A Necessary and Sufficient Symbolic Condition for the Existence of Incomplete Cholesky Factorization

This paper presents a suucient condition on sparsity patterns for the existence of the incomplete Cholesky factorization. Given the sparsity pattern P(A) of a matrix A, and a target sparsity pattern P satisfying the condition, incomplete Cholesky factorization successfully completes for all symmetric positive deenite matrices with the same pattern P(A). This condition is also necessary in the sense that for a given P(A) and target pattern P, if P does not satisfy the condition then there is at least one symmetric positive deenite matrix B whose Cholesky factor has the same sparsity pattern as the Cholesky factor of A, for which incomplete Cholesky factorization fails because of a nonpositive pivot.