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#### Transfer Residual Errors to Backward Errors.

It turns out that the computed eigenvalue and eigenvector(s) can always be interpreted as the exact one of nearby matrices, i.e.,

and

if is available, where the error matrices are generally small in norm relative to . Such an interpretation serves two purposes: first, it reflects indirectly how accurately the eigenproblem has been solved; and second, it can be used to derive error bounds for the computed eigenvalues and eigenvectors to be discussed below. Ideally, we would like to be zero matrices, but this hardly ever happens at all in practice. There are infinitely many error matrices that satisfy the above equations, we would like to know only the optimal or nearly optimal error matrices in the sense that certain norms (usually the 2-norm or the Frobenius norm ) are minimized among all feasible error matrices. In fact, practical purposes will be served if we can determine upper bounds for the norms of these (nearly) optimal matrices. The following collection of results indeed shows that if (and if available) is small, the error matrix is small, too [425].

We distinguish two cases.

1. Only is available but is not. Then the optimal error matrix (in both 2-norm and the Frobenius norm) for which and are an exact eigenvalue and its corresponding eigenvector of , i.e.,
 (210)

satisfies

2. Both and are available. Then the optimal error matrices (in 2-norm) and (in the Frobenius norm) for which , , and are an exact eigenvalue and its corresponding eigenvectors of , i.e.,
 (211)

for , satisfy

and

See [256,431].

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