On Computing Stable Lagrangian Subspaces of Hamiltonian Matrices and Symplectic Pencils
Wen‐Wei Lin, Chern-Shuh Wang · SIAM Journal on Matrix Analysis and Applications · 1997
This paper presents algorithms for computing stable Lagrangian invariant subspaces of a Hamiltonian matrix and a symplectic pencil, respectively, having purely imaginary and unimodular eigenvalues. The problems often arise in solving continuous- or discrete-time $H^{\infty}$-optimal control, linear-quadratic control and filtering theory, etc. The main approach of our algorithms is to determine an isotropic Jordan subbasis corresponding to purely imaginary (unimodular) eigenvalues by using the associated Jordan basis of the square of the Hamiltonian matrix (the $S+S^{-1}$-transformation of the symplectic pencil). The algorithms preserve structures and are numerically efficient and reliable in that they employ only orthogonal transformations in the continuous case.