Sparse Gaussian Elimination

In this paper we consider two structure prediction problems of interest in Gaussian elimination with partial pivoting of sparse matrices. First, we consider the problem of determining the nonzero structure of the factors $L$ and $U$ during the factorization. We present an exact prediction of the structure that identifies some numeric cancellations appearing during Gaussian elimination. The numeric cancellations are related to submatrices of the input matrix $A$ that are structurally singular, that is, singular due to the arrangements of their nonzeros, and independent of their numerical values. Second, we consider the problem of estimating upper bounds for the structure of $L$ and $U$ prior to the numerical factorization. We present tight exact bounds for the nonzero structure of $L$ and $U$ of Gaussian elimination with partial pivoting $PA = LU$ under the assumption that the matrix $A$ satisfies a combinatorial property, namely, the Hall property, and that the nonzero values in $A$ are algebraically independent of each other. This complements existing work showing that a structure called the row merge graph represents a tight bound for the nonzero structure of $L$ and $U$ under a stronger combinatorial assumption, namely, the strong Hall property. We also show that the row merge graph represents a tight symbolic bound for matrices satisfying only the Hall property.