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Solving two generalized nonlinear matrix equations Peter Chang-Yi Weng1 Received: 18 August 2020 / Revised: 9 October 2020 / Accepted: 14 October 2020 © Korean Society for Informatics and Computational Applied Mathematics 2020

Abstract In this paper, we consider the numerical solutions of two generalized nonlinear matrix equations. Newton’s method is applied to compute one of the generalized nonlinear matrix equations and a generalized Stein equation is obtained, then we adapt the generalized Smith method to find the maximal Hermitian positive definite solution. Furthermore, we consider the properties of the solution for the generalized nonlinear matrix equation. Newton’s method is also applied to the other generalized nonlinear matrix equation to find the minimal Hermitian positive definite solution. Finally, two numerical examples are presented to illustrate the effectiveness of the theoretical results and the convergence behaviour of the considered methods for two generalized nonlinear matrix equations, respectively. Keywords Generalized nonlinear matrix equations · Newton’s method · Maximal and minimal Hermitian positive definite solutions · Generalized Smith method · Perturbation analysis Mathematics Subject Classification 15A18 · 15A22 · 15A24 · 65F15 · 65F50

1 Introduction Consider F(X ) ≡ Q − X −

m 

Ai∗ X −1 Ai = 0,

(1)

Ai∗ X −1 Ai = 0,

(2)

i=1

and  )≡Q−X+ F(X

m  i=1

B 1

Peter Chang-Yi Weng [email protected] Guangdong-Taiwan College of Industrial Science and Technology, Dongguan University of Technology, Dongguan 523808, Guangdong, China

123

P. C.-Y. Weng

where X , Q, Ai ∈ Cn×n , X = X ∗ and Q is a Hermitian positive definite matrix (Q > 0). Writing the generalized nonlinear matrix equations (GNMEs) in (1) and m Ai∗ X −1 Ai = Q, we shall refer them respectively as (2) in orthodox forms X ± i=1 GNME+ and GNME− . Firstly, we are interested in the (maximal) positive definite solution of the GNME+ with m ≥ 2, later we discuss the (minimal) positive definite solution of the GNME− with m ≥ 2 in the Sect. 3. Solvability, condition and perturbation of (1) have been considered in [10,24,33,34]. The simpler case where m = 1 has been considered in [3,13,19,21,23,30,39,41,44,45], and the case where m = 2 was studied in [25,27,40]. Generalizations have been considered in [12,26,36]. For m = 1, the GNME+ arises in various applications [3,13,19,21,23,30,39,41,44, 45], such as the solution of palindromic eigenvalue problems [8], with applications in the computation of GreenÕs function in nano research [16–18,20] and surface acoustic simulations. For m = 1, the GNME− arises from, e.g., ladder networks, dynamic programming, control theory, stochastic filtering and statistics [1,2,6,32,35,43]. For m > 1, (2) comes from the nonlinear matrix equation Q − X + A∗ ( Xˆ − C)−1 A = 0,

(3)

where Q is an n × n positive definite matrix, C is an mn × mn positive semidefinite matrix, A is an arbitrary mn × n matrix and Xˆ is the m × m block diagonal matrix with on each diagonal entry the n × n matrix X . In [37],