A unitary symplectic similarity transformation for a special class of Hamiltonian matrices to extended Hamiltonian Hessenberg form is presented. Whereas the classical Hessenberg form links to Krylov subspaces, the extended Hessenberg form links to extended Krylov subspaces. The presented algorithm generalizes thus the classic reduction to Hamiltonian Hessenberg form and offers more freedom in the choice of Hamiltonian condensed forms, to be used within an extended Hamiltonian QR algorithm. Theoretical results identifying the structure of the extended Hamiltonian Hessenberg form and proofs of uniqueness of the reduction process are included. Numerical experiments confirm the validity of the approach. Original language EnglishPages (from-to) 1-31

An algorithm for computing the singular value decomposition of normal matrices using intermediate complex symmetric matrices is proposed. This algorithm, as most eigenvalue and singular value algorithms, consists of two steps. It is based on combining the unitarily equivalence of normal matrices to complex symmetric tridiagonal form with the symmetric singular value decomposition of complex symmetric matrices.Numerical experiments are included comparing several algorithms, with respect to speed and accuracy, for computing the symmetric singular value decomposition (also known as the Takagi factorization). Next we compare the novel approach with the classical Golub-Kahan method for computing the singular value decomposition of normal matrices: it is faster, consumes less memory, but on the other hand the results are significantly less accurate.

An extended QR algorithm specifically tailored for Hamiltonian matrices is presented. The algorithm generalizes the customary Hamiltonian QR algorithm with additional freedom in choosing between various possible extended Hamiltonian Hessenberg forms. We introduced in [Ferranti et al., An extended Hessenberg form for Hamiltonian matrices, Calcolo] an algorithm to transform certain Hamiltonian matrices to such forms. Whereas the convergence of the classical QR algorithm is related to classical Krylov subspaces, convergence in the extended case links to extended Krylov subspaces, resulting in a greater flexibility, and possible enhanced convergence behavior. Details on the implementation, covering the bidirectional chasing and the bulge exchange based on rotations are presented. The numerical experiments reveal that the convergence depends on the selected extended forms and illustrate the validity of the approach.

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We show that extended Hamiltonian Hessenberg matrices arise naturally in projection-based model order reduction. Therefore we reduce a large dynamical system by projecting it on an extended Krylov subspace. The eigenvalues of the reduced order model can then be computed directly by applying the extended Hamiltonian QR algorithm.