Fast Spectral Projection Algorithms for Density-Matrix Computations

We present a fast algorithm for the construction of a spectral projector. This algorithm allows us to compute the density matrix, as used in e.g the Kohn-Sham iteration, and so obtain the electron density. We compute the spectral projector by constructing the matrix sign function through a simple polynomial recursion. We present several matrix representations for fast computation within this recursion, using bases with controlled space-spatial frequency localization. In particular we consider wavelet and local cosine bases. Since spectral projectors appear in many contexts, we expect many additional applications of our approach. Subject Classification: 65D15 65T99 81C06 42C99 Keywords: spectral projectors, density matrix, fast algorithms, wavelets, partitioned SVD. 1 Introduction The goal of this paper is to introduce fast algorithms for computing spectral projectors. Although spectral projectors have a wide range of applications, we deal here primarily with density-matrix computati..