## Abstract

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(n^{3}) computational complexity per iteration and an O(n^{2}) 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(n^{6}) complexity or the slower modified Newton’s methods of O(n^{3}) complexity. The convergence and error analysis will be considered and numerical examples provided.

Original language | English |
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Pages (from-to) | 245-272 |

Number of pages | 28 |

Journal | Numerical Algorithms |

Volume | 71 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1 Feb 2016 |

## Keywords

- 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