Schur parameter pencils for the solution of the unitary eigenproblem

Abstract

Let U− λV be an n× n pencil with unitary matrices U and V. An algorithm is presented which reduces U and V simultaneously to unitary block diagonal matrices G o = Q H UP and G e = Q H VP with block size at most two. It is an O( n 3) process using Householder eliminations, and it is backward stable. In the special case V= I the block diagonal matrices G o , G H e can be normalized so that their entries are just the Schur parameters of the Hessenberg condensed form of U. We call G o − λG e a Schur parameter pencil. It can also be derived from U,V by a Lanczos-like process. For the solution of the eigenvalue problem for G o − λG e a QR-type algorithm can be developed based on this unitary reduction of a pencil U− λV to a Schur parameter pencil. The condensed form is preserved throughout the process. Each iteration step needs only O( n) operations. This method of solving the unitary eigenvalue problem seems to be the closest possible analogy to the QR method for the Hermitian eigenvalue problem.