Eigenvalue bounds of the shift-splitting preconditioned singular nonsymmetric saddle-point matrices
- Quan Shi^{1},
- Qin-Qin Shen^{1, 2}Email author and
- Lin-Quan Yao^{2}
https://doi.org/10.1186/s13660-016-1193-y
© Shi et al. 2016
Received: 18 July 2016
Accepted: 4 October 2016
Published: 19 October 2016
Abstract
For singular nonsymmetric saddle-point problems, a shift-splitting preconditioner was studied in (Appl. Math. Comput. 269:947-955, 2015). To further show the efficiency of the shift-splitting preconditioner, we provide eigenvalue bounds for the nonzero eigenvalues of the shift-splitting preconditioned singular nonsymmetric saddle-point matrices. For real parts of the eigenvalues, the bound is provided by valid inequalities. For eigenvalues having nonzero imaginary parts, the bound is a combination of two inequalities proving their clustering in a confined region of the complex plane. Finally, two numerical examples are presented to verify the theoretical results.
Keywords
MSC
1 Introduction
Since the spectral distribution of the preconditioned matrix is closely related to the convergence rate of Krylov subspace iteration methods [8], we hope that the resulting preconditioned saddle-point matrices have desired eigenvalue distributions, that is, tightly clustered spectra or positive real spectra, and so on; see, for example, [7, 16–19]. To further show the efficiency of the shift-splitting preconditioner (1.2) and (1.3), in this paper, we provide eigenvalue bounds for the nonzero eigenvalues of the shift-splitting preconditioned singular nonsymmetric saddle-point matrices \(\mathcal {P}_{SS}^{-1}\mathcal{A}\) that depend only on the extremal eigenvalues of A and nonzero extremal singular values of B. We show that all eigenvalues having nonzero imaginary parts are located in an intersection of two circles and all nonzero real eigenvalues are located in a positive interval. Although the analysis is done for the singular case, the theoretical results of the nonsingular case can be obtained as a particular case. These theoretical results are presented in Section 2. In Section 3, we present two numerical examples to verify our theoretical results. Finally, in Section 4, we end this paper with a few concluding remarks.
2 Eigenvalue bounds of the shift-splitting preconditioned matrix \(\mathcal{P}_{SS}^{-1}\mathcal{A}\)
We first present the eigenvalue distribution of the shift-splitting preconditioned matrix \(\mathcal{P}_{SS}^{-1}\mathcal{A}\) and give a bound for its nonzero eigenvalues. To this end, we first give a useful lemma.
Lemma 2.1
[11]
Theorem 2.1
Proof
To further derive a bound for the nonreal eigenvalues of the shift-splitting preconditioned matrix \(\mathcal{P}_{SS}^{-1}\mathcal {A}\), we use the following lemma. Note that the conclusion of the lemma is a particular case of [20], Theorem 2.
Lemma 2.2
[20]
Theorem 2.2
Proof
Theorem 2.2 shows that the bound for the eigenvalues having nonzero imaginary parts of the shift-splitting preconditioned matrix \(\mathcal{P}_{SS}^{-1}\mathcal{A}\) only depends on the parameter α and the smallest eigenvalue of the matrix A. Combining Theorem 2.1 with Theorem 2.2, we can obtain some refined bounds for the nonreal eigenvalues of the shift-splitting preconditioned matrix \(\mathcal{P}_{SS}^{-1}\mathcal{A}\) in the following theorem.
Theorem 2.3
Theorem 2.4
Proof
Remark 2.1
- (1)
Theorem 2.4 also presents a bound for the real eigenvalues of the shift-splitting preconditioned matrix \(\mathcal {P}_{SS}^{-1}\mathcal{A}\). Besides, the bound given in Theorem 2.4 is much tighter than that in Theorem 2.1.
- (2)
Obviously, if \(r=m\), then the nonsymmetric saddle-point matrix \(\mathcal{A}\) is nonsingular. Then, the theoretical results presented in Theorems 2.1-2.4 can be extended to the nonsingular case.
3 Numerical experiments
In this section, we use two numerical examples to verify the estimated eigenvalue bounds of the shift-splitting preconditioned matrices \(\mathcal{P}_{SS}^{-1}\mathcal{A}\) shown in Section 2.
