Numerical Methods For Solving Nonsymmetric Algebraic Riccati Equation

Numerical method for solving nonsymmetric algebraic Riccati equation (NARE) is an important subject in the area of numerical algebra. Researchers pay much attention to the computation of the minimal nonnegative solution of NARE. There have been a lot of papers discussed the properties of NARE and presented iterative methods, forming a theoretical system. These methods can be improved, and furthermore, new methods will be constructed by some ideas and techniques. All the above is the original intention of this paper.The paper is organized as follows:●Section 1, the problem of NARE and the resent work are presented.●Section 2, firstly we introduce some useful definitions and lemmas, and then discuss how the choice of the parameter in ALI algorithm influences the convergence rate and find its optimal value in the sense of monotone convergence.●Section 3, The ALI algorithm divides the original equation into two linear equations, constituting alternate iterations. This iteration is considered as the outer iteration. Based on the splitting of the coefficient matrices of both equations, we gain the inner iteration. By combing the two separate iterations, the integrated two-stage iteration method is established.●Section 4, when the matrix K composed of the four coefficient matrices of NARE is irreducible singular M-matrix, it is proved that the ALI algorithm based on shift technique is still an effective solver, and the effectiveness is testified by the numerical experiment.