Algorithm for Decomposition of Integers and Smooth Approximation of Functions

• Дата: 2022
• Автори: Абрамчук, Василь, Абрамчук, Ігор
• Формат: Стаття
• Мова: Українська
• Опубліковано: Кам'янець-Подільський національний університет імені Івана Огієнка 2022
• Онлайн доступ: http://mcm-math.kpnu.edu.ua/article/view/274029
• Назва журналу: Mathematical and computer modelling. Series: Physical and mathematical sciences

The problem of expansion in powers is generalized into decomposition of positive integers in the sequence of degrees of different orders, the conditions of decomposition are determined, and the algorithm for decomposition is constructed. The algorithm is based on two procedures: 1) achievement a minimum of residual at each algorithm step; 2) speeding of decomposition through expanding the local base by reducing decomposition index, which ensures finiteness of algorithm. The algorithm has such efficiency factors as high rate of decomposition, ease of implementation, availability of different options for the decomposition of numbers as in extended, narrowed, sparse bases, which protects the encoded information from external influences. The algorithm can be used to encode large amounts of digital information under basic systems of small dimensions. Decomposition of positive integers into a sequence of powers is optimal and correct. Optimality of decomposition follows from the condition that at each step of algorithm the minimum value of disjunction in the space of mixed parameters x ∈ N, y ∈ R is achieved. Correctness of algorithm is due to the fact that when the disjunction is reduced, the algorithm expands the basis of decomposition by reducing the degree indicators by one. By switching from a discrete model to a continuous model by replacing the degrees with power functions, we obtain a smooth approximation of the ill-conditioned function in the neighborhood of decomposition. The construction of posinomial polynomials on the basis of smooth polynomials is one of the promising directions of integration of ill-conditioned nondifferentiable functions and smooth replacement of variables in the catastrophe theory. Posinomials (functions with a variable exponent) predict the step of splitting the integration interval into parts, since they determine the logarithmic rate of change of an arbitrary monotonic function. The method of decomposition of positive integers provides an optimal decomposition into the sum of powers, and therefore the transition from a discrete model to a continuous model in the neighborhood of decomposition by replacing powers with power functions as well as allows to achieve the high accuracy of approximation.