title: ÐÐ°Ð»Ð¾ÑÐ°Ð½Ð³Ð¾Ð²Ð¾Ðµ ÑÐ°Ð·Ð»Ð¾Ð¶ÐµÐ½Ð¸Ðµ Ð¸Ð½ÑÐµÑÐ²Ð°Ð»ÑÐ½ÑÑ Ð¼Ð°ÑÑÐ¸Ñ
title: Ð²ÑÐ¿ÑÑÐºÐ½Ð°Ñ ÐºÐ²Ð°Ð»Ð¸ÑÐ¸ÐºÐ°ÑÐ¸Ð¾Ð½Ð½Ð°Ñ ÑÐ°Ð±Ð¾ÑÐ° Ð±Ð°ÐºÐ°Ð»Ð°Ð²ÑÐ°
titleType: Subtitle
title: Low-rank decomposition of interval matrices
Creators: name: Ð¡ÑÑÐ°Ð½Ð¾Ð², ÐÐ¸ÐºÐ¸ÑÐ°
Publisher: name: Ð¡Ð°Ð½ÐºÑ-ÐÐµÑÐµÑÐ±ÑÑÐ³ÑÐºÐ¸Ð¹ Ð¿Ð¾Ð»Ð¸ÑÐµÑÐ½Ð¸ÑÐµÑÐºÐ¸Ð¹ ÑÐ½Ð¸Ð²ÐµÑÑÐ¸ÑÐµÑ ÐÐµÑÑÐ° ÐÐµÐ»Ð¸ÐºÐ¾Ð³Ð¾
Year: 2020
Subjects: ÑÐ¸Ð½Ð³ÑÐ»ÑÑÐ½Ð¾Ðµ ÑÐ°Ð·Ð»Ð¾Ð¶ÐµÐ½Ð¸Ðµ
ÑÐ¸ÑÐ»Ð¾ Ð¾Ð±ÑÑÐ»Ð¾Ð²Ð»ÐµÐ½Â­Ð½Ð¾ÑÑÐ¸
Ð»Ð¸Ð½ÐµÐ¹Ð½Ð¾Ðµ Ð¿ÑÐ¾Ð³ÑÐ°Ð¼Ð¼Ð¸ÑÐ¾Ð²Ð°Ð½Ð¸Ðµ
Ð¸Ð½ÑÐµÑÐ²Ð°Ð»ÑÐ½ÑÐ¹ Ð°Ð½Ð°Ð»Ð¸Ð·
Ð¼ÐµÑÐ¾Ð´ Ð³Ð»Ð°Ð²Ð½ÑÑ ÐºÐ¾Ð¼Ð¿Ð¾Ð½ÐµÐ½Ñ
singular value decomposition
condition number
linear programming
interval analysis
principal component analysis

description: ÐÐ°Ð½Ð½Ð°Ñ ÑÐ°Ð±Ð¾ÑÐ° Ð¿Ð¾ÑÐ²ÑÑÐµÐ½Ð° ÑÐ°Ð·Ð»Ð¸ÑÐ½ÑÐ¼ Ð¿Ð¾Ð´ÑÐ¾Ð´Ð°Ð¼ Ðº Ð¸ÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ñ Ð²Ð¾Ð¿ÑÐ¾ÑÐ° Ð¾Ð± Ð¾Ð¿ÑÐ¸Ð¼Ð°Ð»ÑÐ½Ð¾Ð¼ ÐºÐ¾Ð»Ð¸ÑÐµÑÑÐ²Ðµ ÑÐ»ÐµÐ¼ÐµÐ½ÑÐ¾Ð² ÑÐ°Ð·Ð±Ð¸ÐµÐ½Ð¸Ñ Ð¾Ð±ÑÑÐ¼Ð° Ð¿Ð»Ð°Ð·Ð¼Ñ Ð² ÑÑÐµÑÐ¸ÑÐµÑÐºÐ¾Ð¼ ÑÐ¾ÐºÐ°Ð¼Ð°ÐºÐµ. Ð ÑÐ°Ð¼ÐºÐ°Ñ ÑÐ°Ð±Ð¾ÑÑ Ð½Ð° ÑÐ°Ð·Ð½ÑÑ ÑÑÐ°Ð¿Ð°Ñ Ð¿ÑÐ¸Ð¼ÐµÐ½ÑÐ»Ð¸ÑÑ Ð¿Ð¾Ð½ÑÑÐ¸Ñ Ð¸ Ð¼ÐµÑÐ¾Ð´Ñ ÑÐ°Ð·Ð»Ð¸ÑÐ½ÑÑ Ð¾Ð±Ð»Ð°ÑÑÐµÐ¹ Ð¼Ð°ÑÐµÐ¼Ð°ÑÐ¸ÐºÐ¸: Ð»Ð¸Ð½ÐµÐ¹Ð½Ð¾Ð¹ Ð°Ð»Ð³ÐµÐ±ÑÑ, Ð²ÑÑÐ¸ÑÐ»Ð¸ÑÐµÐ»ÑÐ½Ð¾Ð¹ Ð¼Ð°ÑÐµÐ¼Ð°ÑÐ¸ÐºÐ¸, Ð»Ð¸Ð½ÐµÐ¹Ð½Ð¾Ð³Ð¾ Ð¿ÑÐ¾Ð³ÑÐ°Ð¼Ð¼Ð¸ÑÐ¾Ð²Ð°Ð½Ð¸Ñ, Ð¸Ð½ÑÐµÑÐ²Ð°Ð»ÑÐ½Ð¾Ð³Ð¾ Ð°Ð½Ð°Ð»Ð¸Ð·Ð°, Ð° ÑÐ°ÐºÐ¶Ðµ Ð¼ÐµÑÐ¾Ð´ Ð³Ð»Ð°Ð²Ð½ÑÑ ÐºÐ¾Ð¼Ð¿Ð¾Ð½ÐµÐ½Ñ Ð¸ ÐµÐ³Ð¾ Ð¸Ð½ÑÐµÑÐ²Ð°Ð»ÑÐ½ÑÐµ Ð¾Ð±Ð¾Ð±ÑÐµÐ½Ð¸Ñ. Ð ÑÐ¾Ð´Ðµ Ð¸ÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ð¹ Ð±ÑÐ» ÑÑÐ¾ÑÐ¼ÑÐ»Ð¸ÑÐ¾Ð²Ð°Ð½ Ð¸ Ð´Ð¾ÐºÐ°Ð·Ð°Ð½ ÑÑÐ´ ÑÐµÐ¾ÑÐµÐ¼ Ð¾ ÑÐ²Ð¾Ð¹ÑÑÐ²Ð°Ñ Ð·Ð°Ð´Ð°Ñ Ð»Ð¸Ð½ÐµÐ¹Ð½Ð¾Ð³Ð¾ Ð¿ÑÐ¾Ð³ÑÐ°Ð¼Ð¼Ð¸ÑÐ¾Ð²Ð°Ð½Ð¸Ñ, Ð¸ÑÐ¿Ð¾Ð»ÑÐ·ÑÑÑÐ¸ÑÑÑ Ð¿ÑÐ¸ ÑÐµÑÐµÐ½Ð¸Ð¸ Ð¸Ð½ÑÐµÑÐ²Ð°Ð»ÑÐ½ÑÑ ÑÐ¸ÑÑÐµÐ¼ Ð»Ð¸Ð½ÐµÐ¹Ð½ÑÑ Ð°Ð»Ð³ÐµÐ±ÑÐ°Ð¸ÑÐµÑÐºÐ¸Ñ ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ð¹. Ð ÑÐ°ÑÑÐ½Ð¾ÑÑÐ¸, ÑÑÐ¾ÑÐ¼ÑÐ»Ð¸ÑÐ¾Ð²Ð°Ð½Ð½ÑÐµ ÑÐµÐ¾ÑÐµÐ¼Ñ Ð¿Ð¾Ð·Ð²Ð¾Ð»ÑÑÑ ÑÐ¾ÐºÑÐ°ÑÐ¸ÑÑ ÑÐ¸ÑÐ»Ð¾ Ð²ÐµÑÐ¾Ð² Ð·Ð°Ð´Ð°ÑÐ¸, Ð½Ðµ Ð¼ÐµÐ½ÑÑ Ð¿ÑÐ¸ ÑÑÐ¾Ð¼ Ð¼Ð½Ð¾Ð¶ÐµÑÑÐ²Ð¾ ÐµÑ ÑÐµÑÐµÐ½Ð¸Ð¹. Ð ÑÐµÐ·ÑÐ»ÑÑÐ°ÑÐµ Ð±ÑÐ» ÑÐ°Ð·ÑÐ°Ð±Ð¾ÑÐ°Ð½ Ð¸ Ð¿ÑÐ¸Ð¼ÐµÐ½ÐµÐ½ Ð¿Ð¾Ð´ÑÐ¾Ð´ Ðº Ð½Ð°ÑÐ¾Ð¶Ð´ÐµÐ½Ð¸Ñ ÑÑÑÐµÐºÑÐ¸Ð²Ð½Ð¾Ð³Ð¾ (ÑÐµÐ°Ð»ÑÐ½Ð¾Ð³Ð¾) ÑÐ°Ð½Ð³Ð° Ð¼Ð°ÑÑÐ¸ÑÑ, Ð¾Ð¿Ð¸ÑÐ°ÑÑÐ¸Ð¹ÑÑ Ð½Ð° Ð°Ð½Ð°Ð»Ð¸Ð· ÑÐ°ÑÐ¿ÑÐµÐ´ÐµÐ»ÐµÐ½Ð¸Ñ ÐµÑ ÑÐ¸Ð½Ð³ÑÐ»ÑÑÐ½ÑÑ Ð·Ð½Ð°ÑÐµÐ½Ð¸Ð¹, ÑÐ¸ÑÐ»Ð° Ð¾Ð±ÑÑÐ»Ð¾Ð²Ð»ÐµÐ½Ð½Ð¾ÑÑÐ¸ Ð¸ Ð¿ÑÐ¸Ð¼ÐµÐ½ÐµÐ½Ð¸Ðµ Ð¼ÐµÑÐ¾Ð´Ð° Ð³Ð»Ð°Ð²Ð½ÑÑ ÐºÐ¾Ð¼Ð¿Ð¾Ð½ÐµÐ½Ñ. ÐÑÐ»Ð¸ ÑÐ°ÑÑÐ¼Ð¾ÑÑÐµÐ½Ñ Ð¸Ð½ÑÐµÑÐ²Ð°Ð»ÑÐ½ÑÐµ Ð¾Ð±Ð¾Ð±ÑÐµÐ½Ð¸Ñ Ð¼ÐµÑÐ¾Ð´Ð° Ð³Ð»Ð°Ð²Ð½ÑÑ ÐºÐ¾Ð¼Ð¿Ð¾Ð½ÐµÐ½Ñ Ð¸ Ð¾ÑÐ¾Ð±ÐµÐ½Ð½Ð¾ÑÑÐ¸ Ð¸Ñ Ð¿ÑÐ¸Ð¼ÐµÐ½ÐµÐ½Ð¸Ñ. Ð¢Ð°ÐºÐ¶Ðµ Ð±ÑÐ»Ð° Ð¿ÑÐµÐ´Ð»Ð¾Ð¶ÐµÐ½Ð° Ð¼Ð¾Ð´Ð¸ÑÐ¸ÐºÐ°ÑÐ¸Ñ Ð¾Ð¿ÑÐ¸Ð¼Ð¸Ð·Ð°ÑÐ¸Ð¾Ð½Ð½Ð¾Ð³Ð¾ Ð¿Ð¾Ð´ÑÐ¾Ð´Ð° Ðº ÑÐµÑÐµÐ½Ð¸Ñ Ð¸Ð½ÑÐµÑÐ²Ð°Ð»ÑÐ½ÑÑ ÑÐ¸ÑÑÐµÐ¼, Ð¸ÑÐ¿Ð¾Ð»ÑÐ·ÑÑÑÐ°Ñ Ð¼ÐµÑÐ¾Ð´ Ð³Ð»Ð°Ð²Ð½ÑÑ ÐºÐ¾Ð¼Ð¿Ð¾Ð½ÐµÐ½Ñ.
description: This paper is devoted to various approaches to the study of the optimal number of elements for splitting the plasma volume in a spherical tokamak. In this paper, concepts and methods of various areas of mathematics were applied at different stages: linear algebra, numerical mathematics, linear programming, interval analysis, as well as the principal component analysis and its interval generalizations. During the research, a number of theorems about the properties of linear programming problems used for solving interval systems of linear algebraic equations were formulated and proved. In particular, the formulated theorems allow reducing the number of weights of the problem without changing the set of its solutions. As a result, an approach to finding the effective (real) rank of the matrix was developed and applied, based on the analysis of the distribution of its singular values, the condition number, and the application of the principal component analysis. Interval generalizations of the principal component analysis and their application features were considered. A modification of the optimization approach to solving interval systems using the principal component analysis was also proposed.