A rational $QR$-iteration

Raf Vandebril 1 - Marc Van Barel 1 - Nicola Mastronardi 2

Date: 10 December 2007

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Key words and phrases: keywords

Abstract:

In this manuscript a new type of $QR$-iteration will be presented. Each step of this new iteration consists of two substeps. In the explicit version, first an $RQ$-factorization of the initial matrix $A-\kappa I=RQ$ will be computed, followed by a $QR$-factorization of the matrix $(A-\sigma I)Q^H$. Applying the unitary similarity transformation defined by the $QR$-factorization of the transformed matrix $(A-\sigma I)Q^H$, will yield interesting convergence properties. It will be shown that the convergence behavior is related to a subspace iteration based on a rational function in $A$ namely $(A-\sigma I)(A-\kappa I)^{-1}$. Convergence properties of this new iteration will be investigated and examples will be presented, illustrating the effectiveness of this approach with respect to some specific classes of matrices.

$QR$-algorithm, eigenvalues, rational functions