On the numerical solution of large-scale sparse discrete-time Riccati equations

  • Peter BennerEmail author
  • Heike Faßbender


We discuss the numerical solution of large-scale discrete-time algebraic Riccati equations (DAREs) as they arise, e.g., in fully discretized linear-quadratic optimal control problems for parabolic partial differential equations (PDEs). We employ variants of Newton’s method that allow to compute an approximate low-rank factor of the solution of the DARE. The principal computation in the Newton iteration is the numerical solution of a Stein (aka discrete Lyapunov) equation in each step. For this purpose, we present a low-rank Smith method as well as a low-rank alternating-direction-implicit (ADI) iteration to compute low-rank approximations to solutions of Stein equations arising in this context. Numerical results are given to verify the efficiency and accuracy of the proposed algorithms.


Riccati equation  Discrete-time control Large Sparse Algebraic Riccati equation Newton’s method Smith iteration ADI iteration Low-rank factor 

Mathematics Subject Classifications (2010)

65F30 15A24 49M15 93C20 93C05 


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© Springer Science+Business Media, LLC. 2011

Authors and Affiliations

  1. 1.Research Group Computational Methods in Systems and Control TheoryMax Planck Institute for Dynamics of Complex Technical SystemsMagdeburgGermany
  2. 2.Carl-Friedrich-Gauß-Fakultät, Institut Computational Mathematics AG NumerikTU BraunschweigBraunschweigGermany

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