An overview on the eigenvalue computation for matrices
Neural, Parallel & Scientific Computations
An algorithm for the complete solution of quadratic eigenvalue problems
ACM Transactions on Mathematical Software (TOMS)
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The implicitly shifted (bulge-chasing) QZ algorithm is the most popular method for solving the generalized eigenvalue problem $Av=\lambda Bv$. This paper explains why the QZ algorithm functions well even in the presence of infinite eigenvalues. The key to rapid convergence of QZ (and QR) algorithms is the effective transmission of shifts during the bulge chase. In this paper the mechanism of transmission of shifts is identified, and it is shown that this mechanism is not disrupted by the presence of infinite eigenvalues. Both the QZ algorithm and the preliminary reduction to Hessenberg-triangular form tend to push the infinite eigenvalues toward the top of the pencil. Thus they should be deflated at the top.