### Abstract

We consider the numerical solution of the rational algebraic Riccati equations in ℝ^{n}, arising from stochastic optimal control in continuous and discrete time. Applying the homotopy method, we continue from the stabilizing solutions of the deterministic algebraic Riccati equations, which are readily available. The associated differential equations require the solutions of some generalized Lyapunov or Stein equations, which can be solved by the generalized Smith methods, of O(n^{3}) computational complexity and O(n^{2}) memory requirement. For large-scale problems, the sparsity and structures in the relevant matrices further improve the efficiency of our algorithms. In comparison, the alternative (modified) Newton's methods require a difficult initial stabilization step. Some illustrative numerical examples are provided.

Original language | English |
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Pages (from-to) | B103-B125 |

Journal | SIAM Journal on Scientific Computing |

Volume | 37 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2015 Jan 1 |

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### Keywords

- Generalized Stein equation
- Generalized lyapunov equation
- Rational algebraic Riccati equation
- Stochastic algebraic Riccati equation
- Stochastic optional control

### ASJC Scopus subject areas

- Computational Mathematics
- Applied Mathematics

### Cite this

*SIAM Journal on Scientific Computing*,

*37*(1), B103-B125. https://doi.org/10.1137/140953204