Perturbation of eigenpairs of factored symmetric tridiagonal matrices (Q1405733)

The solution of the tridiagonal eigenproblem is discussed for an indefinite symmetric matrix allowing a triangular factorization \(T=LDL^t\), where \(D\) is a diagonal matrix and \(L\) has \(1\)'s on the diagonal and is lower bidiagonal, \(L=I+ \tilde{L}\). From the study of the behaviour of the eigenvalues and eigenvectors under small changes in the nonzero entries of \(D\) and \(\tilde{L}\) the condition numbers are obtained. For the element growth in the factorization, it is shown that some eigenpairs are robust and others are sensitive. A \(4 \times 4\) example with huge element growth for which the standard multiplicative perturbation theory fails to predict that the two very small eigenvalues are determined to high relative accuracy is given. This example shows the advantage of the new method.