Let be a sparse symmetric saddle-point matrix of the form
where is symmetric positive deﬁnite, is rectangular and of full rank (), and is symmetric positive semi-deﬁnite. HSL_MI30 computes a signed incomplete Cholesky factorization. The matrix is optionally reordered, scaled and, if necessary, shifted to avoid breakdown of the factorization so that the incomplete factorization of
is computed, where is a permutation matrix, is a diagonal scaling matrix and are non-negative shifts.
The incomplete factorization may be used for preconditioning when solving the saddle-point system . A separate entry performs the preconditioning operation
where , with , is the incomplete signed Cholesky factorization preconditioner. An option exists to use as the preconditioner.
The incomplete factorization is based on a matrix decomposition of the form
where is lower triangular with unit diagonal entries, is a strictly lower triangular matrix with small entries that is used to stabilize the factorization process, is a diagonal matrix, and has the structure
where is strictly lower triangle. is discarded while is used in the computation of but is then discarded. The user controls the dropping of small entries from and and the maximum number of entries within each column of and (and thus the amount of memory for and the intermediate work and memory used in computing the incomplete factorization).
Note: If an incomplete Cholesky factorization preconditioner for a positive-deﬁnite system is required, HSL_MI28 should be used.