Different Versions of ILU and IUL Factorizations Obtained from Forward and Backward Factored Approximate Inverse Processes—Part I

We present an incomplete UL (IUL) decomposition of matrix which is extracted as a by-product of BFAPINV (backward factored approximate inverse) process. We term this IUL factorization as IULBF. We have used ILUFF [3] and IULBF as left preconditioner for linear systems. Different versions of ILUFF and IULBF preconditioners are computed by using different dropping techniques. In this paper, we compare quality of different versions of ILUFF and IULBF preconditioners. 1. Introduction Consider the linear system of equations where the coefficient matrix is nonsymmetric, nonsingular, large, sparse, and . Suppose . Linear system is termed left preconditioned system of system (1.1) and matrix is called left preconditioner matrix [1]. System (1.2) is solved by Krylov subspace methods [1]. Suppose that matrix is nonsymmetric. Also, suppose that and are unit lower and upper triangular matrices, respectively, and is a diagonal matrix. FFAPINV (forward factored approximate inverse) Algorithm [2], computes matrices , , and such that relation holds. It is possible to obtain an incomplete (ILU) decomposition of matrix , as a by-product of FFAPINV process, such that is an unit lower triangular and is an upper triangular matrix and Matrix in (1.4) is called ILUFF preconditioner (ILU factorization obtained from forward factored approximate inverse process) [3]. The approximate inverse factors , and in (1.3) and , matrices in (1.4) satisfy the two following relations: In Algorithms 1 and 2, and refer to th column and th row of matrix A, respectively. Algorithm 1: ILUFF algorithm. Algorithm 2: IULBF algorithm. In Section 2 of this paper, we present different dropping strategies for , and , factors of ILUFF preconditioner. In Section 3, we first introduce the IULBF preconditioner and then, we present different dropping strategies for this preconditioner. In Section 4, we present numerical results. 2. Different Versions of ILUFF Preconditioner Algorithm 1,which has been presented in the next page, computes the ILUFF preconditioner. Suppose that and are the drop tolerance parameters for and matrices, respectively. We have used two strategies to drop entries of and vectors in ILUFF algorithm. (i) First Dropping Strategy In this strategy, only line 8 of Algorithm 1 will be run and line 10 will not. In this case, entries and , for are dropped when (ii) Second Dropping Strategy In this strategy, only line 10 of Algorithm 1 will be run and line 8 will not. In this case, the whole vectors and are computed as and then, entries and , for , are dropped when criterions (2.1) are