Examining a Mathematical Apparatus of Z-approximation of Functions for the Construction of an Adaptive Algorithm

The result of this research is the proposed to mathematical apparatus and a procedure for constructing adaptive algorithm based on Z-approximation of functions. A given study is required to improve approaches to constructing algorithms that change their performance in response to changes in input information. This, in turn, significantly improves results in solving the problems that can be implemented using such an algorithm. For example, solving nonlinear problems, description of complex surfaces, search for information.It has been shown that the solutions derived in the current study are in agreement with the application of the same algorithms for separate groups of functions used for approximation. These functions are used when constructing a direction to search for and provide an opportunity to build a model of error in Z-approximation using the initial or final approximations.The definition of Zm-approximation has been given as the approximation with a multiple interval reduction that simplifies recurrent formulae and is a feature of the presented approach. The proposed methodology and the basic algorithm make it possible to directly determine a series of common and hyperbolic functions using Zm-approximations and parallel computing. Based on the research results, an adaptive algorithm has been presented to calculate arctg x as a function inverse to tg x.The above can be used when constructing an adaptive search algorithm in the arrays of unstructured or poorly structured information. Such a search is employed for books and textbooks, uploaded to the Internet in formats jpeg, pdf, or as fragments of the specified formats. In this case, based on the adaptive algorithm, a special model is constructed, which can be implemented according to several variants with a change in direction