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Performance of the QZ algorithm in the presence of infinite eigenvalues. (English) Zbl 0969.65029
The paper is inspired by the fact that the generalized eigenvalue problem can have infinite eigenvalues and that they can be dealt with by the QZ (i.e., generalized QR) algorithm [cf. C. B. Moler and G. W. Stewart, SIAM J. Numer. Anal. 10, 214-256 (1973; Zbl 0225.65046)]. An explanation of the latter “paradox” is provided and found in the mechanism of the transmission of implicit shifts which pushes the infinite eigenvalues towards the top of the pencil. The same explanation applies also to the preliminary Hessenberg + triangular reduction of the pencil and is in fact a next-step development of the author’s older results, viz., papers concerning the regular QR algorithm [J. Linear Algebra Appl. 241-243, 877-896 (1996; Zbl 0871.65027); SIAM J. Matrix Anal. Appl. 19, No. 4, 1074-1096 (1998; Zbl 0916.65034)].
As long as the key theorems are just presented in the absence of roundoff errors, an inseparable part of the argument lies in the numerical experiments performed using MATLAB in double precision arithmetics. In particular, two tables illustrate the necessity of avoiding the theoretical zerodivide, i.e., the obligatory deflation of the infinite eigenvalues once they get at the top of the pencil.

65F15 Numerical computation of eigenvalues and eigenvectors of matrices
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