Algebraic solution of tropical best approximation problems
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We consider discrete best approximation problems formulated and solved in the framework of tropical algebra that deals with semirings and semifields with idempotent addition. Given a set of samples each consisting of input and output of an unknown function defined on an idempotent semifield, the problems are 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 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 end 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 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 correspond to the best Chebyshev approximation by piecewise linear functions.
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