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#### Transfer Residual Error to Backward Error.

There are Hermitian matrices such that and are an exact eigenvalue and its corresponding eigenvector of , i.e.,

One such is
 (61)

We are interested in such matrices with smallest possible norms. It turns out the best possible for the spectral norm and the best possible for Frobenius norm satisfy
 (62)

see, e.g., [256,431]. In fact, is given explicitly by (4.52). So if is small, the computed and are an exact eigenpair of nearby matrices. Error analysis of this kind is called backward error analysis, and matrices are backward errors.

We say an algorithm that delivers an approximate eigenpair is -backward stable for the pair with respect to the norm if it is an exact eigenpair for with . With this definition in mind, statements can be made about the numerical stability of the algorithm which computes the eigenpair . By convention, an algorithm is called backward stable if , where is the machine precision.

Next: Error Bounds for Computed Up: Stability and Accuracy Assessments Previous: Residual Vector.   Contents   Index
Susan Blackford 2000-11-20