The design and implementation of the MRRR algorithm
ACM Transactions on Mathematical Software (TOMS)
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Let $T = L \Omega L^t$ be an invertible, unreduced, indefinite tridiagonal symmetric matrix with $\Omega$ a diagonal signature matrix. We provide error bounds on the (relative) change in an eigenvalue and the angular change in its eigenvector when the entries in $L$ suffer small relative changes. Our results extend those of Demmel and Kahan for $\Omega = I$. The relative condition number for an eigenvalue exceeds by 1 its absolute condition number as an eigenvalue of $\Omega L^t L$. The condition number of an eigenvector is a weighted sum of the relative separations of the eigenvalue from each of the others.A small example shows that very small eigenpairs can be robust even when the large eigenvalues are extremely sensitive. When $L$ is well conditioned for inversion, then all eigenvalues are robust and the eigenvectors depend only on the relative separations.