The Ehrlich-aberth method for the nonsymmetric tridiagonal eigenvalue problem

Dario A. Bini, Luca Gemignani, Françoise Tisseur

    Research output: Contribution to journalArticlepeer-review


    An algorithm based on the Ehrlich-Aberth iteration is presented for the computation of the zeros of p(λ) = det(T-λI), where T is a real irreducible nonsymmetric tridiagonal matrix. The algorithm requires the evaluation of p(λ)/p′(λ) = -1/trace(T - λI) -1, which is done by exploiting the QR factorization of T - λI and the semiseparable structure of (T - λI) -1. The choice of initial approximations relies on a divide-and-conquer strategy, and some results motivating this strategy are given. Guaranteed a posteriori error bounds based on a running error analysis are proved. A Fortran 95 module implementing the algorithm is provided and numerical experiments that confirm the effectiveness and the robustness of the approach are presented. In particular, comparisons with the LAPACK subroutine dhseqr show that our algorithm is faster for large dimensions. © 2005 Society for Industrial and Applied Mathematics.
    Original languageEnglish
    Pages (from-to)153-175
    Number of pages22
    JournalSIAM Journal on Matrix Analysis and Applications
    Issue number1
    Publication statusPublished - 2006


    • Divide and conquer
    • Nonsyrnmetric eigenvalue problem
    • QR decomposition
    • Root finder
    • Symmetric indefinite generalized eigenvalue problem
    • Tridiagonal matrix


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