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# Incomplete Cholesky factorization

In numerical analysis, a field within mathematics, 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.

## References

• . See Section 10.3.2.

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