Gradient based iterative solutions for general linear matrix equations

Abstract

a b s t r a c tIn this paper, we present a gradient based iterative algorithm for solving general linear matrix equations by extending the Jacobi iteration and by applying the hierarchical identification principle. Convergence analysis indicates that the iterative solutions always converge fast to the exact solutions for any initial values and small condition numbers of the associated matrices. Two numerical examples are provided to show that the proposed algorithm is effective.

General Recurrent Neural Network for Solving Generalized Linear Matrix Equation

“…If the linear activation function is adopted, the general recurrent neural network model reduces to the linear model (10). Such linear model (10) possesses global exponential convergence property.…”

“…Correspondingly, we will have the following theorems on the two neural network models' convergence properties. (2) is activated by the power sum function ( ) = ∑ =1 2 −1 , the state matrix ( ) ∈ × of (2) can globally and superiorly converge to the unique theoretical solution * ∈ × , as compared with linear model (10).…”

“…The proof is thus complete. (2) is activated by hyperbolic sine function ( ) = exp( )/2 − exp(− )/2 with coefficient ⩾ 1, the state matrix ( ) ∈ × of (2) can globally and superiorly converge to the unique theoretical solution * ∈ × , as compared with the linear model (10).…”

“…In many cases, the number of solutions of (1) can be multiple or even none, depending on what kind of combinations matrices and would make to associate with the unknown matrix . A lot of conventional serial approaches may be not efficient enough to solve online GLME due to their inherent drawbacks, and parallel computational approaches seem more preferable [8][9][10][11][12][13].…”

“…(10) is employed to solve GLME (1), starting from initial condition (0) ∈ × , the state matrix ( ) ∈ × of (10) can globally exponentially converge to unique theoretical solution * ∈ × .…”