Solving a nonlinear matrix equation by Newton's method

Abstract

Nonlinear matrix equations are often used in the quality theory of ordinary differential, functional differential, differential-algebraic and integro-differential equations, in the theory of motion stability, control theory, and in image reconstruction problems. In this paper, we study a nonlinear matrix equation with respect to an unknown rectangular matrix. In general, the linearization of a nonlinear matrix equation with respect to an unknown rectangular matrix defines a linear matrix operator that has no inverse. For such a nonlinear matrix equation, it is not possible to use the classical Newton method, but the Newton-Kantorovich method is applicable. The paper proposes original conditions for solvability and a scheme for finding solutions to a nonlinear matrix equation. To find approximations to solutions of nonlinear matrix equations in the case of an unknown rectangular matrix and to verify the convergence of the constructed iterative scheme, the paper uses the Newton method. To verify the effectiveness of the constructed iterative scheme, we find the nonconformities of the obtained approximations in the solution of a nonlinear matrix algebraic equation.