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The Lanczos recursion is constructed to make the basis orthogonal,
but this is true only for infinite precision computation. In the
algorithm we only make sure that the new vector is
orthogonal to working
precision to the two latest vectors and ,
and orthogonality to the earlier vectors follows from the symmetry of
and the recursion (4.10). As soon as one
eigenvalue converges, i.e., the Ritz pair has a small residual (), all the basis vectors get perturbations in the
direction of the eigenspace of the converged eigenvalue. As a result of this,
a duplicate copy of that eigenvalue will soon show up in
the tridiagonal matrix .
Paige  was the first to discover this; the reader is
referred to the monograph  for a detailed discussion.
Let us consider three different strategies to handle this.