HSL_MI35 computes an incomplete factorization preconditioner that may be used in the solution of weighted sparse linear least squares problems. Given an () sparse matrix and an diagonal matrix of weights , HSL_MI35 computes an incomplete factorization of the matrix
The matrix may be optionally reordered, scaled and, if necessary, shifted to avoid breakdown of the factorization. The computed lower triangular matrix is such that is an incomplete factorization of
where is a permutation matrix, is a diagonal scaling matrix and is a non-negative shift. The incomplete factorization may be used for preconditioning when solving the normal equations
A separate entry performs the preconditioning operation
where is the incomplete factorization preconditioner and .
The incomplete factorization is based on a matrix decomposition of the form
where is lower triangular with positive diagonal entries, is a strictly lower triangular matrix with small entries that is used to stabilize the factorization process, 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).
The incomplete factorization may be computed with or without forming explicitly; the advantage of the latter option being that less memory is required.