Numerical methods for palindromic eigenvalue problems: Computing the anti-triangular Schur form

We present structure-preserving numerical methods for the eigenvalue problem of complex palindromic pencils. Such problems arise in control theory, as well as from palindromic linearizations of higher degree palindromic matrix polynomials. A key ingredient of these methods is the development of an appropriate condensed form — the anti-triangular Schur form. Ill-conditioned problems with eigenvalues near the unit circle, in particular near ±1, are discussed. We show how a combination of unstructured methods followed by a structured refinement can be used to solve such problems accurately.