Numerical methods for the minimal non‐negative solution of the non‐symmetric coupled algebraic Riccati equation

Abstract

In this paper, base on the theory of the non‐symmetric algebraic Riccati equation and the coupled Riccati equation, we give the general form of the non‐symmetric coupled algebraic Riccati equation (NCARE). In order to effectively solve the minimal non‐negative solution of the NCARE, two numerical iteration methods are improved: inexact Newton method (INewton) and alternate linear implicit method (ALI). Further, we give the convergence theory of the two iteration methods. Finally, we offer numerical examples to show the effectiveness of the derived methods.

Newton’s method for the positive solution of the coupled algebraic Riccati equation applied to automatic control

“…2, it can be seen that MNM and RMNM are convergent faster than NM. Experiments in Zhang and Tan (2019) show that the equation has a positive solution. In Table 1, we have recorded IT, Error, and CPU time in second (denoted by "CPU") for three methods with ω = 1.03.…”

On the Iterative Methods for the Solution of Three Types of Nonlinear Matrix Equations

“…Our investigation follows the ideas of Bai and coauthors in [9], Ma and Lu [10], Guan and Lu [11], Guan [12], and Ivanov and Yang [13]. In fact, we propose a new modification of the alternate linear implicit method introduced in [14] and modified in [15]. We propose a different iterative method to compute the minimal nonnegative solution and derive convergence properties of the new iteration.…”

Iterative Computing the Minimal Solution of the Coupled Nonlinear Matrix Equations in Terms of Nonnegative Matrices

“…We will show the effectiveness of the proposed new iterations. We compare the numerical behaviour of the new iterations with the ALI iteration method (ALI) in [17]. Two hundred runs are executed for each example for a fixed value of n (the size on matrix coefficients).…”

“…All iterative methods are executed in MATLAB (version R2014a) on a personal computer. The iterations stop when the current iterative step satisfies RES i ≤ 1.0e − 12, where RES i is defined as [17]:…”

“…Research on numerical methods to compute the minimal nonnegative solution of the set of nonsymmetric coupled Riccati equations (SNCRE) associated with M-matrices is important topic in recent years. Zhang and Tan [17] have investigated two numerical iteration methods for solving a SNCRE, i.e. the inexact Newton method and the alternate linear implicit method (ALI).…”