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  "body": "\n## Definition\n\nLet $A$ be a real symmetric [[positive definite|0KX66CZMBB]] $n \\times n$ [[sparse matrix|JGACZYEEY7]].\nLet $S \\subseteq \\set{(i, j) : 1 \\le j \\le i \\le n}$ be a set of positions that contains every diagonal position $(i, i)$.\nAn *incomplete Cholesky factorization* of $A$ with pattern $S$ is a [[lower triangular|GTG8MJGDKC]] matrix $L$ with positive diagonal such that:\n\n1. $l_{ij} = 0$ for every $i \\ge j$ with $(i, j) \\notin S$;\n2. $(L L^T)_{ij} = a_{ij}$ for every $(i, j) \\in S$.\n\nWhen $S$ is the set of positions $(i, j)$ with $i \\ge j$ and $a_{ij} \\ne 0$, the factorization is called *IC(0)*.\n\n## Remarks\n\nIf $S$ contains all positions with $i \\ge j$, the incomplete factor is the exact [[Cholesky factor|0WKG8CYY6A]].\nExample: let $A$ be tridiagonal and symmetric positive definite.\nIts exact Cholesky factor is lower bidiagonal, so it vanishes outside the pattern of $A$.\nHence the IC(0) factor exists and is the exact factor.\nNon-example: for the $2 \\times 2$ identity matrix, the matrix $L$ with rows $(1, 0)$ and $(1, 1)$ and the pattern of all positions satisfies clause 1.\nIt fails clause 2, since $(L L^T)_{21} = 1 \\ne 0 = a_{21}$.\nThe factor is computed by the Cholesky recurrences with each update outside $S$ discarded.\nFor a general symmetric positive definite $A$, the process can meet a nonpositive pivot, and then no such $L$ is produced.\nIt exists for every pattern when $A$ is a symmetric [[M-matrix|DNKXHMGQY1]], a result of Meijerink and van der Vorst.\nVariants choose $S$ by levels of fill or drop entries below a threshold.\n",
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