I am now continuing my work on direction fields (wrote a topic here a few weeks ago and received very nice feedback and help).

This time around I am doing some energy minimization to find optimal places to place singularities for my direction field. I am trying to implement the following paper:
http://www.cs.columbia.edu/~keenan/Proj ... /paper.pdf

Note that since we need only the smallest eigenvalue, we do not require a sophisticated eigensolver like ARPACK, making our method particularly easy to implement. To minimize alignment energy we performed factorization and a single back-substitution.

As the quote above says (from the paper above) I wish to find the smallest eigenvalue for A x = lambda B x where A and B are sparse matrixes, A is positive definite, B is symmetric and x is a dense initial guess.

I've seen that Eigen has some eigensolvers, would it be possible to use one of them for my problem?

You cannot use Eigen's eigensolver directly. However, writing a solver based on Power iterations is straightforward. There are examples there:

viewtopic.php?f=74&t=108033#p251878

and there:

viewtopic.php?f=74&t=117176&p=290601&hilit=power+iteration#p290620

However, here you need the smallest eigenvalue of a generalized eigenvalue problem, so you need some adaptations of these example to apply the inverse of A instead of A by prefactoring it with, e.g., SparseLDLT and then normalize y with respect to the bilinear form B.