Hongguo Xu

Canonical Forms for Hamiltonian and Symplectic Matrices and Pencils

Abstract We study canonical forms for Hamiltonian and symplectic matrices or pencils under equivalence transformations which keep the class invariant. In contrast to other canonical forms our forms are as close as possible to a triangular structure in the same class. We give necessary and sufficient conditions for the existence of Hamiltonian and symplectic triangular Jordan, Kronecker and Schur forms. The presented results generalize results of Lin and Ho W.-W. Lin, T.-C. Ho, On Schur type decompositions for Hamiltonian and symplectic pencils, Technical report, Institute of Applied Mathematics, National Tsing Hua University, Taiwan, 1990 and simplify the proofs presented there.

Numerical Computation of Deflating Subspaces of Skew-Hamiltonian/Hamiltonian Pencils

We discuss the numerical solution of structured generalized eigenvalue problems that arise from linear-quadratic optimal control problems,

A new method for computing the stable invariant subspace of a real Hamiltonian matrix

H_{\infty}

Existence, Uniqueness, and Parametrization of Lagrangian Invariant Subspaces

optimization, multibody systems, and many other areas of applied mathematics, physics, and chemistry. The classical approach for these problems requires computing invariant and deflating subspaces of matrices and matrix pencils with Hamiltonian and/or skew-Hamiltonian structure. We extend the recently developed methods for Hamiltonian matrices to the general case of skew-Hamiltonian/Hamiltonian pencils. The algorithms circumvent problems with skew-Hamiltonian/Hamiltonian matrix pencils that lack structured Schur forms by embedding them into matrix pencils that always admit a structured Schur form. The rounding error analysis of the resulting algorithms is favorable. For the embedded matrix pencils, the algorithms use structure-preserving unitary matrix computations and are strongly backwards stable, i.e., they compute the exact structured Schur form of a nearby matrix pencil with the same structure.

Perturbation Analysis for the Eigenvalue Problem of a Formal Product of Matrices

Abstract A new backward stable, structure preserving method of complexity O(n3) is presented for computing the stable invariant subspace of a real Hamiltonian matrix and the stabilizing solution of the continuous-time algebraic Riccati equation. The new method is based on the relationship between the invariant subspaces of the Hamiltonian matrix H and the extended matrix 0 H H 0 and makes use of the symplectic URV-like decomposition that was recently introduced by the authors.

HAMILTONIAN SQUARE ROOTS OF SKEW-HAMILTONIAN MATRICES

The existence, uniqueness, and parametrization of Lagrangian invariant subspaces for Hamiltonian matrices is studied. Necessary and sufficient conditions and a complete parametrization are given. Some necessary and sufficient conditions for the existence of Hermitian solutions of algebraic Riccati equations follow as simple corollaries.

A New Scaling for Newton's Iteration for the Polar Decomposition and its Backward Stability

We study the perturbation theory for the eigenvalue problem of a formal matrix product A1s1 ··· Apsp, where all Ak are square and sk ∈ {−1, 1}. We generalize the classical perturbation results for matrices and matrix pencils to perturbation results for generalized deflating subspaces and eigenvalues of such formal matrix products. As an application we then extend the structured perturbation theory for the eigenvalue problem of Hamiltonian matrices to Hamiltonian/skew-Hamiltonian pencils.

Perturbation of Purely Imaginary Eigenvalues of Hamiltonian Matrices under Structured Perturbations

Abstract We present a constructive existence proof that every real skew-Hamiltonian matrix W has a real Hamiltonian square root. The key step in this construction shows how one may bring any such W into a real quasi-Jordan canonical form via symplectic similarity. We show further that every W has infinitely many real Hamiltonian square roots, and give a lower bound on the dimension of the set of all such square roots. Some extensions to complex matrices are also presented.

Explicit Solutions for a Riccati Equation from Transport Theory

We propose a scaling scheme for Newtons iteration for calculating the polar decomposition. The scaling factors are generated by a simple scalar iteration in which the initial value depends only on estimates of the extreme singular values of the original matrix, which can, for example, be the Frobenius norms of the matrix and its inverse. In exact arithmetic, for matrices with condition number no greater than

Numerical methods in control

10^{16}