The theory of differential-difference equations stability solutions is currently one of the most important and actively studied sections of their general theory. The problem of studying the stability of linear stationary differential-difference equations is to find the conditions of negativeness of the real parts of asymptotic roots of quasi-polynomials, which can be found, in general, only by approximate methods. For differential-difference equations, the schemes of their approximation are constructed and substantiated by means of special systems of ordinary differential equations. In case of linear differential-difference equations, roots of the approximating system characteristic equation of ordinary differential equations can be taken as approximate values of the non-asymptotic roots of the corresponding quasi-polynomials. In this paper, we consider two algorithms for the approximate finding of non-asymptotic roots of quasi-polynomials of neutral type differential difference equations, based on the Krasovsky-Repin approximation scheme and the higher accuracy approximation scheme. The form of characteristic equations for ordinary differential equations approximating systems is obtained, which are convenient to use for calculation of their roots. Numerical experiments for a model test example were performed and their results were analyzed.

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