Examples of the ``wanted set" specification are:
Other interesting choices of shifts
include the roots of Chebyshev
polynomials [383], harmonic Ritz
values [331,337,349,411], the roots of Leja
polynomials [23], the roots of least squares
polynomials [384], and refined shifts [244]. In
particular, the Leja and harmonic Ritz values have been used to
estimate the interior eigenvalues of
.
An alternate interpretation stems from the fact that
each of these shift cycles results in the implicit application of a
polynomial in of degree
to the starting vector:
v_1 (A) v_1 with () = _j=1^p (- _j ).
The roots of this polynomial are the shifts used in
the QR process and these may be selected to
enhance components of the starting vector in the direction of eigenvectors
corresponding to desired eigenvalues and damp the components in unwanted
directions. Of course, this is desirable because it forces
the starting vector into an invariant subspace associated with the
desired eigenvalues. This in turn forces
to become small, and hence
convergence results. Full details may be found in [419].
There is an alternative way to apply implicit restart,
thick restart, as described by Wu and Simon [463].
There an eigenvalue factorization of (4.20) is
computed, and the part that corresponds to the wanted
eigenvalues is kept as an arrow matrix and the rest is discarded.
Thick restart is mathematically
equivalent to exact shift IRLM, if the same choice of wanted and
unwanted eigenvalues is made.