Asian Journal of Control, Vol. 20, No. 1, pp. 228–235, January 2018 Published online 14 June 2017 in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/asjc.1574
A MODIFIED GRADIENT-BASED ALGORITHM FOR SOLVING EXTENDED SYLVESTER-CONJUGATE MATRIX EQUATIONS Mohamed A. Ramadan and Ahmed M. E. Bayoumi ABSTRACT In this paper, we present a modified gradient-based algorithm for solving extended Sylvester-conjugate matrix equations. The idea is from the gradient-based method introduced in [14] and the relaxed gradient-based algorithm proposed in [16]. The convergence analysis of the algorithm is investigated. We show that the iterative solution converges to the exact solution for any initial value based on some appropriate assumptions. A numerical example is given to illustrate the effectiveness of the proposed method and to test its efficiency and accuracy compared with those presented in [14] and [16]. Key Words:
extended Sylvester, relaxation parameters, gradient-based algorithm, modified gradient.
I. INTRODUCTION Certain matrix equations arise naturally in linear control and system theory. Among those frequently encountered in the analysis and design of continuous-time systems is the extended Sylvester-conjugate matrix equation. The problem under consideration is one of the more general forms of these matrix equations. The interested reader is referred to, for example, [1–3]. In the matrix algebra field, some complex matrix equations have attracted much attention from many researchers since it was shown in [4] that the consistence of the matrix equation AX X B ¼ C was related to the consimilarity [5–7] of two partitioned matrices associated with the matrices A, B, and C. Huang et al. [8] proposed an Iterative method for solving the linear matrix equation AXB = F over skew-symmetric matrix X. Ding et al. [9] derived iterative solutions of matrix equation AXB = F and the Sylvester matrix equation AXB + CXD = F. The gradient-based iterative algorithm in Ding and Chen [10–12] and least squares based iterative algorithm [10] for solving (coupled) matrix equations are novel and efficiently numerical algorithms were presented based on the hierarchical identification principle [11,13]. Niu et al. [14] proposed a relaxed gradient-based iterative algorithm for solving Sylvester equations AX + XB = C. The numerical experiments show that the convergent behavior of Niu’s algorithm is better than Ding’s algorithm [12]. Ramadan
Manuscript received January 9, 2016; revised December 25, 2016; accepted March 14, 2017. M. A. Ramadan (corresponding author) is with Department of Mathematics, Faculty of Science, Menoufia University, Shebeen El- Koom, Egypt (e-mail:
[email protected];
[email protected]). A. M. E. Bayoumi is with Department of Mathematics, Faculty of Education, Ain Shams University, Cairo, Egypt (e-mail:
[email protected]).
et al. [15] proposed a relaxed gradient-based iterative algorithm for solving extended Sylvester-conjugate matrix equations AXB þ CX D ¼ F . These kinds of problems arise in many applications in system and control theory, signal and image processing, model reduction, stability of linear systems, analysis of bilinear systems, and power systems. Wang et al. [16] proposed a modified gradient-based algorithm for solving Sylvester equation AX + XB = C. Recently, iterative algorithms for solving complex matrix equation have also attracted much attention. By applying the hierarchical identification principle, an iterative algorithm was constructed in Wu et al. [17] to solve so-called extended Sylvester-conjugate matrix equations. In [18], the matrix equation X AX B ¼ C is considered. In [19], a more general complex matrix equation was investigated, and a finite iterative algorithm was proposed to solve these kinds of matrix equations. In [20] the development of the conjugate direction (CD) method was constructed to solve the generalized nonhomegeneous Yakubovich-transpose matrix equation. In [21], Hajarian developed the conjugate gradients squared (CGS) and bi-conjugate gradient stabilized (Bi-CGSTAB) methods for obtaining matrix iterative methods for solving the Sylvester-transpose and periodic Sylvester matrix equations. According to many numerical experiments, GI and RGI algorithms are computationally efficient. However, we observe that both the GI algorithm and the RGI algorithm have some limitations. For the GI algorithm, the convergent rate is slow and stagnation will happen for ill-conditioned problems. For the RGI algorithm, the selection of an optimal relaxation factor is a very difficult task. The convergence condition of the new method is given. The performance of the new method is compared with the existing method. Numerical results show that the new method has better performance.
© 2017 Chinese Automatic Control Society and John Wiley & Sons Australia, Ltd
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M. A. Ramadan and A. M. E. Bayoumi: A modified algorithm for solving Sylvester- conjugate matrix equations
This paper is organized as follows: First, in section II, we introduce some notations and lemmas that will be needed to develop this work. In section III, we introduce a brief review of the gradient-based iterative algorithm proposed in [17] and we introduce a brief review of the relaxed gradient-based iterative algorithm for solving the problem under consideration in [15]. In section IV, we propose an iterative algorithm to obtain the solutions to extended Sylvester-conjugate matrix equations by using a modified gradient-based algorithm and give the convergence properties of this iterative algorithm. In section V, a numerical example is given to explore the effectiveness, efficiency, and accuracy of the presented method.
II. PRELIMINARIES The following notations, definitions, and lemmas will be used to develop the proposed work. We use AT ; A ; AH ,‖A‖2 and tr(A) to denote the transpose, conjugate, conjugate transpose, spectral norm of A, and the trace of A, respectively. We denote the set of all m × n complex matrices by ℂm × n. The Frobenius norm pffiffiffiffiffiffiffiffiffiffiffiffiffiffi of the matrix A is denoted by kAk ¼ hA; Air ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Re tr AH A . Lemma 1. [22] Consider the following matrix equation
traceðN Þ ¼
n X
λ i ðN Þ ¼
i¼1
Therefore,
n
n X
σ i ðN Þ:
i¼1
in
this
case
we
have
j traceðMN Þ j ≤ ρ1 ∑ σ i ¼ ρ1 trace ðN Þ . Where λ1 ≥ ⋯ ≥ λn i¼1
denote the eigenvalues of N.
2.1 A brief review of the gradient-based iterative algorithm proposed in [17] By regarding the extended Sylvester-conjugate matrix equation AXB þ CX D ¼ F;
(1)
as two linear matrix equations of the form AXB ¼ F CX D;
(2)
CX D ¼ F AXB;
(3)
Wu et al. [17] presented the following algorithm for solving equation 1. X 1 ðk Þ ¼ X ðk 1Þ þ μAH F AX ðk 1ÞB CX ðk 1ÞD B H ;
AXB ¼ F;
(4)
where A∈ℂ m × r, B∈ℂ s × n and F∈ℂ m × n are known matrices, and X∈ℂ r × s is the matrix to be determined. For this matrix equation, an iterative algorithm is constructed as
X 2 ðk Þ ¼ X ðk 1Þ H H þ μ C F AX ðk 1ÞB CX ðk 1ÞD D ;
(5)
X ðk þ 1Þ ¼ X ðk Þ þ μAH ðF AX ðk ÞBÞBH ; with
where X(k) is the average of X1(k) and X2(k), that is, 0
