On the generalised eigenvalue method and its relation to Prony and generalised pencil of function methods

We discuss the relation of a variety of different methods to determine energy levels in lattice QCD simulations: the generalised eigenvalue, the Prony, the generalised pencil of function and the Gardner methods. All three former methods can be understood as special cases of a generalised eigenvalue problem. We show analytically that the leading corrections to an energy E-l in all three methods due to unresolved states decay asymptotically exponentially like exp(-(E-n - E-l)t). Using synthetic data we show that these corrections behave as expected also in practice. We propose a novel combination of the generalised eigenvalue and the Prony method, denoted as GEVM/PGEVM, which helps to increase the energy gap E-n - E-l. We illustrate its usage and performance using lattice QCD examples. The Gardner method on the other hand is found less applicable to realistic noisy data.