Applying Schur Complements for Handling

We describe a set of procedures for computing and updating an inverse representation of a large and sparse unsymmetric matrix A. The representation is built of two matrices: an easily invertible, large and sparse matrix A 0 and a dense Schur complement matrix S. An eecient heuristic is given that nds this representation for any matrix A and keeps the size of S as small as possible. Equations with A are replaced with a sequence of equations that involve matrices A 0 and S. The former take full advantage of the sparsity of A 0 ; the latter beneet from applying dense linear algebra techniques to a dense representation of the inverse of S. We show how to manage ve general updates of A: row or column replacement, row and column addition or deletion and a rank one correction. They all maintain an easily invertible form of A 0 and reduce to some low rank update of matrix S. An experimental implementation of the approach is described and the preliminary computational results of applying it to handling working basis updates that arise in linear programming are given.