A column approximate minimum degree ordering algorithm

Sparse Gaussian elimination with partial pivoting computes the factorization <b>PAQ</b> = <b>LU</b> of a sparse matrix <b>A</b>, where the row ordering <b>P</b> is selected during factorization using standard partial pivoting with row interchanges. The goal is to select a column preordering, <b>Q</b>, based solely on the nonzero pattern of <b>A</b>, that limits the worst-case number of nonzeros in the factorization. The fill-in also depends on <b>P</b>, but <b>Q</b> is selected to reduce an upper bound on the fill-in for any subsequent choice of <b>P</b>. The choice of <b>Q</b> can have a dramatic impact on the number of nonzeros in <b>L</b> and <b>U</b>. One scheme for determining a good column ordering for <b>A</b> is to compute a symmetric ordering that reduces fill-in in the Cholesky factorization of <b>A<sup>T</sup>A</b>. A conventional minimum degree ordering algorithm would require the sparsity structure of <b>A<sup>T</sup>A</b> to be computed, which can be expensive both in terms of space and time since <b>A<sup>T</sup>A</b> may be much denser than <b>A</b>. An alternative is to compute <b>Q</b> directly from the sparsity structure of <b>A</b>; 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.

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