Backward errors for eigenvalue and singular value decompositions
Bounds on the backward errors for the symmetric eigenvalue decomposition and the singular value decomposition in the two-norm and in the Frobenius norm are presented. When the computed eigenvectors have a small residual and are close to orthonormal then all backward errors tend to be small. Hence it does not matter how exactly a backward error is defined and how exactly residual and deviation from orthogonality are measured. The effects of the error bounds on implementation for eigenvectors and singular vectors are indicated. Finally, it is shown that the distance of an appropriately scaled matrix to its orthogonal QR factor is not much larger than its distance to the closest orthogonal matrix.
scientific article; zbMATH DE number 691537