Implicit QR for rank-structured matrix pencils

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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Correspondence to P. Boito.

Additional information

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

Communicated by Ahmed Salam.

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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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Keywords

  • Rank-structured matrix
  • Quasiseparable matrix
  • QZ algorithm
  • Chebyshev approximation
  • Eigenvalue computation
  • Complexity

Mathematics Subject Classification (2010)

  • 65F15
  • 65H17