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#### Error Bounds for Computed Eigenvalues.

For the Hermitian eigenproblem, the th largest eigenvalue of differs from the th largest eigenvalue of by at most . Therefore a small backward error implies a small (forward) error in the computed eigenvalue, i.e.,
 (63)

for some eigenvalue of .

With more information, a better error bound can be obtained. Let us assume that is an approximation of the eigenpair of . The best'' corresponding to is the Rayleigh quotient , so we assume that has this value. Suppose that is closer to than any other eigenvalues of , and let be the gap between and any other eigenvalue: . Then we have

 (64)

This improves (4.54) if the gap is reasonably big. In practice we can always pick the better one.

Note that (4.55) needs information on , besides the residual error . Usually such information is available after a successful computation by, e.g., a Lanczos algorithm with SI, which usually delivers eigenvalues in the neighborhood of a shift and consequently yields good information on . This comment also applies to the bound in (4.56) below.

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