A new method for computing the stable invariant subspace of a real Hamiltonian matrix (Q1378991)

The problem of the numerical computation of the Lagrangian invariant subspaces of a real Hamiltonian matrix with a strongly backward stable algorithm is thoroughly examined. The solution of the Hamiltonian eigenvalue problem is important because of its close relation with the solution of the continuous-time algebraic Riccati equation. In order to deal with the problem, an extended matrix is introduced for which the authors determine how its eigenvalues and invariant subspaces are related with those of the original matrix. As a byproduct, these results, under the assumption that the matrix has no purely imaginary eigenvalues, give a direct relationship between its sign function and the square root of its square. Then, starting with a Hamiltonian matrix with no eigenvalues on the imaginary axis, a Hamiltonian Schur form is constructed for a convenient permutation of its Hamiltonian extended matrix. An important tool to accomplish this step is the symplectic URV decomposition of a Hamiltonian matrix, which is developed by the authors in another paper. Finally it is abundantly shown that the algorithm constructed on the basis of these results can compete advantageously with other algorithms, such as those based on the QR algorithm, as far as computational cost, i.e. \(O(n^3)\), is concerned. As for the error analysis, the authors prove that their algorithm is strongly backward stable for the extended matrix, which shows that the results so far obtained, although not complete, give a satisfactory solution to the original problem.