Abstract
A new upper bound which involves a parameter for the infinity norm of the inverse of Nekrasov matrices is given. And we determine the optimal value of the parameter such that the bound improves the results of Kolotilina, 2013. Numerical examples are given to illustrate the corresponding results.
1. Introduction
The class of Nekrasov matrices is a subclass of -matrices. Estimating the infinity norm of the inverse of Nekrasov matrices can be used to prove the convergence of matrix splitting and matrix multisplitting iteration methods for solving large sparse systems of linear equations; see [1–4]. Here, we call a matrix an -matrix if its comparison matrix defined by is an -matrix; that is, [1, 5, 6], and a matrix is called a Nekrasov matrix if for each , where and , [2, 6].
In 1975, Varah [7] provided the following upper bound for strictly diagonally dominant (SDD) matrices as one most important subclass of Nekrasov matrices, consequently, -matrices [2, 6, 8]. Here a matrix is called SDD if for each , where .
Theorem 1 (see [7]). Let be SDD. Then
We call the bound in Theorem 1 the Varah’s bound. As Cvetković et al. [2] said, Varah’s bound works only for SDD matrices and even then it is not always good enough. To obtain new upper bounds for the infinity norm of the inverse of a wider class of matrices which sometimes works better in the SDD case, Cvetković et al. [2] give the following bound of Nekrasov matrices.
Theorem 2 (see [2, Theorem 2]). Let be a Nekrasov matrix. Then where and .
In [9, Theorems 2.2 and 2.3], Kolotilina gave an improvement of these upper bounds in Theorem 2 (see Theorems 3 and 4).
Theorem 3 (see [9, Theorem 2.2]). Let be a Nekrasov matrix. Then
Theorem 4 (see [9, Theorem 2.3]). Let be a Nekrasov matrix. Then
In this paper, we also focus on the estimation problem of the infinity norm of the inverse of Nekrasov matrices and give an improvement of the bound in Theorem 3 (Theorem 2.2 in [9]). Numerical example is given to illustrate the corresponding results.
2. Bounds for the Infinity Norm of the Inverse of Nekrasov Matrices
In order to obtain a new bound, we start with the following lemmas and notations. Given a matrix , by we denote the standard splitting of into its diagonal , strictly lower , and strictly upper triangular parts. And by denote the -entry of ; that is, . Furthermore, we denote .
Lemma 5 (see [10]). Let be a nonsingular -matrix. Then
Lemma 6 (see [11]). Given any matrix , , with for all , then where is the vector with all components equal to 1.
Lemma 7 (see [12]). A matrix , , is a Nekrasov matrix if and only if that is, if and only if is an SDD matrix, where is the identity matrix.
Let . Then from Lemma 7, is SDD when is a Nekrasov matrix. Note that , , , and , , which leads to the following lemma.
Lemma 8. Let be a Nekrasov matrix and where and . Then is SDD.
Proof. It is not difficult from (12) to see that for all and for all and . Hence and for , From the fact that is SDD and , we have that is SDD. The proof is completed.
The main result of this paper is the following theorem.
Theorem 9. Let be a Nekrasov matrix. Then for ,
Proof. Let , where . From (12), we have
which implies that
where
Furthermore, since a Nekrasov matrix is an -matrix, we have, from Lemma 5,
First, we estimate . Since is an -matrix and there exists a positive diagonal matrix such that , see [9], we get
Secondly, we estimate . From Lemma 8, is SDD. Obviously, multiplying the left-hand side of by diagonal matrix does not change SDD property, so is also SDD. Thus, Varah’s bound (4) can be applied as follows:
In addition, since and , see [9, 13], we have
Substituting (22) into (21), we get that
Finally, from (20), (23), , and the fact that , we have
The conclusions follow.
Example 10. Consider the Nekrasov matrix in [2, 9], where By computation, , , , , , , , and . By the bound of Theorem 3 (the bound of Theorem 2.2 in [9]), we have By Theorem 9, we have In fact, .
