An orthogonal high relative accuracy algorithm for the symmetric eigenvalue problem
Juan M. Molera
Departamento de Matematicas, Universidad Carlos III de Madrid
Coauthors: Froilan M. Dopico, Julio Moro
Abstract: A high relative accuracy algorithm is proposed for computing eigenvalues and eigenvectors of general symmetric matrices factorized in rank-revealing form. Unlike previous algorithms dealing with indefinite matrices, this algorithm uses transformations which are orthogonal in the usual sense. The algorithm is divided into two independent stages: in the first one, singular values and vectors are computed using any orthogonal high relative accuracy SVD algorithm producing multiplicative backward errors. In the second stage, the appropriate sign is assigned to each singular value, and eigenvectors are obtained from the corrresponding singular vectors.
A detailed error analysis of the algorithm is presented, making use of a new perturbation theory of simultaneous bases of singular subspaces, which is crucial to prove the high relative accuracy of the second stage of the algorithm. Numerical experiments will also be presented, comparing the accuracy of the proposed algorithm with previously existing ones.