The Quadratic Arnoldi Method for the Solution of the Quadratic Eigenvalue Problem

The Quadratic Arnoldi algorithm is an Arnoldi algorithm for the solution of the quadratic eigenvalue problem, that exploits the structure of the Krylov vectors. This allows us to reduce the memory requirements by about a half. The method is an alternative to the Second Order Arnoldi method (SOAR). In the SOAR method it is not clear how to perform an implicit restart. We discuss various choices of linearizations in L1 and DL. We also explain how to compute a partial Schur form of the underlying linearization with respect to the structure of the Schur vectors. We also formulate some open problems.