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Two-sided Grassmann–Rayleigh quotient iteration

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Abstract

The two-sided Rayleigh quotient iteration proposed by Ostrowski computes a pair of corresponding left–right eigenvectors of a matrix C. We propose a Grassmannian version of this iteration, i.e., its iterates are pairs of p-dimensional subspaces instead of one-dimensional subspaces in the classical case. The new iteration generically converges locally cubically to the pairs of left–right p-dimensional invariant subspaces of C. Moreover, Grassmannian versions of the Rayleigh quotient iteration are given for the generalized Hermitian eigenproblem, the Hamiltonian eigenproblem and the skew-Hamiltonian eigenproblem.

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Authors and Affiliations

  1. Département d’ingénierie mathématique, Université catholique de Louvain, 1348, Louvain-la-Neuve, Belgium

    P.-A. Absil & P. Van Dooren

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  1. P.-A. Absil
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  2. P. Van Dooren
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Correspondence to P.-A. Absil.

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This paper presents research results of the Belgian Network Dynamical systems, control, and optimization (DYSCO), funded by the Interuniversity Attraction Poles Programme, initiated by the Belgian State, Science Policy Office. The scientific responsibility rests with its authors. This work was supported in part by the US National Science Foundation under Grant OCI-0324944 and by the School of Computational Science of Florida State University through a postdoctoral fellowship.

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Absil, PA., Van Dooren, P. Two-sided Grassmann–Rayleigh quotient iteration. Numer. Math. 114, 549–571 (2010). https://doi.org/10.1007/s00211-009-0266-y

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