Example 3.1
[1]
For this example, we have \(n=2p^{2}\), \(m=p^{2}+2\), and the total number of variables is \(m+n=3p^{2}+2\). The matrix B is an augmentation of the full rank matrix B̂ with two linearly independent vectors \(b_{1}\) and \(b_{2}\). Since \(b_{1}\) and \(b_{2}\) are linear combinations of the columns of the matrix B̂, B is a rank-deficient matrix. Thus, the corresponding nonsymmetric saddle-point matrix \(\mathcal{A}\) is singular. In order to verify the theoretical results, we only choose the case \(q=16\) and vary the parameters α and β. We choose four cases for the parameter α, that is, \(\alpha =0.1\eta_{n}, 0.4\eta_{n}, 0.7\eta_{n}, \eta_{n}\). The parameter β is obtained by the formula \(\beta=\|B\|_{2}^{2}/\|\alpha I_{n}+A\| _{2}\), which is often used in the shift-splitting preconditioners [11, 15].
Bounds of nonzero real eigenvalues of the preconditioned matrix \(\pmb{\mathcal{P}_{SS}^{-1}\mathcal{A}}\) for Example 3.1
( α , β ) | The estimated bound | The exact bound |
---|---|---|
(1.9683, 6.6748) | [3.4169e−4, 0.9993] | [0.0536, 0.9991] |
(7.8732, 6.6572) | [3.4172e−4, 0.9970] | [0.0541, 0.9966] |
(13.7782, 6.6397) | [3.4174e−4, 0.9948] | [0.0545, 0.9940] |
(19.6831, 6.6223) | [3.4176e−4, 0.9926] | [0.0550, 0.9915] |
By actual computation we know that two eigenvalues of the shift-splitting preconditioned matrix \(\mathcal{P}_{SS}^{-1}\mathcal {A}\) are zero, which confirms Theorem 2.1. From Table 1 and Figure 1 we can see that the estimated bounds sharply contain the exact bounds of real eigenvalues of \(\mathcal {P}_{GSS}^{-1}\mathcal{A}\) and all eigenvalues having nonzero imaginary parts are located in an intersection of two circles. These results are in good agreement with our theoretical results in Theorem 2.3 and Theorem 2.4.
Example 3.2
The IFISS software package developed by Elman et al. [22] is used to discretize the two-dimensional Stokes equation (3.1) on the unit square domain. In actual computation, the Q2-Q1 mixed finite element method on uniform grid is used to generate discretizations. For simplicity, we set the viscosity values \(\nu=1\) and take \(16\times16\) grids to obtain the test matrix. Note that the rank of the matrix B in the test nonsymmetric saddle-point matrix is \(m-1\), which means that the discretized nonsymmetric saddle-point matrix \(\mathcal{A}\) is singular.
Bounds of nonzero real eigenvalues of the preconditioned matrix \(\pmb{\mathcal{P}_{SS}^{-1}\mathcal{A}}\) for Example 3.2
( α , β ) | The estimated bound | The exact bound |
---|---|---|
(0.0077, 0.0072) | [1.1108e−3, 0.9995] | [0.1356, 0.9990] |
(0.0307, 0.0071) | [1.1108e−3, 0.9981] | [0.1361, 0.9960] |
(0.0538, 0.0071) | [1.1108e−3, 0.9966] | [0.1365, 0.9930] |
(0.0768, 0.0071) | [1.1108e−3, 0.9952] | [0.1370, 0.9900] |
4 Conclusion
To further show the efficiency of the shift-splitting preconditioner for singular nonsymmetric saddle-point problems (1.1), the eigenvalue bounds of the shift-splitting preconditioned saddle-point matrix are studied in detail in this paper. Theoretical analysis shows that all eigenvalues having nonzero imaginary parts are located in an intersection of two circles and all nonzero real eigenvalues are located in a positive interval. Two numerical examples are presented to confirm the theoretical results. The numerical results show that the eigenvalue bounds are very sharp.
Declarations
Acknowledgements
This work is supported by the National Natural Science Foundation of China (No. 11572210), the Natural Science Foundation of Jiangsu Province (No. BK20151272), the ‘333’ Program Talents of Jiangsu Province (No. BRA2015356), and the Six Top Talents of Jiangsu Province (No. 2014-WLW-029).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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