Key Aspects of Using Numerical Methods in the Mechanism of Investment of Small and Medium-sized Business Enterprises

In today's realities of investment activities, small and medium enterprises are faced with the dilemma of choosing areas of investment of monetary and non-monetary resources. Determining the feasibility of a particular investment is based on the use of appropriate mathematical apparatus and economic-mathematical models. The article considers some of the most important aspects of the use of numerical methods in the mechanism of investment of small and medium enterprises, developed on the basis of research on the development of such economic entities. The logical basis of the proposed methods is a proven division of the term of operation of enterprises into several stages of the life cycle, which differ significantly in the characteristics of incoming and outgoing cash flows. It is proposed to determine the absolute expediency of founding small and medium enterprises by calculating the probabilistic values of profitability of the enterprise based on the calculation of the area of geometric shapes limited by the functions of income and expenses during all stages of the life cycle. It is proved that such calculations are expedient by calculating the indefinite integral and applying the Newton-Leibniz formula.With regard to the functions that characterize the income and expenses of small and medium enterprises throughout their existence, the article proves that the most optimal in this case is the use of production functions. Among the whole set of existing production functions, we have identified the most optimal for the task Cobb-Douglas production function, CES production function, Linear production function, Leontief production function, adapted to the conditions of the microlevel. To calculate the elementary areas limited by the corresponding production functions, the article examines the relevant existing methods. Studies have been conducted on the possibility of using to calculate the areas limited by the production functions of income and expenses of small and medium enterprises during each stage of the life cycle, the methods of rectangles, trapezoids and parabolas. It is determined that they have limited conditions for use due to the need to divide the time period of operation of enterprises at intervals that do not coincide with the stages of the life cycle. Therefore, studies were continued, as a result of which it was proved that the most optimal method of calculating the elementary areas limited by the target functions of income and expenditure of small and medium enterprises is the Gaussian method with different variations