Research output: Contribution to journal › Article › peer-review
Algebraic solution of tropical best approximation problems. / Кривулин, Николай Кимович.
In: Mathematics, Vol. 11, No. 18, 3949, 17.09.2023.Research output: Contribution to journal › Article › peer-review
}
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