Conditions for the solvability of nonlinear equations systems in Euclidean spaces
Open Access
- 1 January 2021
- journal article
- Published by Taras Shevchenko National University of Kyiv in Bulletin of Taras Shevchenko National University of Kyiv. Series: Physics and Mathematics
- No. 1,p. 74-80
- https://doi.org/10.17721/1812-5409.2021/1.9
Abstract
The solution of many applied problems is to find a solution of nonlinear equations systems in finite- dimensional Euclidean spaces. The problem of finding 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 find 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 finite-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.Keywords
This publication has 3 references indexed in Scilit:
- Operator Interpolation and Systems of Linear Equations and Inequalities in Euclidean SpacesUkrainian Mathematical Journal, 2021
- A generalization of the Newton-Kantorovich method for systems of nonlinear real equationsReports of the National Academy of Sciences of Ukraine, 2020
- Simultaneous diagonalization of three real symmetric matricesRussian Mathematics, 2014