Methods of Linear Algebra in the Analysis of Certain Classes of Nonlinear Discretely Transformative Systems. II. Systems
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METHODS OF LINEAR ALGEBRA IN THE ANALYSIS OF CERTAIN CLASSES OF NONLINEAR DISCRETELY TRANSFORMATIVE SYSTEMS. II. SYSTEMS WITH ADDITIVELY SELECTED NONLINEARITY
UDC 510 + 004.94
V. A. Stoyan
Abstract. Pseudo-solutions of discretely transformative systems are generated; their linear part is complemented with nonlinearities obtained after the Cartesian transformation of input vector or iterative specification of matrix kernel of the transformer. Sets of root-mean-square approximations to inversion of mathematical model of the transformer are analyzed for accuracy and uniqueness. Quadratically nonlinear systems and systems with arbitrary order of nonlinearity are considered. Keywords: pseudo-inversion, nonlinear discretely transformative systems, nonlinear algebraic systems, nonlinear iteratively specified systems. INTRODUCTION The paper is a continuation of [1], where ideas of pseudo inversion of classical linearly definite transformers [2–4] were used for problems of generating pseudo-inversions of nonlinear discretely transformative systems [5, 6]. Algorithms of construction and analysis of the accuracy and uniqueness of pseudo-solutions of nonlinear algebraic systems obtained by double and multiple Cartesian multiplication of linearly transformed input vector and by its iterative specification were proposed. We will formulate and solve problems of pseudo-inversion of discretely transformative systems in which the nonlinearities considered in [1] are additions to the classical algebraically definite transformer. As well as in [1], sets of solutions (if any) or of its best root-mean-square approximations (if there is no exact solution) are constructed for systems under study. As well as in [1], conditions of uniqueness of the generated sets are written. The obtained mathematical results are simple and available for computer implementations and can be applied (according to the technique [4]) to the problem of pseudo-inversion of integral and functional transformers and hence to problems [6] of mathematical modeling of nonlinear incompletely definite spatially distributed systems. MATHEMATICAL STATEMENT OF THE PROBLEM AND FUNDAMENTALS OF ITS SOLUTION Let us consider processes and phenomena whose discretely definite input u Î R M can be transformed into output vector y Î R L . We will proceed from the classical algebraically linear model of such transformer, which we will write as
A0 u = y ,
(1)
where A0 Î R L´ M is matrix kernel of the transformer. Taras Shevchenko National University of Kyiv, Kyiv, Ukraine, [email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 2, March–April, 2019, pp. 102–107. Original article submitted February 8, 2018. 1060-0396/19/5502-0259 ©2019 Springer Science+Business Media, LLC
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Let us consider the case where the mathematical model (1), which is analyzed quite completely, is supplemented with nonlinear part F ( u ) so that (2) A0 u + F ( u ) = y , for the given ( L ´ M )-dimensional matrices A0 , K , A N ,
F ( u ) = A1 u Ä A2 u ,
(3)
F ( u ) = ( A1 u , K , A M
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