A KQZ algorithm for solving linear-response eigenvalue equations (Q1184477)

An algorithm is proposed for the numerical solution of eigenvalue problems of the form \[ \left({\left[A\;B\atop B\;A\right]}-\lambda \left[{\Sigma\atop -\Delta}{\Delta\atop -\Sigma}\right]\right){x_ 1\choose x_ 2}=0. \] Its properties are examined theoretically and checked numerically. The algorithm preserves the block structure of the matrices during computation, thus preserving properties of the solution which the \(QZ\) algorithm loses through roundoff, and it saves about half of the computational cost and storage of the \(QZ\) algorithm.