A further study on a nonlinear matrix equation
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A further study on a nonlinear matrix equation Jie Meng1 · Hongjia Chen2 · Young‑Jin Kim1 · Hyun‑Min Kim3 Received: 2 October 2019 / Revised: 26 March 2020 © The JJIAM Publishing Committee and Springer Japan KK, part of Springer Nature 2020
Abstract The nonlinear matrix equation X p = R + M T (X −1 + B)−1 M , where p is a positive integer, M is an arbitrary n × n real matrix, R and B are symmetric positive semidefinite matrices, is considered. When p = 1 , this matrix equation is the well-known discrete-time algebraic Riccati equation (DARE), we study the convergence rate of an iterative method which was proposed in Meng and Kim (J Comput Appl Math 322:139–147, 2017). For the generalized case p ≥ 1 , a structured condition number based on the classic definition of condition number is defined and its explicit expression is obtained. Finally, we give some numerical examples to show the sharpness of the structured condition number. Keywords Matrix equation · Symmetric positive definite · Cyclic reduction · Structured condition number Mathematics Subject Classification 15A24 · 65F10 · 65H10
* Hyun‑Min Kim [email protected] Jie Meng [email protected] Hongjia Chen [email protected] Young‑Jin Kim [email protected] 1
Finance·Fishery·Manufacture Industrial Mathematics Center on Big Data, Pusan National University, Busan 46241, Republic of Korea
2
Department of Mathematics, School of Science, Nanchang University, Nanchang 30031, People’s Republic of China
3
Department of Mathematics and Finance·Fishery·Manufacture Industrial Mathematics Center on Big Data, Pusan National University, Busan 46241, Republic of Korea
13
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1 Introduction We consider the nonlinear matrix equation
X p = R + M T (X −1 + B)−1 M,
(1)
where p is a positive integer, M ∈ ℝn×n , B and R are symmetric positive semidefinite matrices. For the special case p = 1 , Eq. (1) is exactly
X = R + M T (X −1 + B)−1 M,
(2)
which, under certain condition, is a simplified symmetric form of the well-known discrete-time algebraic Riccati equation (DARE)
X = M T XM − M T XE(G + ET XE)−1 ET XM + CT C,
(3)
where M ∈ ℝn×n , E ∈ ℝn×m , C ∈ ℝq×n and G ∈ ℝm×m is a symmetric positive definite matrix. It has been proved that if (M, E) in DARE (3) is a stabilizable pair and (C, M) is a detectable pair,1 then DARE (3) has a unique symmetric positive definite solution, see [7, 12, 22]. Under this assumption, with B = EG−1 ET and R = CT C , the DARE (3) can be rewritten in the symmetric form as Eq. (2). The symmetric positive definite solution of Eq. (2) or, equivalently, the DARE (3) is of theoretical and practical importance in some control problems, see [1, 3, 8, 13, 15, 16, 23] and the references therein. For finding the unique symmetric positive definite solution, Komaroff [11] proposed a fixed-point iteration
Xk+1 = M T (Xk−1 + B)−1 M + R,
k = 0, 1, 2, … ,
(4)
and proved that the matrix sequence {Xk } converges to the unique positive definite solution when M is nonsingular, R > 0 and B > 0 . Later, Dai
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