Global Convergence of the $QR$ Algorithm for Unitary Matrices with Some Results for Normal Matrices
P. J. Eberlein, C. P. Huang · SIAM Journal on Numerical Analysis · 1975
An algorithm is devised for choosing linear or quadratic shifts so that the $QR$ algorithm applied to a unitary upper Hessenberg matrix converges globally in the sense that $| {a_{nn - 1}^{(t)} } | \to 0$. Some additional results are obtained for normal matrices.