STRUCTURE-PRESERVING FLOWS OF SYMPLECTIC MATRIX PAIRS

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) = ((sic)S-2, (sic)S-1)| X = [X-i j](1<i,j<2) 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) over dot (t) = [-W(t), I]H[I, W(t)(inverted perpendicular)](inverted perpendicular), 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: S-1 = S-2 = I-2n and S-1 = J, 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 is an element of 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 J. The asymptotic analysis of the structure-preserving flows and RDEs is studied by using e(Jt). Some asymptotic dynamics of the SDA are investigated, including the linear and quadratic convergence.