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

A sparse approximate inverse for triangular matrices based on Jacobi iteration