Structure-Preserving Flows of Symplectic Matrix Pairs
Yueh‐Cheng Kuo, Wen-Wei Lin, Shih-Feng Shieh · SIAM Journal on Matrix Analysis and Applications · 2016
We construct a nonlinear differential equation of matrix pairs $(\mathcal{M}(t),\mathcal{L}(t))$ that are invariant (structure-preserving property) in the class of symplectic matrix pairs $ \left\{ \left(\mathcal{M},\mathcal{L}\right)= \scriptstyle\left.\left(\left[ \begin{smallmatrix} X_{12} && 0 \\ X_{22} && I \\ \end{smallmatrix} \right]\mathcal{S}_2, \left[ \begin{smallmatrix} I && X_{11} \\ 0 && X_{21} \\ \end{smallmatrix} \right]\mathcal{S}_1\right) \right| \ X=[X_{ij}]_{1\le i,j\le2}\ \textstyle{\rm is\ Hermitian} \right\},$ where $\mathcal{S}_1$ and $\mathcal{S}_2$ 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 $\dot{W}(t)=[-W(t),I]\mathscr{H}[I ,W(t)^{\top} ]^{\top}$, $W(0)=W_0$, for some suitable Hamiltonian matrix $\mathscr{H}$. We then utilize the Grassmann manifolds to extend the domain of the structure-preserving flow to the whole $\mathbb{R}$ 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: $\mathcal{S}_1=\mathcal{S}_2=I_{2n}$ and $\mathcal{S}_1=\mathcal{J}$, $\mathcal{S}_2=-I_{2n}$ as well as the associated algorithms SDA-1 and SDA-2. It is shown that at $t=2^{k-1},k\in \mathbb{Z}$ 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 $\mathscr{H}$ to a Hamiltonian Jordan canonical form $\mathfrak{J}$. The asymptotic analysis of the structure-preserving flows and RDEs is studied by using $e^{\mathfrak{J}t}$. Some asymptotic dynamics of the SDA are investigated, including the linear and quadratic convergence.