A METHOD FOR SOLVING GENERALIZED ONE-PARAMETRIC QUADRATIC MATRIX EQUATIONS

Abstract

A spectral method for solving one-parametric generalized quadratic matrix equations with squared matrix-coefficients based on the symbolic apparatus of G.E. Pukhov’s differential transformations is proposed. The considered one-parametric generalized quadratic matrix equations generates partial matrix equations, such as a one-parametric quadratic matrix equation, one-parametric modified quadratic matrix equation, Sylvester’s one-parametric continuous matrix equation, Lyapunov’s one-parametric matrix equation, one-parametric reduced generalized quadratic matrix equation, Riccati’s one-parametric matrix equation, etc. The spectral model of one-parametric generalized quadratic matrix equations in the field of differential transformations, allowing to define the initial matrix discrete solutions of the problem is obtained. In that case, a nonlinear algebraic quadratic matrix equation is used in the zero-step of iterative procedures, as well as the rest of matrix discrete solutions of the problem, using, in that case, recurrent computational procedures with invariant hypermatrix, relative to the numbers of unknown matrix discrete. A model example of the second order is considered. Its solution, based on using the proposed method and the Newton’s method, shows the high computational efficiency of the first method. The conditions of the problem solvability are obtained.