Methods of Linear Algebra in the Analysis of Certain Classes of Nonlinear Discretely Transformative Systems. I. Multipli
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METHODS OF LINEAR ALGEBRA IN THE ANALYSIS OF CERTAIN CLASSES OF NONLINEAR DISCRETELY TRANSFORMATIVE SYSTEMS. I. MULTIPLICATIVE NONLINEAR SYSTEMS
UDC 519.6
V. A. Stoyan
Abstract. The ideas and methods of pseudo-inversion of linear algebraic systems are propagated to problems of constructing the best root mean square approximation to solutions of nonlinear discretely transformative systems. The cases are considered where the form of nonlinearity is defined by a Cartesian product or iterative specification of linearly transformed input. Pseudo-solutions of quadratic nonlinear systems and systems of arbitrary order of nonlinearity are constructed and analyzed for accuracy and uniqueness. Keywords: pseudo-inversion, nonlinear discretely transformative systems, nonlinear algebraic systems, multiplicative nonlinear systems. INTRODUCTION The pseudo-inverse approach to mean square inversion of linear algebraic transformations, proposed by N. F. Kirichenko [1, 2] and generalized in [3] to linear integral and functional systems, has allowed successful solution by root mean square criterion [4, 5] to a series of direct and inverse problems of the dynamics of incompletely observed linear spatially distributed processes and phenomena. In applying the ideas from [4, 5] for the analysis of quasilinear and nonlinear systems [6, 7], the fundamental results from [1, 2] on pseudo-inversion of linear algebraic transformations also obtained further development. In the present paper, we will use the ideas and methods from [7] to generate root mean square approximations to solution of some classes of nonlinear discretely transformative systems with vector inputs–outputs and analyze their accuracy and uniqueness. We will consider the cases of multiplicative and iteratively specified quadratic nonlinearities without linearly transformative part and will generalize the results for systems with arbitrary order of nonlinearity as well. These studies of the considered nonlinearly transformative systems can be used [6] to solve problems of mathematical modeling of the state of some classes of nonlinear spatially distributed systems and their control. MATHEMATICAL RESULTS ON PSEUDO-INVERSION OF LINEARLY TRANSFORMATIVE ALGEBRAIC SYSTEMS We will not list the mathematical problems whose solution involves the apparatus of classical linear algebra to solve linear algebraic equations (1) Ax = b , Taras Shevchenko National University of Kyiv, Kyiv, Ukraine, [email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 1, January–February, 2019, pp. 127–134. Original article submitted February 14, 2018. 1060-0396/19/5501-0109 ©2019 Springer Science+Business Media, LLC
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where A and b are given ( L ´ M )- and L-dimensional matrix and vector and x Î R M needs to be found. The number of such problems is great and ability to work with system (1) is priceless. We will proceed on the basis that system (1) may have a solution (one or several) and may not have it. The latter case is not considered in classical algebra. However, in Kirichenko’s
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