Abstract: The subject of the article is the consideration and analysis of a set of algorithms for numerically finding the roots of polynomials, primarily complex ones based on methods for searching for an approximate decomposition of the initial polynomials into multipliers. If the numerical finding of real roots usually does not cause difficulties, then a number of difficulties arise with finding complex roots. This article proposes a set of algorithms for sequentially finding multiple roots of polynomials with real roots, then real roots by highlighting intervals that potentially contain roots and obviously do not contain them, and then complex roots of polynomials. To find complex roots, an iterative approximation of the original polynomial by the product of a trinomial by a polynomial of a lesser degree is used, followed by the use of the tangent method in the complex domain in the vicinity of the roots of the resulting trinomial. To find the roots of a polynomial with complex coefficients, we propose a solution to an equivalent problem with real coefficients.  The implementation of the tasks is carried out by step-by-step application of a set of algorithms. After each stage, a group of roots is allocated and the same problem is solved for a polynomial of lesser degree. The sequence of the proposed algorithms makes it possible to find all the real and complex roots of the polynomial. To find the roots of a polynomial with real coefficients, an algorithm is constructed that includes the following main steps: determining multiple roots with a corresponding decrease in the degree of the polynomial; allocating a range of roots; finding intervals that are guaranteed to contain roots and finding them, after their allocation, it remains to find only pairs of complex conjugate roots; iterative construction of trinomials that serve as an estimate of the values of such pairs with minimal the accuracy sufficient for their localization; the actual search for roots in the complex domain by the tangent method. The computational complexity of the proposed algorithms is polynomial and does not exceed the cube of the degree of the polynomial, which makes it possible to obtain a solution for almost any polynomials arising in real problems. The field of application, in addition to the polynomial equations themselves, is the problems of optimization, differential equations and optimal control that can be reduced to them.

Abstract: The subject of this research is the model of business development that describes the dependence of ongoing volume of production from previous investments and intensity of wear of production capacities. The investment process is characterized by a delay between the moment of investment, actual return and its continuation, gradual decrease in the level of return, and discreetness of investments. In the process of modeling, discrete investment were replaced by an integral, which leads to integral-differential equation, and in terms of facile assumption to the linear standard differential equation of second order or their system, solved by the disharmonious fluctuations on the background of an aperiodic trend. As the method of analysis of correspondence of the model data with the actual dynamics of business development, the research utilizes computational solution of the emerging differential equations. Comparison of the model data with the known statistics reveals their adequacy to the current economic processes. Statistical data contains noise component, which consists of various economic and political factors and principally limits the precision of forecasting. Differences in the length of fluctuation periods by industries impedes analysis of the economic behavior as a whole. At the same time, forecast of crisis phenomena that emerge in superposition of the phases of industry fluctuations can be executed with sufficient level of precision.

Abstract: The article discusses the methodology for estimating the noise component in time series with variable pitch, its justification, and suggests an algorithm for removing noise from data. The analysis is based on the requirement of smoothness of a function representing the original data and having continuous derivatives up to the third order. The proposed method and algorithms for estimating and eliminating noise in the data under the assumption of smoothness, the function they represent, allow reasonably determining both absolute and relative noise in the data, regardless of the uniformity of the measurement step in the source data, the noise level in the data, remove the noise component from the data . The algorithm for solving the problem is based on minimizing the deviations of the calculated values from the smooth function, provided that the deviations from the source data correspond to the noise level. The proposed method and algorithms for estimating and eliminating noise in the data under the assumption of smoothness, the function they represent, allow reasonably determining both absolute and relative noise in the data, regardless of the uniformity of the measurement step in the source data and their noise, and remove the noise component from the data. Considering the smoothness of the data obtained as a result of noise elimination, the data obtained by noise elimination are suitable for detecting both analytical and differential dependencies in them.

Abstract: The subject of research is the method of estimating the noise component in the time series and its removal, the selection of the trend and fluctuations with different periods, the concept of T-ε and T-h-ε almost periods for the final series is introduced. The analysis is based on the requirement of smoothness of a function representing the original data and having derivatives up to the fourth order inclusive and the allocation of almost periods based on functions of the Alter-Johnson type. Separately, the trend of the length of the periods identified in the data of a number of fluctuations. The algorithm for solving the problem is based on minimizing the deviations of the calculated values from the smooth function, provided that the deviations from the source data correspond to the noise level. To identify the oscillatory component and the trend of almost periods, the modified Alter-Johnson function is used. The proposed methodology and algorithms for estimating and eliminating noise in the data allow us to reasonably determine the noise level in the data, remove the noise component from the data, identify almost the periods in the data in the sense of the definitions introduced in the article, highlight the trend and oscillation components in the data, identify, if necessary, the trend of changes almost periods.