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Algebraic solution of tropical best approximation problems. / Кривулин, Николай Кимович.

In: Mathematics, Vol. 11, No. 18, 3949, 17.09.2023.

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@article{05f4b12f0c614edca70b6e2b41b0ebc3,
title = "Algebraic solution of tropical best approximation problems",
abstract = "We introduce new discrete best approximation problems, formulated and solved in the framework of tropical algebra, which deals with semirings and semifields with idempotent addition. Given a set of samples, each consisting of the input and output of an unknown function defined on an idempotent semifield, the problem is to find a best approximation of the function, by tropical Puiseux polynomial and rational functions. A new solution approach is proposed, which involves the reduction of the problem of polynomial approximation to the best approximate solution of a tropical linear vector equation with an unknown vector on one side (a one-sided equation). We derive a best approximate solution to the one-sided equation, and we evaluate the inherent approximation error in a direct analytical form. Furthermore, we reduce the rational approximation problem to the best approximate solution of an equation with unknown vectors on both sides (a two-sided equation). A best approximate solution to the two-sided equation is obtained in numerical form, by using an iterative alternating algorithm. To illustrate the new technique developed, we solve example approximation problems in terms of a real semifield, where addition is defined as maximum and multiplication as arithmetic addition (max-plus algebra), which corresponds to the best Chebyshev approximation by piecewise linear functions.",
keywords = "tropical semifield, tropical Puiseux polynomial, best approximate solution, discrete best approximation, Chebyshev approximation",
author = "Кривулин, {Николай Кимович}",
note = "Krivulin N. Algebraic solution of tropical best approximation problems // Mathematics. 2023. Vol. 11, N 18. P. 3949. DOI: 10.3390/math11183949. URL: https://www.mdpi.com/2227-7390/11/18/3949",
year = "2023",
month = sep,
day = "17",
doi = "10.3390/math11183949",
language = "English",
volume = "11",
journal = "Mathematics",
issn = "2227-7390",
publisher = "MDPI AG",
number = "18",

}

RIS

TY - JOUR

T1 - Algebraic solution of tropical best approximation problems

AU - Кривулин, Николай Кимович

N1 - Krivulin N. Algebraic solution of tropical best approximation problems // Mathematics. 2023. Vol. 11, N 18. P. 3949. DOI: 10.3390/math11183949. URL: https://www.mdpi.com/2227-7390/11/18/3949

PY - 2023/9/17

Y1 - 2023/9/17

N2 - We introduce new discrete best approximation problems, formulated and solved in the framework of tropical algebra, which deals with semirings and semifields with idempotent addition. Given a set of samples, each consisting of the input and output of an unknown function defined on an idempotent semifield, the problem is to find a best approximation of the function, by tropical Puiseux polynomial and rational functions. A new solution approach is proposed, which involves the reduction of the problem of polynomial approximation to the best approximate solution of a tropical linear vector equation with an unknown vector on one side (a one-sided equation). We derive a best approximate solution to the one-sided equation, and we evaluate the inherent approximation error in a direct analytical form. Furthermore, we reduce the rational approximation problem to the best approximate solution of an equation with unknown vectors on both sides (a two-sided equation). A best approximate solution to the two-sided equation is obtained in numerical form, by using an iterative alternating algorithm. To illustrate the new technique developed, we solve example approximation problems in terms of a real semifield, where addition is defined as maximum and multiplication as arithmetic addition (max-plus algebra), which corresponds to the best Chebyshev approximation by piecewise linear functions.

AB - We introduce new discrete best approximation problems, formulated and solved in the framework of tropical algebra, which deals with semirings and semifields with idempotent addition. Given a set of samples, each consisting of the input and output of an unknown function defined on an idempotent semifield, the problem is to find a best approximation of the function, by tropical Puiseux polynomial and rational functions. A new solution approach is proposed, which involves the reduction of the problem of polynomial approximation to the best approximate solution of a tropical linear vector equation with an unknown vector on one side (a one-sided equation). We derive a best approximate solution to the one-sided equation, and we evaluate the inherent approximation error in a direct analytical form. Furthermore, we reduce the rational approximation problem to the best approximate solution of an equation with unknown vectors on both sides (a two-sided equation). A best approximate solution to the two-sided equation is obtained in numerical form, by using an iterative alternating algorithm. To illustrate the new technique developed, we solve example approximation problems in terms of a real semifield, where addition is defined as maximum and multiplication as arithmetic addition (max-plus algebra), which corresponds to the best Chebyshev approximation by piecewise linear functions.

KW - tropical semifield

KW - tropical Puiseux polynomial

KW - best approximate solution

KW - discrete best approximation

KW - Chebyshev approximation

UR - https://www.mendeley.com/catalogue/19182993-c507-3d4b-8efe-68dcf00762be/

U2 - 10.3390/math11183949

DO - 10.3390/math11183949

M3 - Article

VL - 11

JO - Mathematics

JF - Mathematics

SN - 2227-7390

IS - 18

M1 - 3949

ER -

ID: 111057604