Examples of the ``wanted set" specification are:
Other interesting choices of shifts include the roots of Chebyshev polynomials , harmonic Ritz values [331,337,349,411], the roots of Leja polynomials , the roots of least squares polynomials , and refined shifts . 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 .
There is an alternative way to apply implicit restart, thick restart, as described by Wu and Simon . 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.