# Incomplete Cholesky factorization

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## Cholesky Factorization

When the square matrix A is symmetric and positive definite then it has an efficient triangular decomposition. Symmetric means that aij = aji for i,j = 1, ... , N. While positive definite means that

$v \bullet A \bullet v > 0$ $\forall v$

In cholesky factorization we construct a lower triangular matrix L whose transpose LT can itself serve as upper triangular part.
In other words we have
L $\bullet$LT = A

### Algorithm for full matrix A

We have by definition $L_{ij}^T = L_{ji}$
From this we can easily obtain

for := 1 step 1 until N do

$L_{ii} = \left( {a_{ii} - \sum\limits_{k = 1}^{i - 1} {L_{ik}^2 } } \right)^{{1 \over 2}}$
and
$L_{ji} = {1 \over {L_{ii} }}\left( {a_{ij} - \sum\limits_{k = 1}^{i - 1} {L_{ik} L_{jk} } } \right)$ ; where j = i+1, i+2, ..., N

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