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arXiv · 2512.04983

A tangential low-rank ADI method for solving indefinite Lyapunov equations

Rudi Smith·Steffen W. R. Werner
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Abstract

Continuous-time algebraic Lyapunov equations have become an essential tool in various applications. In the case of large-scale sparse coefficient matrices and indefinite constant terms, indefinite low-rank factorizations have successfully been used to allow methods like the alternating direction implicit (ADI) iteration to efficiently compute accurate approximations to the solution of the Lyapunov equation. However, classical block-type approaches quickly increase in computational costs when the rank of the constant term grows. While the regular truncation of solution approximations during the iteration may be a remedy, it does not resolve the issue of computing high-dimensional updates and may even compromise the numerical accuracy of the approximation. In this paper, we propose a novel tangential reformulation of the ADI iteration that preserves the exactness of the iterative updates while allowing for the efficient construction of low-rank approximations to the solution of Lyapunov equations with indefinite right-hand sides even in the case of constant terms with higher ranks. We provide adaptive methods for the selection of the corresponding ADI parameters, namely shifts and tangential directions, which allow for the automatic application of the method to any relevant problem setting. The effectiveness of the developed algorithms is illustrated in several numerical examples.

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Rudi Smith, Steffen W. R. Werner. 2026-09-13. A tangential low-rank ADI method for solving indefinite Lyapunov equations. https://arxiv.org/abs/2512.04983

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