I was wondering how the FEAST algorithm determines the multiplicity of the eigenvalues, for the case of repeated eigenvalues. More precisely, I would like to know if there exists a special tolerance for when two or more eigenvalues are treated as repeated.
The reason behind the questions is that I need the derivative of eigenvalues and thus special treatment is required for the case of repeated eigenvalues, since the eigenvectors of repeated eigenvalue can be linearly combined in an infinite number of ways.
You are correct, in case of repeated eigenvalue there are infinite numbers of sets of eigenvectors but all these sets are correct! So Extended EigenSolver routines will find one of the mentioned set. You can split eigenvalues in case they are not repeated by stopping criteria (fpm) - just set it less than difference between eigenvalue.