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Low-rank updates and divide-and-conquer methods for quadratic matrix equations

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Abstract

In this work, we consider two types of large-scale quadratic matrix equations: continuous-time algebraic Riccati equations, which play a central role in optimal and robust control, and unilateral quadratic matrix equations, which arise from stochastic processes on 2D lattices and vibrating systems. We propose a simple and fast way to update the solution to such matrix equations under low-rank modifications of the coefficients. Based on this procedure, we develop a divide-and-conquer method for quadratic matrix equations with coefficients that feature a specific type of hierarchical low-rank structure, which includes banded matrices. This generalizes earlier work on linear matrix equations. Numerical experiments indicate the advantages of our newly proposed method versus iterative schemes combined with hierarchical low-rank arithmetic.

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Notes

  1. Preliminary results suggest that replacing Algorithm 1 with RADI reduces the time by 10% on average over all used sizes n.

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Acknowledgments

During the larger part of the work on this article, the second author PK was affiliated with the Max Planck Institute for Dynamics of Complex Technical Systems Magdeburg.

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Authors and Affiliations

  1. EPF Lausanne, Lausanne, Switzerland

    Daniel Kressner & Stefano Massei

  2. Department of Electrical Engineering (ESAT/STADIUS), KU Leuven, Heverlee, Belgium

    Patrick Kürschner

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  1. Daniel Kressner
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  3. Stefano Massei
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Correspondence to Stefano Massei.

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The work of Stefano Massei has been supported by the SNSF research project Fast algorithms from low-rank updates, grant number: 200020_178806.

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Kressner, D., Kürschner, P. & Massei, S. Low-rank updates and divide-and-conquer methods for quadratic matrix equations. Numer Algor 84, 717–741 (2020). https://doi.org/10.1007/s11075-019-00776-w

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