Sparse Gaussian elimination with partial pivoting computes the factorization PAQ = LU of a sparse matrix A , where the row ordering P is selected during factorization using standard partial pivoting with row interchanges. The goal is to select a column preordering, Q , based solely on the nonzero pattern of A , that limits the worst-case number of nonzeros in the factorization. The fill-in also depends on P , but Q is selected to reduce an upper bound on the fill-in for any subsequent choice of P . The choice of Q can have a dramatic impact on the number of nonzeros in L and U . One scheme for determining a good column ordering for A is to compute a symmetric ordering that reduces fill-in in the Cholesky factorization of A T A . A conventional minimum degree ordering algorithm would require the sparsity structure of A T A to be computed, which can be expensive both in terms of space and time since A T A may be much denser than A . An alternative is to compute Q directly from the sparsity structure of A ; this strategy is used by MATLAB's COLMMD preordering algorithm. A new ordering algorithm, COLAMD, is presented. It is based on the same strategy but uses a better ordering heuristic. COLAMD is faster and computes better orderings, with fewer nonzeros in the factors of the matrix.

Two codes are discussed, COLAMD and SYMAMD, that compute approximate minimum degree orderings for sparse matrices in two contexts: (1) sparse partial pivoting, which requires a sparsity preserving column pre-ordering prior to numerical factorization, and (2) sparse Cholesky factorization, which requires a symmetric permutation of both the rows and columns of the matrix being factorized. These orderings are computed by COLAMD and SYMAMD, respectively. The ordering from COLAMD is also suitable for sparse QR factorization, and the factorization of matrices of the form A T A and AA T , such as those that arise in least-squares problems and interior point methods for linear programming problems. The two routines are available both in MATLAB and C-callable forms. They appear as built-in routines in MATLAB Version 6.0.