A multishift, multipole rational QZ method with aggressive early deflation

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. The result is a multishift, multipole iteration on block Hessenberg pencils. In combination with tightly-packed shifts and advanced deflation techniques such as aggressive early deflation we obtain an efficient method for the dense generalized eigenvalue problem. Numerical experiments demonstrate the level 3 BLAS performance and compare the results with LAPACK routines for the generalized eigenvalue problem. We show that our methods can outperform all in terms of speed and accuracy and observe an empirical time complexity significantly lower than $O(n^3)$.