Accurate Eigensystem Computations by Jacobi Methods

Demmel and Veselic showed that, subject to a minor proviso, Jacobi's method computes the eigenvalues and eigenvectors of a positive definite matrix more accurately than methods that first tridiagonalize the matrix. We extend their analysis and thereby: 1. We remove the minor proviso in their results and thus guarantee the accuracy of Jacobi's method. 2. We show how to cheaply check, a posteriori, whether tridiagonalizing a particular matrix has caused a large relative perturbation in the eigenvalues on the matrix. This can be useful when dealing with graded matrices. 3. We derive hybrid Jacobi algorithms that have the same accuracy of Jacobi's method but are faster, at least on a serial computer. 4. We show that if $G$ is an $m \times n$ matrix and $m > > n$ then Jacobi's method computes the singular values almost as quickly as standard methods, but potentially much more accurately.