An Orthogonal High Relative Accuracy Algorithm for the Symmetric Eigenproblem

We propose a new algorithm for the symmetric eigenproblem that computes eigenvalues and eigenvectors with high relative accuracy for the largest class of symmetric, definite and indefinite, matrices known so far. The algorithm is divided into two stages: the first one computes a singular value decomposition (SVD) with high relative accuracy, and the second one obtains eigenvalues and eigenvectors from singular values and vectors. The SVD, used as a first stage, is responsible both for the wide applicability of the algorithm and for being able to use only orthogonal transformations, unlike previous algorithms in the literature. Theory, a complete error analysis, and numerical experiments are presented.