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Suppose that is an eigenpair of
It follows easily from equation (7.29) that
Thus, if the last components of are small relative to the size of
, then the Ritz pair
is an exact eigenpair for a matrix near
This follows since (7.30) may be rewritten as
The iteration in BIRAM terminates at the value of when the wanted
eigenvalues of satisfy (7.30). The eigenvalues
are partitioned as in (7.31) so that the wanted ones correspond to
the eigenvalues of desired.