Linear matrix equations, such as the Sylvester and Lyapunov equations, play an important role in various applications, including the stability analysis and dimensionality reduction of linear dynamical control systems and the solution of partial differential equations. In this work, we present and analyze a new algorithm, based on tensorized Krylov subspaces, for quickly updating the solution of such a matrix equation when its coefficients undergo low-rank changes. We demonstrate how our algorithm can be utilized to accelerate the Newton method for solving continuous-time algebraic Riccati equations. Our algorithm also forms the basis of a new divide-and-conquer approach for linear matrix equations with coefficients that feature hierarchical low-rank structure, such as hierarchically off-diagonal low-rank structures, hierarchically semiseparable, and banded matrices. Numerical experiments demonstrate the advantages of divide-and-conquer over existing approaches, in terms of computational time and memory consumption.


  1. Sylvester equation
  2. Lyapunov equation
  3. low-rank update
  4. divide-and-conquer
  5. hierarchical matrices

MSC codes

  1. 15A06
  2. 93C20

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Information & Authors


Published In

cover image SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Pages: A848 - A876
ISSN (online): 1095-7197


Submitted: 12 December 2017
Accepted: 7 February 2019
Published online: 26 March 2019


  1. Sylvester equation
  2. Lyapunov equation
  3. low-rank update
  4. divide-and-conquer
  5. hierarchical matrices

MSC codes

  1. 15A06
  2. 93C20



Funding Information

Gruppo Nazionale per il Calcolo Scientifico

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