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Spectral Transformation
If the extreme eigenvalues are not well separated and when we want interior
eigenvalues,
it is greatly advantageous to replace
by a shiftandinvert operator

(28) 
for an appropriately chosen shift , for instance, in the interval
, where we are interested in knowing the
eigenvalues. The shiftandinvert operator has
the eigenvalues

(29) 
and now the eigenvalues that correspond to eigenvalues
close to the shift will be at the ends of the
spectrum and well separated from the rest; see [162].
If we use a shiftandinvert operator, we start the algorithm by factoring

(30) 
using some appropriate sparse Gaussian
elimination scheme. Here, is a permutation and is unit lower triangular.
If there are eigenvalues at both sides
of the shift , we cannot use
a scalar diagonal , but we have to make a symmetric
indefinite factorization, as in MA47 of Duff and Reid
[141]. Here is block diagonal with one by one
and two by two blocks, and we get the inertia of
as a byproduct. We can get a count of the number of
eigenvalues in an interval by recording the inertia of for two values at the ends of the interval. Obtaining this count is used
to make sure that no multiple eigenvalues are missed.
See §10.3 for further information about
sparse matrix factorizations.
During the actual iteration, we use the factors , , and , computing

(31) 
in the order indicated by the parentheses,
in step (5) of Algorithm 4.6.
Next: Reorthogonalization
Up: Lanczos Method A.
Previous: Multiple Eigenvalues.
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Susan Blackford
20001120