Conditions for the solvability of nonlinear equations systems in Euclidean spaces

O. F. Kashpur

doi:10.17721/1812-5409.2021/1.9

Authors

O. F. Kashpur Taras Shevchenko National University of Kyiv https://orcid.org/0000-0001-7091-8851

DOI:

https://doi.org/10.17721/1812-5409.2021/1.9

Keywords:

interpolation polynomial, system of nonlinear equations, Euclidean space, solvability conditions

Abstract

The solution of many applied problems is to ?nd a solution of nonlinear equations systems in ?nite- dimensional Euclidean spaces. The problem of ?nding the solution of a nonlinear system is divided into two problems: 1. The existence of a solution of a nonlinear equations system; in the case of nonunique of the solution, it is necessary to ?nd the number of these solutions and their surroundings. 2. Finding the solution of a system of nonlinear equations with a given accuracy. Many publications are devoted to solving problem 2, namely the construction of iterative methods, their convergence and estimates of the solution accuracy. In contrast to problem 2, for problem 1 there is no general algorithm for solving this task, there are no constructive conditions for the existence of a solution of a nonlinear equations system in Euclidean spaces. In this article, in ?nite-dimensional Euclidean spaces, the constructive conditions for the existence of a solution of nonlinear systems of polynomial form are found. The connection of these conditions with the linear polynomial interpolant of the minimum norm, generated by a scalar product with Gaussian measure and the conditions of its existence, is given.

Pages of the article in the issue: 74 - 80

Language of the article: Ukrainian

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