J. Comp. Math., 3 (1985), pp. 252-261. On The Convergence Of Diagonal Elements And Asymptotic Convergence Rates For The Shifted Tridiagonal QL Algorithm Er-Xiong Jiang 11 Fudan University, Shanghai, China Received 1984-9-28 Revised Online 2006-11-19 Abstract 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 $\alpha_1-\sigma$-›0 and $\beta_j$-›0, j=1,2,...,m, then $\alpha_j$-›$lambada_j$ where $\lambada_j$ are m eigenvalues of the matrix, and $\sigma$ is the origin shift. The asymptotic convergence rates of three kinds of shift, Rayleigh quotient shift Wilkinson's shift and RW shift, are analysed. Key words: