Performance of the QZ Algorithm in the Presence of Infinite Eigenvalues

The implicitly shifted (bulge chasing) QZ algorithm is the most popular method for solving the generalized eigenvalue problem Av = Bv. This paper explains why the QZ algorithm functions well even in the presence of innnite eigenvalues. The key to rapid convergence of QZ (and QR) algorithms is the eeective transmission of shifts during the bulge chase. In this paper the mechanism of transmission of shifts is identiied, and it is shown that this mechanism is not disrupted by the presence of innnite eigenvalues. Both the QZ algorithm and the preliminary reduction to Hessenberg-triangular form tend to push the innnite eigenvalues toward the top of the pencil. Thus they should be deeated at the top.