  We present an algorithm where only the Cholesky basis is determined in the
decomposition procedure. This allows for improved screening and a partitioned
matrix decomposition scheme, both of which significantly reduce memory usage
and computational cost. After the basis has been determined, an inner
projection technique is used to construct the Cholesky vectors. The algorithm
extends the application range of the methodology and is well suited for
multilevel methods. We apply the algorithm to systems with up to 80000 atomic
orbitals.
