The Theory of Elimination Trees for Sparse Unsymmetric Matrices

Abstract

The elimination tree of a symmetric matrix plays an important role in sparse matrix factorization. By using paths instead of edges to define the tree, we generalize this structure to unsymmetric matrices while retaining many of its properties. If we use a tree traversal to reorder a matrix into a bordered block triangular form, the structure has further desirable properties relevant to a sparse LU factorization of the reordered matrix. When pivoting is required for stability, the tree changes only locally if the choice of pivot is suitably restricted.