A New Recursive Implementation of Sparse Cholesky Factorization

Consider the Cholesky factorization of a sparse symmetric positive de nite matrix, A = LL . The rst two steps use symbolic, graph-theoretic techniques to order A to reduce ll in L, and to determine the exact sparsity structure of L. The factor L is computed in a third \numeric factorization" step. The two leading schemes for numeric factorization are a blocked column-oriented scheme, and a multifrontal implementation. We propose a new recursive implementation that could be viewed as a hybrid of these two schemes. The new scheme seeks to eÆciently access the memory hierarchy on modern computers by a simple recursion on a \supernodal tree" associated with L. Consider diagonal blocks in L numbered in post-order on the supernodal tree; now the recursive formulation is equivalent to processing a sequence of dense diagonal blocks in L from the top left to the bottom right. Unlike the multifrontal scheme, the new scheme does not require extra stack storage.