Two modified least-squares iterative algorithms for the Lyapunov matrix equations

Abstract

In this paper two modified least-squares iterative algorithms are presented for solving the Lyapunov matrix equations. The first algorithm is based on the hierarchical identification principle, which can be viewed as a surrogate of the least-squares iterative algorithm proposed by Ding et al., whose convergence has not been proved until now. The second one is motivated by a new form of fixed point iterative scheme. With the tool of a new matrix norm, the proof of both algorithms' global convergence is offered. Furthermore, the feasible sets of their convergence factors are analyzed. Finally, a numerical example is presented to illustrate the rationality of theoretical results.

General Six-Step Discrete-Time Zhang Neural Network for Time-Varying Tensor Absolute Value Equations

“…However, to the best of the authors' knowledge, there is no research on the discrete-time Zhang neural network equipped with nonlinear activation function in the literature, and designing such a discrete-time Zhang neural network maybe an interesting research direction. (3) It is worth to research the application of discrete-time Zhang neural network in the nonsmooth LASSO problem [7], the multi-augmented Sylvester matrix problem [9,24].…”

“…ZNN is an interesting research topic in the literature, and has been extensively studied in past years. Many e cient DTZNN models have been proposed in the literature, including the general two-step DTZNN model in [6] (its steady-state residual error changes in an O 2 manner), the general three or four-step DTZNN models in [7,8] (their steady-state residual errors change in an O 3 or O 4 manner), and the general ve-step DTZNN model in [9] (its steady-state residual error changes in an O 4 manner), where > 0 is the sampling gap. Other special DTZNN models, which are included in the above general DTZNN models, can be found in [10][11][12] and the references therein.…”

“…Other special DTZNN models, which are included in the above general DTZNN models, can be found in [10][11][12] and the references therein. Note that the step size is an important parameter which determines the convergence of the corresponding DTZNN, and the feasible region of the step size in [6][7][8][9][10] is rigorously proved, while [11,12] only present the feasible region without rigorous proof. Meanwhile, the corresponding optimal step size is also investigated in [6,7,9] by establishing some optimization model.…”

A novel noise-tolerant Zhang neural network for time-varying Lyapunov equation

“…Due to the important role that the time-varying Lyapunov equation plays in a broad spectrum of areas, there has been a rapid increase in its algorithm design, and many numerical methods and neural dynamics have been proposed to solve this problem and its timeinvariant version; see, e.g., [1][2][3][4][5][6] on this subject. Let t 0 ∈ R and t f ∈ R denote the start and the final time instant of the solving process, respectively.…”

Gradient-descent iterative algorithm for solving a class of linear matrix equations with applications to heat and Poisson equations

“…There are several ideas to formulate an iterative procedure, namely, one can use matrix sign function [5], block recursion [6,7], Krylov subspace [8,9], Hermitian and skew-Hermitian splitting [10,11], and other related research works; see, e.g., [12][13][14][15]. In the recent decade, the ideas of gradients, hierarchical identification and minimization of associated norm-error functions have encouraged and brought about many researches; see, e.g., [16][17][18][19][20][21][22][23][24][25][26][27][28]. Such iterative schemes turn out to have wide applications in many engineering problems, especially in systems identification for parameter estimation; see, e.g., [29][30][31].…”