On the modified Hermitian and skew-Hermitian splitting iteration methods for a class of weakly absolute value equations
- Cui-Xia Li^{1}Email author
DOI: 10.1186/s13660-016-1202-1
© Li 2016
Received: 12 January 2016
Accepted: 11 October 2016
Published: 21 October 2016
Abstract
In this paper, based on the modified Hermitian and skew-Hermitian splitting (MHSS) iteration method, the nonlinear MHSS-like iteration method is presented to solve a class of the weakly absolute value equations (AVE). By using a smoothing approximate function, the convergence properties of the nonlinear MHSS-like iteration method are presented. Numerical experiments are reported to illustrate the feasibility, robustness, and effectiveness of the proposed method.
Keywords
absolute value equation MHSS iteration convergence smoothing approximate functionMSC
90C05 90C30 65F101 Introduction
In recent years, some efficient methods have been proposed to solve the AVE (1.1), such as the smoothing Newton method [6], the generalized Newton method [7–11], the sign accord method [12]. For other forms of the iteration method, one can see [13–15] for more details.
When the involved matrix A in (1.1) is a non-Hermitian positive definite matrix, based on the Hermitian and skew-Hermitian splitting (HSS) iteration method [16], the Picard-HSS iteration method for solving the AVE (1.1) has been proposed in [17]. Numerical results show that the Picard-HSS iteration method outperforms the Picard and generalized Newton methods under certain conditions. Although the Picard-HSS iteration method is efficient and competitive, the numbers of the inner HSS iteration steps are often problem-dependent and difficult to be obtained in actual computations. To overcome this disadvantage and improve the convergence of the Picard-HSS iteration method, the nonlinear HSS-like iteration method in [18] has been presented and its convergent conditions are established. Numerical experiments demonstrate that the nonlinear HSS-like iteration method is feasible and robust.
When the involved matrix A in (1.1) is \(A = W + iT\), the convergent rate of the aforementioned Picard-HSS and nonlinear HSS-like methods maybe reduce. This is reason that each step of Picard-HSS and HSS-like iterations needs to solve two linear subsystems with the symmetric positive definite matrix \(\alpha I +W\) and the shifted skew-Hermitian matrix \(\alpha I + iT\). It is well known that the solution of the linear system with the coefficient matrix \(\alpha I + iT\) is not easy to obtain [19]. To overcome this defect, based on MHSS iteration method [20], we will establish the nonlinear MHSS-like iteration method to solve the AVE (1.1). Compared with the nonlinear HSS-like iteration method, the potential advantage of the nonlinear MHSS-like iteration method is that only two linear subsystems with coefficient matrices \(\alpha I +W\) and \(\alpha I +T\), both being real and symmetric positive definite, need to be solved at each step. This shows that the nonlinear MHSS-like iteration method can avoid a shifted skew-Hermitian linear subsystem with coefficient matrix \(\alpha I + iT\). Therefore, in this case, these two linear subsystems can be solved either exactly by a sparse Cholesky factorization or inexactly by conjugated gradient scheme. The convergent conditions of the nonlinear MHSS-like iteration method are obtained by using a smoothing approximate function.
The remainder of the paper is organized as follows. In Section 2, the MHSS iteration method is briefly reviewed. The nonlinear MHSS-like iteration method is discussed in Section 3. Numerical experiments are reported in Section 4. Finally, in Section 5 we draw some conclusions.
2 The MHSS iteration method
To establish the nonlinear MHSS-like iteration method for solving the AVE (1.1), a brief review of MHSS iteration is needed.
Theoretical analysis in [20] shows that the MHSS method converges unconditionally to the unique solution of the complex symmetric linear system (2.1) when \(W\in\mathbb{R}^{n\times n}\) is symmetric positive definite and \(T\in\mathbb{R}^{n\times n}\) is symmetric positive semidefinite.
3 The nonlinear MHSS-like iteration method
Evidently, each step of the nonlinear MHSS-like iteration alternates between two real symmetric positive definite matrices \(\alpha I+W\) and \(\alpha I+T\). Hence, these two linear subsystems involved in each step of the nonlinear MHSS-like iteration can be solved effectively by exactly using the Cholesky factorization. On the other hand, in the Picard-HSS and nonlinear HSS-like methods, a shifted skew-Hermitian linear subsystem with coefficient matrix \(\alpha I+iT\) needs to be solved at every iterative step.
Definition 3.1
[22]
Lemma 3.1
Ostrowski theorem [22]
Suppose that \(G:\mathbb{D}\subset \mathbb{R}^{n}\rightarrow \mathbb{R}^{n}\) has a fixed point \(x^{\ast}\in \operatorname {int}(\mathbb{D})\) and is F-differentiable at \(x^{\ast}\). If the spectral radius of \(G'(x^{\ast})\) is less than 1, then \(x^{\ast}\) is a point of attraction of the iteration \(x^{k+1}=Gx^{k}\), \(k=0,1,2,\ldots\) .
