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The generalized null space decomposition (GNSD) is a unitary reduction of a general matrix A of order n to a block upper triangular form that reveals the structure of the Jordan blocks of A corresponding to a zero eigenvalue. The reduction was introduced by Kublanovskaya. It was extended first by Ruhe and then by Golub and Wilkinson, who based the reduction on the singular value decomposition. Unfortunately, if A has large Jordan blocks, the complexity of these algorithms can approach the order of n4. This paper presents an alternative algorithm, based on repeated updates of a QR decomposition of A, that is guaranteed to be of order n³. Numerical experiments confirm the stability of this algorithm, which turns out to produce essentially the same form as that of Golub and Wilkinson. The effect of errors in A on the ability to recover the original structure is investigated empirically. Several applications are discussed, including the computation of the Drazin inverse. [ABSTRACT FROM AUTHOR]

In this paper we discuss the deflation criterion used in the extended QR algorithm based on the chasing of rotations. We provide absolute and relative perturbation bounds for this deflation criterion. Further, we present a generalization of aggressive early deflation to the extended QR algorithms. Aggressive early deflation is the key technique for the identification and deflation of already converged, but hidden, eigenvalues. Often these possibilities for deflation are not detected by the standard technique. We present numerical results underpinning the power of aggressive early deflation also in the context of extended QR algorithms. We further generalize these ideas by the transcription of middle deflations. [ABSTRACT FROM AUTHOR]

Recently a generalization of Francis's implicitly shifted QR algorithm was proposed, notably widening the class of matrices admitting low-cost implicit QR steps. This unifying framework covered the methods and theory for Hessenberg and inverse Hessenberg matrices and furnished also new, single-shifted, QR-type methods for, e.g., CMV matrices. Convergence of this approach was only suggested by numerical experiments. No theoretical claims supporting the results were presented. In this paper we present multishift variants of these new algorithms. We also provide a convergence theory that shows that the new algorithm performs nested subspace iterations on rational Krylov subspaces. Numerical experiments confirm the validity of the theory. [ABSTRACT FROM AUTHOR]