  In this paper, we propose a simple sparse approximate inverse for triangular
matrices (SAIT). Using the Jacobi iteration method, we obtain an expression of
the exact inverse of triangular matrix, which is a finite series. The SAIT is
constructed based on this series. We apply the SAIT matrices to iterative
methods with ILU preconditioners. The two triangular solvers in the ILU
preconditioning procedure are replaced by two matrix-vector multiplications,
which can be fine-grained parallelized. We test this method by solving some
linear systems and eigenvalue problems with preconditioned iterative methods.
