An Estimate of the Difference Between a Diagonal Element and the Corresponding Eigenvalue of a Symmetric Tridiagonal Matrix

Abstract

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 $\beta_1$ is replaced by zero and when two neighboring off-diagonal elements $ \beta_{i-1},\beta_i$ are replaced by zero.