Computing partial spectra with least-squares rational filters

We present a method for computing partial spectra of Hermitian matrices, based on a combination of subspace iteration with rational filtering. In contrast with classical rational filters derived from Cauchy integrals or from uniform approximations to a step function, we adopt a least-squares (LS) viewpoint for designing filters. One of the goals of the proposed approach is to build a filter that will lead to linear systems that are easier to solve by iterative methods. Among the main advantages of the proposed LS filters is their flexibility. Thus, we can place poles in more general locations than with the formulations mentioned above, and we can also repeat these poles a few times for better efficiency. This leads to a smaller number of required poles than in existing methods. As a consequence, factorization costs are reduced when direct solvers are used and the scheme is also beneficial for iterative solvers. The paper discusses iterative schemes to solve the linear systems resulting from the filtered subspace iteration that take advantage of the nature and structure of the proposed LS filters. The efficiency and robustness of the resulting method is illustrated with a few model problems as well as several Hamiltonian matrices from electronic structure calculations.