Abstract
Circulant and block circulant type matrices are important tools in solving networked systems. In this paper, based on the style spectral decomposition of the basic circulant matrix and the basic skew circulant matrix, the block style spectral decomposition of the BCSCB matrix is obtained. And then, the structure perturbation is analysed, which includes the condition number and relative error of the BCSCB linear system. Then the optimal backward perturbation bound of the BCSCB linear system is discussed. Simultaneously, the algorithm for the optimal backward perturbation bound is given. Finally, a numerical example is provided to verify the effectiveness of the algorithm.
1. Introduction
It is an active objective that circulant and block circulant type matrices are applied to networks engineering. The stability region in the parameters space is extended by the breaking of a delayed ring neural network where the form of time-delay systems is , where is a circulant matrix, if the number of the neurons is sufficiently large in [1]. In [2], the question of when circulant quantum spin networks with nearest-neighbor couplings can give perfect state transfer is solved. The properties of linear diffusion algorithm are investigated both by a worst-case analysis and by a probabilistic analysis and are shown to depend on the spectral properties of the circulant matrix in [3]. A viable option for increasing the lifetime of the sensor network for a small loss in accuracy of the query results whose matrices are circulant is offered in [4]. In [5], the authors considered the kinetics of an autocatalytic reaction network in which replication and catalytic actions are separated by a translation step. They found that the behavior of such a system is closely related to second-order replicator equations, where the second-order replicator equations are circulant interaction matrices. In order to obtain the optimal routing in double loop networks, the problem of finding the shortest path in circulant graphs with an arbitrary number of jumps is studied in [6].
A block circulant with skew circulant blocks matrix with the first row has the following form:and for any , The matrix is denoted by BCSCB.
Rigal and Gaches [7] considered a posteriori analysis of the compatibility of a computed solution to the uncertain data of a linear system by some new theorems generalizing a result of Oettli and Prager. In [8], the style spectral decomposition of the skew circulant matrix is given and the optimal backward perturbation analysis for the skew circulant linear system is discussed. Liu and Guo [9] obtained the bound of the optimal backward perturbation for a block circulant linear system. J.-G. Sun and Z. Sun [10] studied the optimal backward perturbation bounds for undetermined systems. In [11], the optimal backward perturbation analysis for the block skew circulant linear system with skew circulant blocks is given by Li et al.
2. The Block Style Spectral Decomposition of the BCSCB Matrix
Let matrix be a BCSCB matrix as the form (1); then by using the properties of Kronecker products in [12], the can be decomposed as where is a square matrix of order , and it has the following form:
Based on and in [9], the style spectral decomposition of the matrix is where is an orthogonal matrix.
When is even,
When is odd,
Taking (3) and (5) into consideration, the matrix can be decomposed as is an orthogonal matrix obviously. So (8) is the block style spectral decomposition of the matrix .
3. Analysis of the Structured Perturbation
The structured perturbation analysis for BCSCB linear system is given in this section. We discuss the condition number and the relative error of the BCSCB linear system. The optimal backward perturbation bound of the BCSCB linear system is analysed. And, at the end of the section, we give the algorithm for the optimal backward perturbation bound.
3.1. Condition Number and Relative Error of BCSCB Linear System
Consider where is defined in (1).
From (8) and the property of Kronecker products in [12], the matrix can be expressed by using the elements in its first row as where is a square matrix of order , and it has the following form:
Based on and in [8], the style spectral decomposition of the matrix is where is an orthogonal matrix.
When is even,
When is odd,
Furthermore, (10) can be expressed as follows: and here , where and are identity matrices with orders and , respectively.
The problem will be discussed at two different situations.
(1) When is even,
(2) When is odd,
We denote by the eigenvalues of matrix [9], and are denoted as the eigenvalues of matrix [8], and then the eigenvalues of are obtained (refer to [12, 13]). Consider
Lemma 1. is a nonsingular matrix if and only if , where
Let
Theorem 2. If , then the singular values of the matrix are .
Proof. Assume the conjugate transpose of is By a direct calculation, is a normal matrix as . Then matrix is a unitarily diagonalizable matrix based on Theorem in [14]. Then there exists a unitary matrix , such that where are the eigenvalues of matrix . Taking the conjugate transpose at both sides of (22), we get then And are the eigenvalues of the matrix , for any . Therefore, the singular values of are Recall (19) and (20); the proof is completed.
As the definition of the spectral norm of matrix is via Theorem 2, the following corollary is obtained.
Corollary 3. Let ; then the spectrum norm of matrix is
Let be a perturbation of the coefficient matrix and let be a perturbation of the vector , where has the following form: and for any , Let If then through Lemma 1, is a nonsingular matrix. Let By , we obtain where is a BCSCB matrix apparently, and . So The following theorem can be obtained.
Theorem 4. Let , , , , and be defined as above. If , then where
Remark 5. The condition number of the BCSCB matrix can be easily computed with the basis of (37) and (38), the same as the bound of perturbation (37).
3.2. Optimal Backward Perturbation Bound of the BCSCB Linear System
In this part, a new method is given to obtain the minimal value of the perturbation bound, which is only related to the perturbation of the coefficient matrix and the vector. At the end of this part, the algorithm for the optimal backward perturbation bound is given.
Let be an approximate solution to and let which is equal to According to [7], we can get
Let be an approximate solution to , where is defined in (1): So (as is a BCSCB matrix, ). Hence, Since the question will be analysed in two different conditions.
(1) When is even, where
(2) When is odd, where
Let and then which is equal to As is a convex function of , the point of the minimal value is Substituting it back into (49), we obtain the following.
Theorem 6. Consider
Let be the singular value decomposition of , where and are unitary matrices, , and ; then where , , .
Remark 7. As , then can be obtained; hence, .
From what we analysed above, the following algorithm can be obtained.
Algorithm 8. We have the following steps.
Step 1. Form the style spectral decomposition of the matrixes and :Step 2. Form the block style spectral decomposition of the BCSCB matrix.
Step 3. Compute .
Step 4. Compute .
Step 5. Compute Step 6. Form .
Step 7. Compute the singular value decomposition of .
Step 8. Compute .
4. Numerical Example
In this section, a simple numerical example is given to verify the conclusion above. Suppose that , in the following example.
If the coefficient matrix of the BCSCB linear system is and the constant vector , now, three perturbations are given as follows: From the equation , where , are defined as above, the approximate solution of can be obtained as follows:where is the solution of and , , is the solution of , , respectively.
Based on Algorithm 8, we obtain Table 1, where is the relative error of the BCSCB linear system, is the condition number, , and can be obtained from the algorithm.
From the numerical example, the accuracy of the conclusion and the effectiveness of the algorithm are verified.
5. Conclusion
In this paper, we consider the problems associated with the BCSCB matrix. The BCSCB matrix is an extension of the circulant matrix and skew circulant matrix. We give the form of the BCSCB matrix and obtain its block style spectral decomposition. The algorithm of the optimal backward perturbation is given. Furthermore, by circulant matrices technology, we will develop solving problems in [15–17].
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This research was supported by the Development Project of Science & Technology of Shandong Province (Grant no. 2012GGX10115) and the AMEP of Linyi University, China.