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A generalization of the method of Chu, Liu and Mehrmann
for the computation of the Hamiltonian real Schur form is presented.
The new method avoids some of the difficulties that may arise when
a Hamiltonian matrix has tightly clustered groups of eigenvalues.
A detailed analysis of the method is presented and several numerical examples demonstrate the superior behavior of the method.
In the spirit of the Hamiltonian QR algorithm and other bidirectional chasing algorithms, a structure-preserving variant of the implicit QR algorithm for palindromic eigenvalue problems is proposed.
This new palindromic QR algorithm is strongly backward stable and requires less operations than the standard QZ algorithm, but
is restricted to matrix classes where a preliminary reduction to structured Hessenberg form can be performed. By an extension of the
implicit Q theorem, the palindromic QR algorithm is shown to be equivalent to a previously developed explicit version.
Also, the classical convergence theory for the QR algorithm can be extended to prove local quadratic convergence.
We briefly demonstrate how even eigenvalue problems can be addressed by similar techniques.