On the Relation between the AINV and the FAPINV Algorithms

Abstract

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.

1. 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 [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 the factored form. In fact both of these methods compute lower unitriangular matrices 𝑊 and 𝑍^𝑇 and a diagonal matrix 𝐷=diag(𝑑₁,𝑑₂,…,𝑑_𝑛) such that 𝑊𝐴𝑍≈𝐷. In this case, the matrix 𝑀=𝑍𝐷⁻¹𝑊≈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].

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 𝑀-matrices. It can be easily seen that the FFAPINV algorithm is free from breakdown for 𝑀-matrices, 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. A Review of the FFAPINV Algorithm

Let 𝑊 and 𝑍 be the inverse factors of 𝐴=(𝑎_𝑖𝑗), that is, 𝑊𝐴𝑍=𝐷, (2.1) where 𝑊=(𝑤^𝑇₁,𝑤^𝑇₂,…,𝑤^𝑇_𝑛)^𝑇, 𝑍=(𝑧₁,𝑧₂,…,𝑧_𝑛), and 𝐷=diag(𝑑₁,𝑑₂,…,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 𝑤_𝑖𝐴𝑧_𝑗=𝑑_𝑖,𝑖=𝑗,0,𝑖≠𝑗. (2.2) From the structure of the matrices 𝑊 and 𝑍, we have 𝑧₁=𝑒₁,𝑧_𝑗=𝑒_𝑗−^𝑗−1_𝑖=1𝛼_𝑖𝑧_𝑖𝑤,𝑗=2,…,𝑛,₁=𝑒^𝑇₁,𝑤_𝑗=𝑒^𝑇_𝑗−^𝑗−1_𝑖=1𝛽_𝑖𝑤_𝑖,𝑗=2,…,𝑛, (2.3)(2.4) 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 𝑑₁=𝑧^𝑇₁𝐴𝑧₁=𝑒^𝑇₁𝐴𝑒₁=𝑎₁₁. (2.5) Now let 2≤𝑗≤𝑛 be fixed. Then from (2.2) and (2.3) and for 𝑘=1,…,𝑗−1, we have 0=𝑤_𝑘𝐴𝑧_𝑗=𝑤_𝑘𝐴𝑒_𝑗−^𝑗−1_𝑖=1𝛼_𝑖𝑤_𝑘𝐴𝑧_𝑖=𝑤_𝑘𝐴_∗𝑗−𝛼_𝑘𝑤_𝑘𝐴𝑧_𝑘=𝑤_𝑘𝐴_∗𝑗−𝛼_𝑘𝑑_𝑘, (2.6) where 𝐴_∗𝑗 is the 𝑗th column of 𝐴. Therefore 𝛼_𝑖=1𝑑_𝑖𝑤_𝑖𝐴_∗𝑗,𝑖=1,…,𝑗−1. (2.7) In the same manner 𝛽_𝑖=1𝑑_𝑖𝐴_𝑗∗𝑧_𝑖,𝑖=1,…,𝑗−1, (2.8) where 𝐴_𝑗∗ is the 𝑗th row of 𝐴. Putting these results together gives the Algorithm 1 for computing the inverse factors of 𝐴.

(1) 𝑧1∶=𝑒1, 𝑤1∶=𝑒𝑇1 and 𝑑1∶=𝑎11 (2) For 𝑗=2,…,𝑛, Do (3)  𝑧𝑗∶=𝑒𝑗;  𝑤𝑗∶=𝑒𝑇𝑗 (4) For 𝑖=1,…,𝑗−1, Do (5)   𝛼𝑖∶=(1/𝑑𝑖)𝑤𝑖𝐴∗𝑗;  𝛽𝑖∶=(1/𝑑𝑖)𝐴𝑗∗𝑧𝑖 (6)   𝑧𝑗∶=𝑧𝑗−𝛼𝑖𝑧𝑖;  𝑤𝑗∶=𝑤𝑗−𝛽𝑖𝑤𝑖 (7) EndDo (8)  𝑑𝑗∶=𝑤𝑗𝐴𝑧𝑗 (9) EndDo

Algorithm 1 

The FFAPINV algorithm without dropping.

Some observation can be posed here. It can be easily seen that (see, e.g., Salkuyeh [15]) 𝑑_𝑗=𝑤_𝑗𝐴𝑧_𝑗=𝑧^𝑇_𝑗𝐴𝑧_𝑗=𝑤_𝑗𝐴𝑧_𝑗=𝐴_𝑗∗𝑧_𝑗=𝑤_𝑗𝐴_∗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.

