A Hessenberg Reduction Algorithm for Rank Structured Matrices

In this paper we show how to perform the Hessenberg reduction of a rank structured matrix under unitary similarity operations in an efficient way, using the Givens-weight representation which we introduced in an earlier paper. This reduction can be used as a first step for eigenvalue computation. We also show how the algorithm can be modified to compute the bidiagonal reduction of a rank structured matrix, the latter being a preprocessing step for computing the singular values of the matrix. Numerical experiments demonstrate the stability of this approach.