Towards a mixed-precision ADI method for Lyapunov equations
Abstract
We apply mixed-precision to the low-rank Lyapunov ADI (LR-ADI) by performing certain aspects of the algorithm in a lower working precision. Namely, we accumulate the overall solution, solve the linear systems comprising the ADI iteration, and store the inner low-rank factors of the residuals in various combinations of IEEE 754 single and double precision. We empirically test our implementation on Lyapunov equations arising from first- and second-order descriptor systems. For the first-order examples, accumulating the solution in single-precision yields an almost-as-small residual as for the double-precision solution. For certain applications, like computing the H2 norm of a descriptor system, low- or mixed-precision variants of the ADI can be quite competitive
Keywords
Cite
@article{arxiv.2508.00722,
title = {Towards a mixed-precision ADI method for Lyapunov equations},
author = {Jonas Schulze and Jens Saak},
journal= {arXiv preprint arXiv:2508.00722},
year = {2026}
}
Comments
11 pages, 3 figures, 1 table; submitted to PAMM 2025; using larger Steel Profile and reltol=1e-10 instead of 1e-8