Implicit QR for rank-structured matrix pencils

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 $$N\times N$$N×N rank structured matrix pencils using $$O(N^2)$$O(N2) flops and $$O(N)$$O(N) memory storage. Numerical experiments and comparisons confirm the effectiveness and the stability of the proposed method.

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