On the Relation between the Ainv and the Fapinv Algorithms

The approximate inverse AINV and the factored approximate inverse FAPINV are two known algorithms in the field of preconditioning of linear systems of equations. Both of these algorithms compute a sparse approximate inverse of matrix A in the factored form and are based on computing two sets of vectors which are A-biconjugate. The AINV algorithm computes the inverse factors W and Z of a matrix independently of each other, as opposed to the AINV algorithm, where the computations of the inverse factors are done independently. In this paper, we show that, without any dropping, removing the dependence of the computations of the inverse factors in the FAPINV algorithm results in the AINV algorithm. distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Introduction
Consider the linear system of equations Ax b, 1.1 where the coefficient matrix A ∈ R n×n is nonsingular, large, sparse, and x, b ∈ R n .Such linear systems are often solved by Krylov subspace methods such as the GMRES see Saad and Schultz 1 , Saad 2 and the BiCGSTAB see van der Vorst 3 , Saad 2 methods in conjunction with a suitable preconditioner.A preconditioner is a matrix M such that Mu can be easily computed for a given vector u and system MAx Mb is easier to solve than 1.1 .Usually, to this end one intends to find M such that matrix M ≈ A −1 MA ≈ I n , where I n is the identity matrix.There are various methods to compute such an appropriate matrix see Benzi 4 , Benzi and Tuma 5 , Saad The main idea of the FAPINV algorithm was first introduced by Luo see Luo 8-10 .Then the algorithm was more investigated by Zhang in 12 .Since in this procedure the factorization is performed in backward direction, we call it BFAPINV for backward FAPINV algorithm.In 11 , Zhang proposed an alternative procedure to compute the factorization in the forward direction, which we call it FFAPINV for forward FAPINV algorithm.In 7 , Lee and Zhang showed that the BFAPINV algorithm is free from breakdown for Mmatrices.It can be easily seen that the FFAPINV algorithm is free from breakdown for Mmatrices, as well.In the left-looking AINV algorithm see Benzi and Tuma 13, 14 , the inverse factors are computed quite independently.In contrast, in the FFAPINV algorithm, the inverse factors W and Z are not computed completely independently of each other.In this paper, from the FFAPINV algorithm without any dropping, we obtain a procedure which bypasses this dependence.Then we show that this procedure is equivalent to the left-looking AINV algorithm.In the same way one can see that the right-looking AINV algorithm see Benzi and Tuma 13 can be obtained from BFAPINV algorithm.
In Section 2, we give a brief description of the FFAPINV algorithm.The main results are given in Section 3. Section 4 is devoted to some concluding remarks.

2.2
From the structure of the matrices W and Z, we have for some α i 's and β i 's, where e j is the jth column of the identity matrix.
First of all, we see that where A * j is the jth column of A. Therefore In the same manner where A j * is the jth row of A. Putting these results together gives the Algorithm 1 for computing the inverse factors of A. Some observation can be posed here.It can be easily seen that see, e.g., Salkuyeh 15 d j w j Az j z T j Az j w j Az j A j * z j w j A * j .

2.9
In this algorithm, the computations for the inverse factors Z and W are tightly coupled.This algorithm needs the columns of the strictly upper triangular part of A for computing Z and the strictly lower triangular part of A for computing W. A sparse approximate inverse of A in the factored form is computed by inserting some dropping strategies in Algorithm 1.

Main Results
At the beginning of this section we mention that all of the results presented in this section are valid only when we do not use any dropping.As we mentioned in the previous section the computations for the inverse factors Z and W are tightly coupled.In this section, we extract a procedure from Algorithm 1 such that the computations for the inverse factors are done independently.We also show that the resulting algorithm is equivalent to the left-looking AINV algorithm.
From WAZ D we have Z T AZ Z T W −1 D. Obviously, the right-hand side of the latter equation is a lower triangular matrix and diag Z T AZ diag Z T W −1 D .Therefore 3.1 Premultiplying both sides of 2.3 by z T k A, k 1, 2, . . ., j − 1, from the left, we obtain Taking into account 3.1 , we obtain Hence we can state a procedure for computing the inverse factor Z without need to the inverse factor W as follows: 1 By some modifications this algorithm can be converted in a simple form, avoiding extra computations.Letting q i z T i A, steps 3 -7 may be written as follows: vii d j : q j z j .
Obviously the parameter α i at step iii of this procedure can be computed via α i 1 d i q i z j .

3.5
We have AZ W −1 D. This shows that the matrix AZ is a lower triangular matrix.Therefore, since Z is a unit upper triangular matrix, we deduce

3.6
On the other hand from 2.9 , in step 7 of this procedure, we can replace d j : q j z j by d j : A j * z j .Now by using the above results we can summarized an algorithm for computing Z as in Algorithm 2. This algorithm is known as the left-looking AINV algorithm .We observe that the left-looking AINV algorithm can be extracted from the FFAPINV algorithm.This algorithm computes Z with working on rows of A. Obviously the factor W can be computed via this algorithm, working on rows of A T .In the same way, one can obtain the right-looking AINV algorithm from the BFAPINV algorithm.
In fact both of these methods compute lower unitriangular matrices W and Z T and a diagonal matrix D diag d 1 , d 2 , . . ., d n such that WAZ ≈ D. In this case, the matrix M ZD −1 W ≈ A may be used as a preconditioner for 1.1 .It is well-known that the AINV algorithm is free from breakdown for the class of H-matrices 13 .
2 .The factored approximate inverse FAPINV Lee and Zhang 6, 7 , Luo 8-10 , Zhang 11, 12 and the approximate inverse AINV see Benzi and Tuma 13, 14 are among the algorithms for computing an approximate inverse of A in International Journal of Mathematics and Mathematical Sciences the factored form.
Let W and Z be the inverse factors of A a ij , that is, 1 , z 2 , . . ., z n , and D diag d 1 , d 2 , . . ., d n , in which w i 's and z i 's are the rows and columns of W and Z, respectively.Using 2.1 we obtain