Performance of the QZ algorithm in the presence of infinite eigenvalues

The implicitly shifted (bulge-chasing) QZ algorithm is the most popular met hod for solving the generalized eigenvalue problem Av = lambda Bv. This pap er explains why the QZ algorithm functions well even in the presence of inf inite eigenvalues. The key to rapid convergence of QZ (and QR) algorithms i s the effective transmission of shifts during the bulge chase. In this pape r the mechanism of transmission of shifts is identified, and it is shown th at this mechanism is not disrupted by the presence of infinite eigenvalues. Both the QZ algorithm and the preliminary reduction to Hessenberg-triangul ar form tend to push the infinite eigenvalues toward the top of the pencil. Thus they should be deflated at the top.