TY - JOUR

T1 - Structure-preserving flows of symplectic matrix pairs

AU - Kuo, Yueh Cheng

AU - Lin, Wen Wei

AU - Shieh, Shih Feng

N1 - Publisher Copyright:
© 2016 Society for Industrial and Applied Mathematics.

PY - 2016

Y1 - 2016

N2 - 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 e3T. Some asymptotic dynamics of the SDA are investigated, including the linear and quadratic convergence.

AB - 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 e3T. Some asymptotic dynamics of the SDA are investigated, including the linear and quadratic convergence.

KW - Riccati differential equations

KW - Structure-preserving doubling algorithm

KW - Structure-preserving flow

KW - Symplectic pairs

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U2 - 10.1137/15M1019155

DO - 10.1137/15M1019155

M3 - Article

AN - SCOPUS:84990841197

SN - 0895-4798

VL - 37

SP - 976

EP - 1001

JO - SIAM Journal on Matrix Analysis and Applications

JF - SIAM Journal on Matrix Analysis and Applications

IS - 3

ER -