DOI: https://doi.org/10.15587/1729-4061.2019.170824

Examining a mathematical apparatus of Z-approximation of functions for the construction of an adaptive algorithm

Olha Kryazhych, Oleksandr Kovalenko

Abstract


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.


Keywords


search algorithm; process distribution; recurrence record; residual; approximation

References


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GOST Style Citations


Entsiklopediya kibernetiki / V. M. Glushkov et. al. (Eds.). Kyiv: Glavnaya redaktsiya ukrainskoy sovetskoy entsiklopedii, 1974. 1228 p.

Filosofskyi entsyklopedychnyi slovnyk. Kyiv: Abrys, 2002. 742 p.

Shennon K. Raboty po teorii informatsii i kibernetike. Moscow: Inostrannaya literatura, 1963. 832 p.

Searching the web: The public and their queries / Spink A., Wolfram D., Jansen M. B. J., Saracevic T. // Journal of the American Society for Information Science and Technology. 2001. Vol. 52, Issue 3. P. 226–234. doi: https://doi.org/10.1002/1097-4571(2000)9999:9999<::aid-asi1591>3.0.co;2-r 

Figueroa A. Exploring effective features for recognizing the user intent behind web queries // Computers in Industry. 2015. Vol. 68. P. 162–169. doi: https://doi.org/10.1016/j.compind.2015.01.005 

Ashmanov I., Ivanov A. Optimizatsiya i prodvizhenie saytov v poiskovyh sistemah. Sankt-Peterburg: Piter, 2011. 464 p.

Glushkov V. M. Kibernetika. Vychislitel'naya tekhnika. Informatika. Izbrannye trudy. Vol. 3. Kibernetika i ee primenenie v narodnom hozyaystve / V. S. Mihalevich et. al. (Eds.). Kyiv: Naukova dumka, 1990. 224 p.

Glushkov V. M. Kibernetika. Voprosy teorii i praktiki. Moscow: Nauka, 1986. 488 p.

Glushkov V. M., Stogniy A. A., Afanas'ev V. N. Avtomatizirovannye informatsionnye sistemy. Moscow: Znanie, 1973. 64 p.

Akademik V. M. Glushkov – pioner kibernetiki. Kyiv: Izdatel'stvo Yunior, 2003. 384 p.

Glushkov V. M. Osnovy bezbumazhnoy informatiki. izd. 2-e, ispr. Moscow: Nauka, Gl. Red. fiz.-mat. lit., 1987. 552 p.

Pleskach V. L, Zatonatska T. H. Informatsiyni systemy i tekhnolohiyi na pidpryiemstvakh: pidruchnyk. Kyiv: Znannia, 2011. 718 p.

Encyclopaedia of Mathematics (set) / M. Hazewinkel (Ed.). Springer, 1994.

Kovalenko O. V. Kontseptualni osnovy stvorennia bazy danykh naukovoho eksperymentu ta sposterezhennia // Matematychni mashyny i systemy. 2016. Issue 2. P. 91–101.

Diaz F. Autocorrelation and Regularization of Query-Based Retrieval Scores. Chap. 3: PhD thesis. Amherst, 2008.

Beer S. Brain of the Firm, Allen Lane. London: Herder and Herder, 1972. 416 р.

Beer S. Designing Freedom. House of Anansi Press, 1993. 110 р.

Algorithms and Data Structures. URL: https://people.inf.ethz.ch/wirth/AD.pdf

Knuth D. E. The Art of Computer Programming. Vo. 1. Fundamental Algorithms. 3-rd ed. Reading, Massachusetts: Addison-Wesley, 1997. 664 p.

Kriazhych O. O., Kovalenko O. V., Ivanchenko V. V. Sposib opysu zabrudnenoi terytoriyi: prohramna realizatsiya // Matematychne modeliuvannia v ekonomitsi. 2016. Issue 2. P. 22–35.

Zoutendijk G. Methods of feasible directions. A study in linear and non-linear programming. Elsevier Pub. Co., 1960. 178 р.

A Resolver-to-Digital Conversion Method Based on Third-Order Rational Fraction Polynomial Approximation for PMSM Control / Wang S., Kang J., Degano M., Buticchi G. // IEEE Transactions on Industrial Electronics. 2019. Vol. 66, Issue 8. P. 6383–6392. doi: https://doi.org/10.1109/tie.2018.2884209 

Knuth D. E. The Art of Computer Programming. Vol. 3: Sorting and Searching. 2-nd ed. Reading, Massachusetts: Addison-Wesley, 1998. 780 p.

Eyler L. Integral'noe ischislenie. Vol. 1. Moscow: GITTL, 1956. 416 p.

Bahvalov N. S., Zhidkov N. P., Kobel'kov G. M. Chislennye metody. 6-e izd. Moscow: BINOM. Laboratoriya znaniy, 2008. 636 p.

Popov B. A., Tesler G. S. Priblizhenie funktsiy dlya tekhnicheskih prilozheniy. Kyiv: Naukova dumka, 1980. 352 p.

Radiatsiyni ta yaderni tekhnolohiyi v Instytuti yadernykh doslidzhen NAN Ukrainy: monohrafiya / Vyshnevskyi I. M., Haidar H. P., Kovalenko O. V. et. al. Kyiv: In-t yadernykh doslidzhen, 2014. 176 p.







Copyright (c) 2019 Olha Kryazhych, Oleksandr Kovalenko

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ISSN (print) 1729-3774, ISSN (on-line) 1729-4061