An Estimate of the Difference Between a Diagonal Element and the Corresponding Eigenvalue of a Symmetric Tridiagonal Matrix (Journal of Computational Mathematics, 1987)
A sharper upperbound of the difference between a diagonal element and the corresponding eigenvalue of a symmetric tridiagonal matrix is given. The bound can be used in QL and QR algorithms and Rayleigh quotient approximation. The change of eigenvalues is estimated when the first off-diagonal element $β_1$ is replaced by zero and when two neighboring off-diagonal elements $ β_{i-1},β_i$ are replaced by zero.