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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 \bulletLT = 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}}

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

end (i-loop)

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