3. 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^𝑇𝐴𝑍=𝑍^𝑇𝑊⁻¹𝐷. Obviously, the right-hand side of the latter equation is a lower triangular matrix and diag(𝑍^𝑇𝐴𝑍)=diag(𝑍^𝑇𝑊⁻¹𝐷). Therefore 𝑧^𝑇_𝑖𝐴𝑧_𝑗=𝑑_𝑖,𝑖=𝑗,0,𝑖<𝑗. (3.1) Premultiplying both sides of (2.3) by 𝑧^𝑇_𝑘𝐴, 𝑘=1,2,…,𝑗−1, from the left, we obtain 𝑧^𝑇_𝑘𝐴𝑧_𝑗=𝑧^𝑇_𝑘𝐴𝑒_𝑗−^𝑗−1_𝑖=1𝛼_𝑖𝑧^𝑇_𝑘𝐴𝑧_𝑖. (3.2) Taking into account (3.1), we obtain 0=𝑧^𝑇_𝑘𝐴𝑒_𝑗−^𝑘−1_𝑖=1𝛼_𝑖𝑧^𝑇_𝑘𝐴𝑧_𝑖−𝛼_𝑘𝑧^𝑇_𝑘𝐴𝑧_𝑘. (3.3) Therefore 𝛼_𝑘=1𝑑_𝑘𝑧^𝑇_𝑘𝐴(𝑒_𝑗−^𝑘−1_𝑖=1𝛼_𝑖𝑧_𝑖). (3.4) Hence we can state a procedure for computing the inverse factor 𝑍 without need to the inverse factor 𝑊 as follows:

(1)𝑧₁∶=𝑒₁, 𝑑₁∶=𝑎₁₁(2)For 𝑗=2,…,𝑛, Do(3)  For 𝑖=1,…,𝑗−1, Do(4)   𝛼_𝑖∶=(1/𝑑_𝑖)𝑧^𝑇_𝑖𝐴(𝑒_𝑗−∑^𝑖−1_𝑘=1𝛼_𝑘𝑧_𝑘)(5)  EndDo(6)  𝑧_𝑗∶=𝑒_𝑗−∑^𝑗−1_𝑖=1𝛼_𝑖𝑧_𝑖(7)  𝑑_𝑗∶=𝑧^𝑇_𝑗𝐴𝑧_𝑗(8)EndDo

By some modifications this algorithm can be converted in a simple form, avoiding extra computations. Letting 𝑞_𝑖=𝑧^𝑇_𝑖𝐴, steps (3)–(7) may be written as follows:

(i)𝑧_𝑗∶=𝑒_𝑗(ii)For 𝑖=1,…,𝑗−1, Do(iii) 𝛼_𝑖∶=(1/𝑑_𝑖)𝑞_𝑖(𝑒_𝑗−∑^𝑖−1_𝑘=1𝛼_𝑘𝑧_𝑘)(iv) 𝑧_𝑗∶=𝑧_𝑗−𝛼_𝑖𝑧_𝑖(v)EndDo(vi)𝑞_𝑗∶=𝑧^𝑇_𝑗𝐴(vii)𝑑_𝑗∶=𝑞_𝑗𝑧_𝑗.

Obviously the parameter 𝛼_𝑖 at step (iii) of this procedure can be computed via 𝛼_𝑖=1𝑑_𝑖𝑞_𝑖𝑧_𝑗. (3.5) We have 𝐴𝑍=𝑊⁻¹𝐷. This shows that the matrix 𝐴𝑍 is a lower triangular matrix. Therefore, since 𝑍 is a unit upper triangular matrix, we deduce 𝛼_𝑖=1𝑑_𝑖𝑞_𝑖𝑧_𝑗=1𝑑_𝑖𝑧^𝑇_𝑖𝐴𝑧_𝑗=1𝑑_𝑖𝑒^𝑇_𝑖𝐴𝑧_𝑗=1𝑑_𝑖𝐴_𝑖∗𝑧_𝑗. (3.6) On the other hand from (2.9), in step (7) of this procedure, we can replace 𝑑_𝑗∶=𝑞_𝑗𝑧_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.

(1) 𝑧1∶=𝑒1, 𝑑1∶=𝑎11 (2) For 𝑗=2,…,𝑛, Do (3)   𝑧𝑗∶=𝑒𝑗 (4)  For 𝑖=1,…,𝑗−1, Do (5)    𝛼𝑖∶=(1/𝑑𝑖)𝐴𝑖∗𝑧𝑗 (6)    𝑧𝑗∶=𝑧𝑗−𝛼𝑖𝑧𝑖 (7)  EndDo (8)   𝑑𝑗∶=𝐴𝑗∗𝑧𝑗 (9) EndDo

Algorithm 2 

Left-looking AINV algorithm without dropping.

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.

4. Conclusions

In this paper, we have shown that the AINV and FAPINV algorithms are strongly related. In fact, we have shown that the AINV algorithm can be extracted from the FAPINV algorithm by some modification. Although, without any dropping, the computation of inverse factors of a matrix by the two algorithms is done in different ways, but the results are the same. Hence many of the properties of each of these algorithms are valid for the other one. For example, in (Benzi and Tuma [13]), it has been shown that the right-looking AINV algorithm without any dropping role is well defined for 𝐻-matrices. Therefore we conclude that the BFAPINV algorithm is well defined for 𝐻-matrices as well.

Acknowledgment

The authors would like to thank one of the referees for helpful suggestions.