Алгоритм розкладу цілих чисел і гладкого наближення функцій

Abstract

The problem of expansion in powers is generalized into decomposition of positive integers in the sequence of degrees of different orders, the con-ditions of decomposition are determined, and the algorithm for decomposi-tion is constructed. The algorithm is based on two procedures: 1) achieve-ment a minimum of residual at each algorithm step; 2) speeding of decom-position through expanding the local base by reducing decomposition in-dex, which ensures finiteness of algorithm. The algorithm has such effi-ciency factors as high rate of decomposition, ease of implementation, availability of different options for the decomposition of numbers as in ex-tended, 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 opti-mal 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∈Ris achieved. Correctness of algo-rithm 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 replac-ing 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 polynomi-als is one of the promising directions of integration of ill-conditioned non-differentiable functions and smooth replacement of variables in the catas-trophe theory.Posinomials (functions with a variable exponent) predict the step of splitting the integration interval into parts, since they determine the loga-rithmic 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 accura-cy of approximation.