Real-Arithmetic Sketched Eigensolvers for Symmetric Matrices
Abstract
In this work, we develop sketched variants of subspace iteration and TraceMIN, called sSI and sTraceMIN, for real symmetric eigenvalue problems by combining basis whitening with the sketched Rayleigh-Ritz (sRR) procedure. The nonsymmetric projected matrix in sRR may have nonreal eigenpairs, leading to complex arithmetic and additional computational costs. We overcome this difficulty using a real Schur formulation of sRR that keeps the iterations in real arithmetic. We further develop procedures for characteristic subspace selection and convergence monitoring within this formulation. On the theoretical side, we discuss suitable subspace embedding assumptions for sketched eigensolvers that use a fixed sketching matrix throughout the iterations. Under a mild embedding assumption, we establish local convergence results for sSI and sTraceMIN. Numerical experiments demonstrate runtime savings of up to 39% for sSI and 21% for sTraceMIN relative to their standard counterparts on the test problems.
Related
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BibTeX
@misc{liu2026realarithmetic,
title = {{Real-Arithmetic Sketched Eigensolvers for Symmetric Matrices}},
author = {Jingyu Liu and Yuqi Liu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22261},
eprint = {2610.22261},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22261},
}
RIS
TY - UNPB
TI - Real-Arithmetic Sketched Eigensolvers for Symmetric Matrices
AU - Jingyu Liu
AU - Yuqi Liu
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.22261
UR - https://arxiv.org/abs/2610.22261
AB - In this work, we develop sketched variants of subspace iteration and TraceMIN, called sSI and sTraceMIN, for real symmetric eigenvalue problems by combining basis whitening with the sketched Rayleigh-Ritz (sRR) procedure. The nonsymmetric projected matrix in sRR may have nonreal eigenpairs, leading to complex arithmetic and additional computational costs. We overcome this difficulty using a real Schur formulation of sRR that keeps the iterations in real arithmetic. We further develop procedures for characteristic subspace selection and convergence monitoring within this formulation. On the theoretical side, we discuss suitable subspace embedding assumptions for sketched eigensolvers that use a fixed sketching matrix throughout the iterations. Under a mild embedding assumption, we establish local convergence results for sSI and sTraceMIN. Numerical experiments demonstrate runtime savings of up to 39% for sSI and 21% for sTraceMIN relative to their standard counterparts on the test problems.
ER -