### Abstract

We construct a nonlinear differential equation of matrix pairs (M(t), L(t)) that are invariant (structure-preserving property) in the class of symplectic matrix pairs {(M,L) = (eqution presented) is Hermitian}, where S_{1} and S_{2} are two fixed symplectic matrices. Furthermore, its solution also preserves deflating subspaces on the whole orbit (Eigenvector-preserving property). Such a flow is called a structure-preserving flow and is governed by a Riccati differential equation (RDE) of the form W(t) = [-W(t), I] H[I, W(t)^{⊥}]^{⊥}, W(0) = W_{0}, for some suitable Hamiltonian matrix H. We then utilize the Grassmann manifolds to extend the domain of the structure-preserving flow to the whole ℝ except some isolated points. On the other hand, the structure-preserving doubling algorithm (SDA) is an efficient numerical method for solving algebraic Riccati equations and nonlinear matrix equations. In conjunction with the structure-preserving flow, we consider two special classes of symplectic pairs: S1 = S2 = I2n and S1 = J, S2 = -I2n as well as the associated algorithms SDA-1 and SDA-2. It is shown that at t = 2, k ∈ ℤ this flow passes through the iterates generated by SDA-1 and SDA-2, respectively. Therefore, the SDA and its corresponding structure-preserving flow have identical asymptotic behaviors. Taking advantage of the special structure and properties of the Hamiltonian matrix, we apply a symplectically similarity transformation to reduce H to a Hamiltonian Jordan canonical form T. The asymptotic analysis of the structure-preserving flows and RDEs is studied by using e^{3T}. Some asymptotic dynamics of the SDA are investigated, including the linear and quadratic convergence.

Original language | English |
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Pages (from-to) | 976-1001 |

Number of pages | 26 |

Journal | SIAM Journal on Matrix Analysis and Applications |

Volume | 37 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2016 Jan 1 |

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### Keywords

- Riccati differential equations
- Structure-preserving doubling algorithm
- Structure-preserving flow
- Symplectic pairs

### ASJC Scopus subject areas

- Analysis

### Cite this

*SIAM Journal on Matrix Analysis and Applications*,

*37*(3), 976-1001. https://doi.org/10.1137/15M1019155