Numerical solution to generalized Lyapunov/Stein and rational Riccati equations in stochastic control

Hung Yuan Fan, Peter Chang Yi Weng*, Eric King Wah Chu

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

18 Citations (Scopus)


We consider the numerical solution of the generalized Lyapunov and Stein equations in ℝn$\mathbb {R}^{n}$, arising respectively from stochastic optimal control in continuous- and discrete-time. Generalizing the Smith method, our algorithms converge quadratically and have an O(n3) computational complexity per iteration and an O(n2) memory requirement. For large-scale problems, when the relevant matrix operators are “sparse”, our algorithm for generalized Stein (or Lyapunov) equations may achieve the complexity and memory requirement of O(n) (or similar to that of the solution of the linear systems associated with the sparse matrix operators). These efficient algorithms can be applied to Newton’s method for the solution of the rational Riccati equations. This contrasts favourably with the naive Newton algorithms of O(n6) complexity or the slower modified Newton’s methods of O(n3) complexity. The convergence and error analysis will be considered and numerical examples provided.

Original languageEnglish
Pages (from-to)245-272
Number of pages28
JournalNumerical Algorithms
Issue number2
Publication statusPublished - 2016 Feb 1


  • Algebraic Riccati equation
  • Bilinear model order reduction
  • Generalized Lyapunov equation
  • Generalized Stein equation
  • Large-scale problem
  • Newton’s method
  • Rational Riccati equation
  • Smith method
  • Stochastic Algebraic Riccati equation
  • Stochastic optimal control

ASJC Scopus subject areas

  • Applied Mathematics


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