Improving projection-based eigensolvers via adaptive techniques

We consider subspace iteration (or projection-based) algorithms for computing those eigenvalues (and associated eigenvectors) of a Hermitian matrix that lie in a prescribed interval. For the case that the projector is approximated with polynomials, we present an adaptive strategy for selecting the degree of these polynomials such that convergence is achieved with near-to-optimum overall work without detailed a priori knowledge about the eigenvalue distribution. The idea is then transferred to the approximation of the projector by numerical integration, which corresponds to FEAST algorithm proposed by E. Polizzi in 2009. [E.Polizzi: Density-matrix-based algorithm for solving eigenvalue problems. Phys.Rev.B 2009; 79:115112]. Here, our adaptation controls the number of integration nodes. We also discuss the interaction of the method with search space reduction methods.