Умови розв’язуваностi систем нелiнiйних рiвнянь в евклiдових просторах

O F Kashpur

doi: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.

Highlights

The solution of many applied problems is to find a solution of nonlinear equations systems in finitedimensional 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
Many publications are devoted to solving problem 2, namely the construction of iterative methods, their convergence and estimates of the solution accuracy

Summary

Допомiжнi результати

Розглянемо систему лiнiйних рiвнянь αj1x1 + αj2x2 + · · · + αjnxn − yj = 0, (1). М. Чернiков) Для того, щоб система лiнiйних рiвнянь (1) мала хоча б один розв’язок строго вiд’ємний (сторого додатнiй) вiдносно сукупностi невiдомих xn, xn2, . Xnk , 1 k n i недодатнiй (невiд’ємний) вiдносно сукупностi iнших невiдомих xp, xp2, . 2. Cистема (1) має хоча б один недодатньо (невiд’ємно) орiєнтований вiдносно сукупностi невiдомих xn , xn2 , . Sm є цiлком ∆-допустимою вiдносно сукупностi цих невiдомих. Але для перевiрки виконання умов цих теорем потрiбно обчилювати Cnr визначникiв порядку r, де r ранг матрицi системи, n - число невiдомих. В подальшому в роботi буде показано зв’язок мiж умовами iснування розв’язку нелiнiйної системи рiвнянь та умовою iснування iнтерполяцiйного полiнома в евклiдовому просторi. В [11] показано, що умова (5) є аналогом теореми Кронекера-Капеллi розв’язуваностi системи лiнiйних алгебраїчних рiвнять (1)

Умови розв’язуваностi систем нелiнiйних рiвнянь

Висновки