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An unfortunate aspect of the Lanczos/Arnoldi process is that
one cannot know in advance how many steps will be required before
eigenvalues of interest are well approximated by the Ritz values.
This is particularly true when the problem has a wide range of
eigenvalues but the eigenvalues of interest are clustered.
Without a spectral transformation, many Lanczos steps are required to obtain
the selected eigenvalues. In order to recover eigenvectors, one is
obliged either to store all of the Lanczos basis
vectors (usually on a peripheral device) or to re-compute them.
Also, very large tridiagonal eigenvalue problems will have to be solved
at each step. In the Arnoldi process that is used in the non-Hermitian
case, not only do the basis vectors have to be stored, but the cost
of the Hessenberg eigenvalue subproblem is at
the k-th step. The obvious need to control this cost has motivated the
development of restarting schemes.