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  "body": "\n## Definition\n\nLet $A$ be a real $n \\times n$ [[sparse matrix|JGACZYEEY7]].\nLet $P \\subseteq \\set{1, \\dots, n}^2$ be a set of positions that contains every diagonal position $(i, i)$.\nAn *incomplete LU factorization* of $A$ with pattern $P$ is a pair $(L, U)$ such that:\n\n1. $L$ is unit [[lower triangular|GTG8MJGDKC]] and $U$ is upper triangular;\n2. $l_{ij} = 0$ and $u_{ij} = 0$ for every position $(i, j) \\notin P$;\n3. $(L U)_{ij} = a_{ij}$ for every $(i, j) \\in P$.\n\nWhen $P$ is the set of positions with $a_{ij} \\ne 0$ together with the diagonal, it is called *ILU(0)*.\n\n## Remarks\n\nIf $P$ is the set of all positions and $A$ has an [[LU decomposition|0RKG5YW66Y]] with unit lower triangular $L$, the two factorizations coincide.\nExample: if $A$ is tridiagonal and has an LU decomposition, its exact factors are bidiagonal.\nThey vanish outside the pattern of $A$, so they form the ILU(0) factorization.\nNon-example: let $A$ be the $2 \\times 2$ identity, let $P$ be all positions, and let $L$ have rows $(1, 0)$ and $(1, 1)$, with $U = I$.\nClauses 1 and 2 hold, but clause 3 fails, since $(L U)_{21} = 1 \\ne 0 = a_{21}$.\nThe factors are computed by Gaussian elimination in which each update outside $P$ is discarded.\nFor a general matrix a zero pivot can occur, and then the factorization does not exist.\nIt exists for every pattern when $A$ is an [[M-matrix|DNKXHMGQY1]].\nFor symmetric positive definite $A$ the symmetric analogue is the [[incomplete Cholesky factorization|QYYCFQ8E5D]].\nVariants use levels of fill, written ILU($k$), or thresholds, written ILUT.\n",
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