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On bounds for rank growth of iterands in conjugate gradients for Lyapunov equation
Abstract
We focus on solving the Lyapunov equation AX + XAT = F, where A, X and F are square matrices, A is symmetric positive definite (SPD) and sparse, and F is symmetric and of a low-rank. The solution X is then also symmetric and in general dense, but it can be approximated by a low-rank matrix. If n is large, X cannot be computed directly, but it is accessible by using the so-called low-rank arithmetics (LRA). The equation is solved by matrix reformulation of the method of conjugate gradients (MCG). We are interested in the behavior of ranks of the solution approximations Xℓ, residuals Rℓ, and direction vectors Pℓ during iterations.
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BibTeX
@misc{lungova2026bounds,
title = {{On bounds for rank growth of iterands in conjugate gradients for Lyapunov equation}},
author = {Jana Lungová and Martin Plešinger},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.39308},
eprint = {2609.39308},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39308},
}
RIS
TY - UNPB
TI - On bounds for rank growth of iterands in conjugate gradients for Lyapunov equation
AU - Jana Lungová
AU - Martin Plešinger
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.39308
UR - https://arxiv.org/abs/2609.39308
AB - We focus on solving the Lyapunov equation AX + XA^T = F, where A, X and F are square matrices, A is symmetric positive definite (SPD) and sparse, and F is symmetric and of a low-rank. The solution X is then also symmetric and in general dense, but it can be approximated by a low-rank matrix. If n is large, X cannot be computed directly, but it is accessible by using the so-called low-rank arithmetics (LRA). The equation is solved by matrix reformulation of the method of conjugate gradients (MCG). We are interested in the behavior of ranks of the solution approximations X_ℓ, residuals R_ℓ, and direction vectors P_ℓ during iterations.
ER -