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  "body": "\n## Summary\n\nAn incomplete LU factorization of a sparse square matrix $A$ is a pair of sparse triangular matrices $L$ and $U$ with $L U \\approx A$.\nIt is computed like the [[LU decomposition|0RKG5YW66Y]], but entries outside a prescribed sparsity pattern are dropped.\nIt is one of the most common [[preconditioners|KE98SW3SBD]] for Krylov methods on nonsymmetric systems.\n\n## Explanations\n\n### L1 human\n\nExact elimination on a sparse matrix creates many new nonzeros, called fill.\nThe incomplete factorization throws away the fill that falls outside allowed positions.\nThe factors stay as sparse as $A$, or nearly so, and $L U$ is close to $A$.\nSolving with $L$ and then $U$ is cheap, so $L U$ acts as an easy approximate inverse inside an iterative solver.\n",
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