Handbook of Linear Algebra, 2nd Edition

Chapter 59

Computing Eigenvalues and Singular Values to High Relative Accuracy

Zlatko Drmač

University of Zagreb

To compute the eigenvalues and singular values to high relative accuracy means to have a guaranteed number of accurate digits in all computed approximate values. If $\tilde{\lambda }$ is the computed approximation of an eigenvalue λ of a given matrix A, then the desirable high relative accuracy means that $\left|\lambda -\tilde{\lambda }\right|\le \epsilon |\lambda |$, where 0 ≤ ε ≪ 1 independent of the ratio |λ|/||A||2. This is not always possible. The proper course of action is to first determine classes of matrices and classes of perturbations under which the eigenvalues (singular values) undergo only small relative changes. This means that the development of highly accurate algorithms is determined ...