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Implicit QR for rank-structured matrix pencils

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Abstract

A fast implicit QR algorithm for eigenvalue computation of low rank corrections of Hermitian matrices is adjusted to work with matrix pencils arising from zerofinding problems for polynomials expressed in Chebyshev-like bases. The modified QZ algorithm computes the generalized eigenvalues of certain \(N\times N\) rank structured matrix pencils using \(O(N^2)\) flops and \(O(N)\) memory storage. Numerical experiments and comparisons confirm the effectiveness and the stability of the proposed method.

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Authors and Affiliations

  1. Faculté des Sciences et Techniques, XLIM–DMI, UMR CNRS 7252, Université de Limoges, 123 av. A. Thomas, 87060 , Limoges, France

    P. Boito

  2. Raymond and Beverly Sackler Faculty of Exact Sciences, School of Mathematical Sciences, Tel-Aviv University, 69978 , Ramat-Aviv, Israel

    Y. Eidelman

  3. Dipartimento di Informatica, Università di Pisa, Largo B. Pontecorvo 3, 56127 , Pisa, Italy

    L. Gemignani

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  1. P. Boito
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  2. Y. Eidelman
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  3. L. Gemignani
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Correspondence to P. Boito.

Additional information

Communicated by Ahmed Salam.

This work was partially supported by MIUR, grant number 20083KLJEZ.

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Boito, P., Eidelman, Y. & Gemignani, L. Implicit QR for rank-structured matrix pencils. Bit Numer Math 54, 85–111 (2014). https://doi.org/10.1007/s10543-014-0478-0

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