Structure-preserving flows of symplectic matrix pairs

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 S₁ and S₂ 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) = W₀, 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.