On the Convergence of Diagonal Elements and Asymptotic Convergence Rates for the Shifted Tridiagonal QL Algorithm (Journal of Computational Mathematics, 1985)
The convergence of diagonal elements of an irreducible symmetric triadiagonal matrix under QL algorithm with some kinds of shift is discussed. It is proved that if $α_1-σ$→0 and $β_j$→0, j=1,2,...,m, then $α_j$→$λ_j$ where $λ_j$ are m eigenvalues of the matrix, and $σ$ is the origin shift. The asymptotic convergence rates of three kinds of shift, Rayleigh quotient shift, Wilkinson's shift and RW shift, are analysed.