This routine forms an incomplete factorization of an sparse unsymmetric matrix . No ﬁll-in is allowed. The entries of are stored by rows. If has zeros on the diagonal, the routine ﬁrst ﬁnds a row permutation which makes the matrix have nonzeros on the diagonal. The incomplete factorization of the permuted matrix is then formed. is lower triangular and is unit upper triangular. The incomplete factorization may be used as a preconditioner when solving the linear system . A second entry performs the preconditioning operations
where is the preconditioner.