On convergence of three iterative methods for solving of the matrix equation $$X+A^{*}X^{-1}A+B^{*}X^{-1}B=Q$$
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On convergence of three iterative methods for solving of the matrix equation X + A∗ X −1 A + B ∗ X −1 B = Q Vejdi I. Hasanov · Aynur A. Ali
Received: 12 October 2014 / Revised: 12 January 2015 / Accepted: 22 January 2015 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2015
Abstract In this paper, we give new convergence results for the basic fixed point iteration and its two inversion-free variants for finding the maximal positive definite solution of the matrix equation X + A∗ X −1 A + B ∗ X −1 B = Q, proposed by Long et al. (Bull Braz Math Soc 39:371–386, 2008) and Vaezzadeh et al. (Adv Differ Equ 2013). The new results are illustrated by numerical examples. Keywords Nonlinear matrix equation · Fixed point iteration · Inversion-free iteration · Convergence rate Mathematics Subject Classification
65F10 · 65F30 · 65H10 · 15A24
1 Introduction In this paper, we study the matrix equation X + A∗ X −1 A + B ∗ X −1 B = Q,
(1)
where A, B are square matrices and Q is a positive definite matrix. Here, A∗ denotes the conjugate transpose of the matrix A. The matrix Eq. (1) can be reduced to Y + C ∗ Y −1 C + D ∗ Y −1 D = I,
(2)
where I is the identity matrix. Moreover, the Eq. (1) is solvable if and only if the Eq. (2) is solvable. For the first time, the Eqs. (2) and (1) are considered by Long et al. (2008) and Vaezzadeh et al. (2013), respectively. Also, the Eqs. (1) and (2) are appeared as particular cases of the equations in El-Sayed and Ran (2001), Ran and Reurings (2002), He and Long (2010),
Communicated by Jinyun Yuan. V. I. Hasanov (B) · A. A. Ali Faculty of Mathematics and Informatics, Konstantin Preslavsky University of Shumen, 9712 Shumen, Bulgaria e-mail: [email protected]
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V. I. Hasanov, A. A. Ali
Duan et al. (2011) and Liu and Chen (2011). El-Sayed and Ran (2001) and Ran and Reurings ∗ F (X )A = Q. He and Long (2010) and Duan et al. (2002) investigated the equation X + A m (2011) investigated the equation X + i=1 Ai∗ X −1 Ai = I . Liu and Chen (2011) studied the equation X s + A∗ X −t1 A + B ∗ X −t2 B = Q. Berzig et al. (2012) considered the equation X = Q − A∗ X −1 A + B ∗ X −1 B. Zhou et al. (2013) and Li et al. (2014) investigated the −1 equation X + A∗ X A = Q. Specifically, if B = 0, the Eq. (1) reduces to X + A∗ X −1 A = Q,
(3)
which has many applications and has been studied recently by several authors (Anderson et al. 1990; Engwerda 1993; Zhan and Xie 1996; Zhan 1996; Guo and Lancaster 1999; Xu 2001; Meini 2002; Sun and Xu 2003; Hasanov and Ivanov 2006; Hasanov 2010). In this paper, we write A > 0 (A ≥ 0) if A is a Hermitian positive definite (semidefinite) matrix. For Hermitian matrices A and B, we write A > B (A ≥ B) if A− B > 0 (A− B ≥ 0). A positive definite solutions X S and X L of a matrix equation is called minimal and maximal, respectively, if X S ≤ X ≤ X L for any positive definite solution X of the equation. Long et al. (2008) presented some conditions for existence of a positive definite solution of (2). They propose two iterative methods: basic f
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