A simulated annealing based algorithm for eigenvalue problems

Abstract

We present a simple, iterative algorithm to find the lowest (few) eigenvalue(s) and eigenvector(s) of a large sparse real Hermitian matrix. The algorithm is an application of the simulated annealing technique implemented via molecular dynamics. We also demonstrate that the slow rate of convergence, characteristic of the late-time behaviour of such annealing based methods, can be considerably improved by using simple gradient minimization schemes. Our studies are based on a large number of real Hermitian matrices of dimensions up to 15876 arising out of our investigations of strongly-correlated systems using the finite-cluster approach. We also investigate the effect of the controlling parameters (time step and fictitious mass) on the convergence and suggest a way to choose these parameters a priori.