Perturbation in eigenvalues of a symmetric tridiagonal matrix

We study the eigenvalue perturbations of an n × n real unreduced symmetric tridiagonal matrix T when one of the off-diagonal element is replaced by zero. We provide both the lower and upper perturbation bounds for every eigenvalue of T. The bounds are described by the jth off-diagonal element (the one that is replaced), and the eigenvalues and eigenvectors of the leading j × j and trailing (n − j) × (n − j) principal submatrices of T. We also provide several simpler perturbation bounds that are easy to estimate in practice. Numerical examples show that the bounds predict the perturbations well. They are sharper than whose classical results only related to the off-diagonal element, especially for extreme eigenvalues. The bounds can also be incorporated with numerical methods, such as the QL(QR) algorithm and the divide-conquer algorithm, to estimates the errors of computed eigenvalues.