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The rational QZ method generalizes the QZ method by implicitly supporting rational subspace iteration. In this paper we extend the rational QZ method by introducing shifts and poles of higher multiplicity in the Hessenberg pencil, which is a pencil consisting of two Hessenberg matrices. The result is a multishift, multipole iteration on block Hessenberg pencils which allows one to stick to real arithmetic for a real input pencil. In combination with optimally packed shifts and aggressive earlyarXiv:1902.10954v2 fatcat:7brqqaqn7vbxlg6wv3fip2svra