A symplectic method for approximating all the eigenvalues of a Hamiltonian matrix
The paper presents a fast method for finding the eigenvalues of a Hamiltonian matrix M, that is of the partitioned form \(\left[ \begin{matrix} A^ T\\ H\end{matrix} \begin{matrix} G\\ -A\end{matrix} \right].\) It does this by squaring M and showing that using a special form of orthogonal matrix \(M^ 2\) can be reduced to the form \(\left[ \begin{matrix} H\\ 0\end{matrix} \begin{matrix} R\\ H^ T\end{matrix} \right].\) The standard QR algorithm is then used to calculate the eigenvalues of H which are the squares of the eigenvalues of M. Algorithms are given explicitly which minimize the work involved and which reduce the storage required. Examples are given and an error analysis is provided.
scientific article; zbMATH DE number 3901911