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#### Transfer Residual Errors to Backward Errors.

It can be shown that the computed eigenvalue and eigenvector(s) are the exact ones of a nearby matrix pair, i.e.,

and

if is available, where error matrices and are small relative to the norms of and .
1. Only is available but is not. Then the optimal error matrix (in both the 2-norm and the Frobenius norm) for which and are an exact eigenvalue and its corresponding eigenvector of the pair satisfies
 (241)

2. Both and are available. Then the optimal error matrices (in the 2-norm) and

(in the Frobenius norm) for which , , and are an exact eigenvalue and its corresponding eigenvectors of the pair satisfy
 (242)

and
 (243)

See [256,431,473].

We say the algorithm that delivers the approximate eigenpair is -backward stable for the pair with respect to the norm if it is an exact eigenpair for with ; analogously the algorithm that delivers the eigentriplet is -backward stable for the triplet with respect to the norm if it is an exact eigentriplet for with . With these in mind, statements can be made about the backward stability of the algorithm which computes the eigenpair or the eigentriplet . Conventionally, an algorithm is called backward stable if .

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