Implicit QR with compression

In this paper, we elaborate on the implicit shifted QR eigenvalue algorithm given in [D.A. Bini, P. Boito, Y. Eidelman, L. Gemignani, I. Gohberg, A fast implicit QR eigenvalue algorithm for companion matrices, Linear Algebra Appl. 432 (2010), 2006-2031]. The algorithm is substantially simplified and speeded up while preserving its numerical robustness. This allows us to obtain a potentially important advance towards a proof of its backward stability together with both cost reductions and implementative benefits.