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

Abstract

We investigate a set of nonlinear matrix equations with nonnegative matrix coefficients which has arisen in applied sciences. There are papers where the minimal nonnegative solution of the set of nonlinear matrix equations is computed applying the different procedures. The alternate linear implicit method and its modifications have intensively investigated because they have simple computational scheme. We construct a new decoupled modification of the alternate linear implicit procedure to compute the minimal nonnegative solution of the considered set of equations. The convergence properties of the proposed iteration are derived and a sufficient condition for convergence is derived. The performance of the proposed algorithm is illustrated on several numerical examples. On the basis of the experiments we derive conclusions for applicability of the computational schemes.

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

“…Here, the computations of the inverse matrices of (γ i I n + D i ) and (γ i I n + A i ) are executed in the beginning of the iterative process, i.e., it operates only one time. This fact significantly reduces the computational cost throughout the iteration process, which is confirmed by the numerical experiments executed in [15].…”

“…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.…”

“…Iteration (3) computes two inverse matrices at each iteration step. In order to avoid the computation of inverse matrices at each step, we have proposed a modification, as in [15]:…”

“…Iteration (4) and its convergence properties are derived in [15]. Here, the computations of the inverse matrices of (γ i I n + D i ) and (γ i I n + A i ) are executed in the beginning of the iterative process, i.e., it operates only one time.…”