Computing Eigenvalues and Singular Values to High Relative Accuracy

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 λ ˜ i https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429138492/cc3be78c-f644-49c2-b40d-db06d778c1a5/content/eq6802.tif"/> is the computed approximation of λ_i , then the desirable high relative accuracy means that |λ_i – λ ˜ i https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429138492/cc3be78c-f644-49c2-b40d-db06d778c1a5/content/eq6803.tif"/> |≤ ε |λ_i |, where 0 ≤ ε ≪ 1 independent of the ratio |λ_i|/max _j |λ _j|. 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 by the framework established by the perturbation theory.