  We introduce the strong recursive skeletonization factorization (RS-S), a new
approximate matrix factorization based on recursive skeletonization for solving
discretizations of linear integral equations associated with elliptic partial
differential equations in two and three dimensions (and other matrices with
similar hierarchical rank structure). Unlike previous skeletonization-based
factorizations, RS-S uses a simple modification of skeletonization, strong
skeletonization, which compresses only far-field interactions. This leads to an
approximate factorization in the form of a product of many block
unit-triangular matrices that may be used as a preconditioner or
moderate-accuracy direct solver, with dramatically reduced rank growth. We
further combine the strong skeletonization procedure with alternating
near-field compression to obtain the hybrid recursive skeletonization
factorization (RS-WS), a modification of RS-S that exhibits reduced storage
cost in many settings. Under suitable rank assumptions both RS-S and RS-WS
exhibit linear computational complexity, which we demonstrate with a number of
numerical examples.
