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.