Remark 11. Example 10 shows that by choosing the value of , the bound in Theorem 9 is better than that in Theorem 3 in some cases. We further observe the bound in Theorem 9 by Figure 1 and find that there is an interval such that for any in this interval, the bound in Theorem 9 for the matrix is always smaller than that in Theorem 3. An interesting problem arises: whether there is an interval of such that the bound in Theorem 9 for any Nekrasov matrix is smaller than that in Theorem 3. In the following section, we will study this problem.
3. The Choice of
In this section, we determine the value of such that the bound for in Theorem 9 is less than or equal to that in [9]. First, we consider the Nekrasov matrix with and give the following lemma.
Lemma 12. Let , , and be positive real numbers, and . Then
Proof. We only need to prove that and . In fact, The proof is completed.
Lemma 13. Let be a Nekrasov matrix with Then
Proof. Let , , and . From (28), we get . Then from Lemma 12, the first and second inequalities in (32) hold.
We now give an interval of such that the bound in Theorem 9 is less than that in Theorem 3.
Lemma 14. Let be a Nekrasov matrix with Then for each ,
Proof. From Lemma 13, we have
and .
For , then
that is,
Therefore,
Consider the function , . It is easy to prove that is a monotonically decreasing function of . Hence, for any ,
that is,
Hence,
For , , then
that is,
Therefore,
Consider the function
Obviously, is a monotonically increasing function of . Hence, for any , ,
that is,
Hence,
The conclusion follows from and .
Lemma 14 provides an interval of such that the bound in Theorem 9 is better than the bound in Theorem 3 (the bound in [9]). Moreover, we can determine the optimal value of by the following theorem.
Theorem 15. Let be a Nekrasov matrix with Then Furthermore,
Proof. From the proof of Lemma 14, we have that is decreasing and that is increasing. Therefore, the minimum of and is which implies that (50) holds. Again by Lemma 14, (51) follows easily.
Remark 16. Theorem 15 provides a method to determine the optimal value of for a Nekrasov matrix with Also consider the matrix in Example 10. By computation, we get Hence, by Theorem 15, we can obtain that the bound in Theorem 9 reaches its minimum at (also see Figure 1).
Next, we study the bound in Theorem 9 for the Nekrasov matrix with
Theorem 17. Let be a Nekrasov matrix with Then we can take such that
Proof. From (59), we get . Then, for , we have
For , then and
that is,
Therefore,
For , then and
that is,
Therefore,
For , then and
Therefore,
The conclusion follows from (I), (II), and (III).
Remark 18. Theorems 15 and 17 provide the value of ; that is, such that the bound in Theorem 9 is not worse than that in Theorem 3 for a Nekrasov matrix . In particular, for the Nekrasov matrix with , the bound in Theorem 9 is better than that in Theorem 3.
Example 19. Consider the following five Nekrasov matrices in [2, 9]: Obviously, , , and are SDD. And it is not difficult to verify that satisfies the conditions in Theorem 15 and , , , and satisfy the conditions in Theorem 17. We compute by Matlab 7.0 the upper bounds for the infinity norm of the inverse of , , which are shown in Table 1. It is easy to see from Table 1 that this example illustrates Theorems 15 and 17.
4. Conclusions
In this paper, we give an improvement on the infinity norm bound for the inverse of a Nekrasov matrix in [9]. In particular, for the Nekrasov matrix with we prove that new bound is better than that in [9]. However, for the Nekrasov matrix with we only obtain that new bound is equal to that in [9]. For this case, we try to found some better bounds in future. On the other hand, our bound only considers one parameter , that is, , which poses an interesting problem: whether we further improve this bound by introducing more parameters. In future, we will research this problem.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This research is supported by the National Natural Science Foundation of China (11361074 and 11326242), the Science Foundation of the Education Department of Yunnan Province of China (2013FD002), and Science and Technology Innovation Fund projects of Yunnan University (ynuy201366).