We note that we cannot directly use the Ostrowski theorem (Theorem 10.1.3 in [22]) to give a local convergence theory for the iteration (3.6). It is reason that the nonlinear term \(\vert x\vert +b\) is non-differentiable. To overcome this defect, based on the smoothing approximate function introduced in [6], we can establish the following local convergence theory for nonlinear MHSS-like iteration method.
Lemma 3.2
Lemma 3.3
Proof
Theorem 3.2
Proof
Observe that the matrices W and T are strongly dominant over the matrix \(\phi'(x^{\ast})\) in certain norm (i.e., \(\Vert W\Vert \gg \Vert \phi'(x^{\ast})\Vert \) and \(\Vert T\Vert \gg \Vert \phi'(x^{\ast})\Vert \) for certain matrix norm). Therefore, matrix A satisfies the condition (3.11) with W be symmetric positive definite and T be symmetric positive semidefinite. In fact, in this case, the matrix \(M_{\alpha,x^{\ast}}\) can be approximated by the matrix \(M_{\alpha}\) and the condition (3.11) reduces to \(\rho(M_{\alpha})<1\). Clearly, \(\rho(M_{\alpha})\leq\theta(\alpha)<1\).
Theorem 3.3
Proof
Theorem 3.4
Proof
From Theorems 3.1 and 3.2, obviously, \(\Vert x^{(k+1)}-x^{\ast} \Vert <\varepsilon\). This completes the proof. □
The convergent speed of the nonlinear MHSS-like method (3.3) may depend on two factors: (1) the nonlinear term \(\vert x\vert +b\); (2) finding the optimal parameters to guarantee that the spectral radius \(\rho(M_{\alpha})\) of the iteration matrix \(M_{\alpha}\) is less than 1. As the former is problem-independent, the latter can be estimated. Based on Corollary 2.1 in [20], the optimal parameter \(\alpha^{\ast}=\sqrt{\lambda_{\max}\lambda_{\min}}\) is obtained to minimize the upper bound on the spectral radius \(\rho(M_{\alpha})\) of the MHSS iteration matrix \(M_{\alpha}\), where \(\lambda_{\max}\) and \(\lambda_{\min}\) are the extreme eigenvalues of the symmetric positive definite matrix W, respectively. It is noted that, although one usually cannot expect to minimize the spectral radius \(\rho(M_{\alpha})\) of the corresponding iteration matrix \(M_{\alpha}\) with the optimal parameter \(\alpha^{\ast}\), it is still helpful for us to choose an effective parameter for the nonlinear MHSS-like method.
4 Numerical experiments
In this section, we give some numerical experiments to demonstrate the performance of the nonlinear MHSS-like method for solving the AVE (1.1). Since the numerical results in [18] show that the nonlinear HSS-like method outperforms the Picard and Picard-HSS methods under certain conditions, here we compare the nonlinear MHSS-like method with the nonlinear HSS-like method [18] from the point of view of the number of iterations (denoted IT) and CPU times (denoted CPU) to show the advantage of the nonlinear MHSS-like method. All the tests are performed in MATLAB 7.0.
Example 1
The optimal parameters α for MHSS-like and MHSS-like methods
m | 8 × 8 | 16 × 16 | 32 × 32 | 48 × 48 |
---|---|---|---|---|
MHSS-like | 3.66 | 2.09 | 1.22 | 0.87 |
HSS-like | 9.55 | 5.32 | 3.01 | 2.21 |
CPU and IT for the MHSS-like and HSS-like methods
m | 8 × 8 | 16 × 16 | 32 × 32 | 48 × 48 | |
---|---|---|---|---|---|
MHSS | IT | 155 | 55 | 66 | 165 |
CPU | 0.062 | 0.078 | 0.563 | 3.875 | |
PMHSS | IT | 353 | 133 | 154 | 231 |
CPU | 0.203 | 0.313 | 1.969 | 7.953 |
From Table 2, the iteration numbers and CPU times of the nonlinear MHSS-like method for solving the AVE (1.1) are less than that of the nonlinear HSS-like method. The presented results in Table 2 show that in all cases the nonlinear MHSS-like method is superior to the nonlinear HSS-like method in terms of the iteration numbers and CPU times. Comparing with the nonlinear HSS-like method, the nonlinear MHSS-like method for solving the AVE (1.1) may be given priority under certain conditions.
5 Conclusions
In this paper, the nonlinear MHSS-like method has been established for solving the weakly absolute value equations (AVE). In the proposed method, two real linear subsystems with symmetric positive definite matrices \(\alpha I + W\) and \(\alpha I + T\) are solved at each step. In contrast, in the nonlinear HSS-like method a shifted skew-Hermitian linear subsystem with the matrix \(\alpha I + iT\) is solved at each iteration. By using a smoothing approximate function, the local convergence of the proposed method has been analyzed. Numerical experiments have shown that the nonlinear MHSS-like method is feasible, robust, and efficient.
Declarations
Acknowledgements
This research was supported by NSFC (No. 11301009) 17HASTIT012, Natural Science Foundations of Henan Province (No. 15A110077) and Project of Young Core Instructor of Universities in Henan Province (No. 2015GGJS-003). The author thanks the anonymous referees for their constructive suggestions and helpful comments which led to significant improvement of the original manuscript of this paper.
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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