A Generalization of the Multishift QR Algorithm | Zendy

Raf Vandebril | Zendy; David S. Watkins | Zendy

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A Generalization of the Multishift QR Algorithm

Author(s) -

Raf Vandebril,

David S. Watkins

Publication year - 2012

Publication title -

siam journal on matrix analysis and applications

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 1.268

H-Index - 101

eISSN - 1095-7162

pISSN - 0895-4798

DOI - 10.1137/11085219x

Subject(s) - krylov subspace , mathematics , linear subspace , qr decomposition , convergence (economics) , generalization , algorithm , symbolic convergence theory , inverse , algebra over a field , eigenvalues and eigenvectors , iterative method , computer science , pure mathematics , key (lock) , mathematical analysis , physics , computer security , geometry , quantum mechanics , economics , economic growth

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.status: publishe

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