Sears–Haack body

In numerical analysis, an incomplete Cholesky factorization of a symmetric positive definite matrix is a sparse approximation of the Cholesky factorization. Incomplete Cholesky factorization are often used as a preconditioner for algorithms like the conjugate gradient method.

The Cholesky factorization of a positive definite matrix A is A = LL* where L is a lower triangular matrix. An incomplete Cholesky factorization is given by a sparse lower triangular matrix K that is in some sense close to L. The corresponding preconditioner is KK*.

One popular way to find such a matrix K is to use the algorithm for finding the exact Cholesky decomposition, except that any entry is set to zero if the corresponding entry in A is also zero. This gives an incomplete Cholesky factorization which is as sparse as the matrix A.

Algorithm

For $i$ from $1$ to $N$:

$L_{ii}=\left({a_{ii}-\sum \limits _{k=1}^{i-1}{L_{ik}^{2}}}\right)^{1 \over 2}$

For $j$ from $i+1$ to $N$:

$L_{ji}={1 \over {L_{ii}}}\left({a_{ij}-\sum \limits _{k=1}^{i-1}{L_{ik}L_{jk}}}\right)$

References

• Incomplete Cholesky factorization] at CFD Online wiki
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