is backward stable, but the underlying symmetries needed to cast out the square roots were not identified, and the question of structured backward stability was left open. In this paper we present a matrix-based derivation of the symmetric unitary core-chasing algorithm and use it to obtain a square-root-free variant. We give a matrix-based proof of backward stability and show that strict structured backward stability is impossible in general. Open-source Fortran code and numerical experiments demonstrate that the combination of exploiting the symmetry and removing the square roots reduces computation time without degrading accuracy.
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BibTeX
@misc{esquivelrosado2026fast,
title = {{A Fast and Stable Square-Root-Free Unitary Core-Chasing Algorithm}},
author = {Mónica Esquivel-Rosado and Jared L. Aurentz and Giovanni Barbarino and María C. Quintana},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12228},
eprint = {2610.12228},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12228},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - A Fast and Stable Square-Root-Free Unitary Core-Chasing Algorithm
AU - Mónica Esquivel-Rosado
AU - Jared L. Aurentz
AU - Giovanni Barbarino
AU - María C. Quintana
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12228
UR - https://arxiv.org/abs/2610.12228
AB - Symmetric and unitary matrices are among the most important classes of structured matrices admitting efficient and stable eigenvalue algorithms. Notable examples include structure-preserving variants of the QR algorithm and their square-root-free counterparts. For unitary matrices, these algorithms were historically derived from recurrence relations for orthogonal polynomials on the unit circle. A matrix-based derivation using core chasing was later given in [3]; this family of algorithms is backward stable, but the underlying symmetries needed to cast out the square roots were not identified, and the question of structured backward stability was left open. In this paper we present a matrix-based derivation of the symmetric unitary core-chasing algorithm and use it to obtain a square-root-free variant. We give a matrix-based proof of backward stability and show that strict structured backward stability is impossible in general. Open-source Fortran code and numerical experiments demonstrate that the combination of exploiting the symmetry and removing the square roots reduces computation time without degrading accuracy.
ER -