  Computing more than one eigenvalue for (large sparse) one-parameter
polynomial and general nonlinear eigenproblems, as well as for multiparameter
linear and nonlinear eigenproblems, is a much harder task than for standard
eigenvalue problems. We present simple but efficient selection methods based on
divided differences to do this. In contrast to locking techniques, it is not
necessary to keep converged eigenvectors in the search space, so that the
entire search space may be devoted to new information. The techniques are
applicable to many types of matrix eigenvalue problems; standard deflation is
possible only for linear one-parameter problems. The methods are easy to
understand and implement. Although divided differences are well-known in the
context of nonlinear eigenproblems, the proposed selection techniques are new
for one-parameter problems. For multiparameter problems, we improve on and
generalize our previous work. We also show how to use divided differences in
the framework of homogeneous coordinates, which may be appropriate for
generalized eigenvalue problems with infinite eigenvalues.
  While the approaches are valuable alternatives for one-parameter nonlinear
eigenproblems, they seem the only option for multiparameter problems.
