= Sparse Gaussian elimination
A sparse direct solver performs <Gaussian elimination> while storing and updating only the predicted nonzero entries. A symbolic phase predicts the factor pattern from an elimination ordering, and a numerical phase evaluates the factors. The ordering can greatly alter <fill-in> and arithmetic work. For symmetric positive-definite systems, <Cholesky decomposition> avoids numerical pivoting; general systems require a balance between sparsity and stable pivot selection.
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