We present a new numerical technique to solve large-scale eigenvalue problems. It is based on the projection technique, used in strongly correlated quantum many-body systems, where first an effective approximate model of smaller complexity is constructed by projecting out high energy degrees of freedom and in turn solving the resulting model by some standard eigenvalue solver. Here we introduce a generalization of this idea, where both steps are performed numerically and which in contrast to the standard projection technique converges in principle to the exact eigenvalues. This approach is not just applicable to eigenvalue problems encountered in many-body systems but also in other areas of research that result in large scale eigenvalue problems for matrices which have, roughly speaking, mostly a pronounced dominant diagonal part. We will present detailed studies of the approach guided by two many-body models.
Gamillscheg, R., Haase, G., & von der Linden, W. (2011). A numerical projection technique for large-scale eigenvalue problems. Computer physics communications, 182(10), 2168-2173. https://doi.org/10.1016/j.cpc.2011.05.016