A Hamiltonian $QR$ Algorithm

Abstract

This paper presents a variant $QR$ algorithm for calculating a Hamiltonian–Schur decomposition [10]. It is defined for Hamiltonian matrices that arise from single input control systems. Numerical stability and Hamiltonian structure are preserved by using unitary symplectic similarity transformations. Following a finite step reduction to a Hessenberg-like condensed form, a sequence of similarity transformations yields a permuted triangular matrix. As the iteration converges, it deflates into problems of lower dimension. Convergence is accelerated by varying a scalar shift. When the Hamiltonian matrix is real, complex arithmetic can be avoided by using an implicit double shift technique. The Hamiltonian-Schur decomposition yields the same invariant subspace information as a Schur decomposition but requires significantly less work and storage for problems of size greater than about 20.