Analysis of linear systems with complex elements by the basis matrix method
DOI:
https://doi.org/10.17721/1812-5409.2026/1.7Keywords:
system of linear algebraic equations, basis matrix method, complex analysisAbstract
This work proposes the application of the mathematical framework of complex variable function theory (specifically, for linear systems with complex elements) to describe the mechanism of interaction between components in a system ("model" – "researcher") during the process of mathematical modeling. The method of basis matrices is extended to linear systems with complex elements. An algorithm and computational procedure are developed for analyzing the properties and solving such systems, taking into account the interaction mechanism between the real and imaginary components in the representation of the mathematical model. A computational experiment is conducted to verify the correctness of the proposed algorithm and computational procedure.
The object of the study is processes of various natures in the context of solving the problem of mathematical modeling.
The subject of the study is methods, algorithms, and computational procedures of mathematical modeling, particularly the mathematical apparatus of complex variable function theory and linear algebra.
Research Objective: to develop algorithms and computational procedures for analyzing the properties of two-component mathematical models of processes (with complex elements), capable of accounting for the researcher's activity in the problem-solving process.
Research Tasks: to analyze known approaches to solving computational problems while considering the "activity" of the human analyst, operations researcher, or decision-maker; to identify basic mathematical methods in the class of linear systems that incorporate the ability to account for (human) activity in the problem-solving process; to develop computational procedures and algorithms adapted to modeling tasks (at various stages) for comprehensive analysis of linear systems (two-component systems), with a component accounting for human activity; to conduct a computational experiment based on the proposed method, algorithm, and procedure to verify their correctness on model examples; to analyze and summarize the experimental results and outline prospects for further research.
The mathematical modeling procedure involves a series of sequential stages (with feedback). Unfortunately, the unavoidable issue of the adequacy of the mathematical model in representing the studied process remains a challenge in solving applied problems. It is well known that modeling problems are expressed using mathematical tools of varying complexity and are often simplified (reduced) to linear models. This includes the discretization of continuous problems and the linearization of nonlinear ones.
It should be noted that the modeling process does not explicitly account for the influence of the researcher's experience, i. e., the human factor, on the quality of the modeling process at all stages. In fact, the two-component structure of the problem ("explicit" component – the mathematical model, and "implicit" – the researcher within the modeling loop) can be "traced".
This underscores the relevance of developing new mathematical problem formulations for modeling with complex elements (two-component systems), in which mechanisms are embedded to account for the researcher's experience in achieving greater adequacy of the model to the studied process at various problem-solving stages.
Pages of the article in the issue: 55 - 62
Language of the article: English
References
BigInt. (2013). Large Integer Package. https://github.com/luckyaibin/BigInt
Blackford, L., Demmel, J., Dongarra, J., Duff, I., Hammarling, S., Henry, G., Heroux, M., Kaufman, L., Lumsdaine, A., Petitet, A., Pozo, R., Remington, K., & Whaley, R. (2002). An Updated Set of Basic Linear Algebra Subprograms (BLAS). ACM Transactions on Mathematical Software, 28(2), 135–151. https://doi.org/10.1145/567806.567807
Demmel, J. (1997). Applied Numerical Linear Algebra. SIAM.
Dul, J., Bruder, R., Buckle, P., Carayon, P., Falzon, P., Marras, W., Wilson, J., & van der Doelen B. (2012). A strategy for human factors/ergonomics: developing the discipline and profession. Ergonomics, 55(4), 377–395. https://doi.org/10.1080/00140139.2012.661087
Fromberger, M. (2005). Multiple Precision Integer Library (MPI). https://github.com/servo/nss/tree/master/lib/freebl/mpi
Granlund, T. (2023). The GNU Multiple Precision Arithmetic Library. Edition 6.3.0. https://gmplib.org/list-archives/gmp-announce/2023-July/000050.html
Korobiichuk, I., Tokar, A., & Danik, Y. (2019). Evaluation methods for the ergatic system reliability operator. Automation 2019: progress in automation, robotics and measurement techniques, 920, 560–570. http://dx.doi.org/10.1007/978-3-030-13273-6_52
Kudin, V., Lyashko, S., Khritonenko, N., & Yatsenko, Y. (2007). Analysis of the properties of a linear system using the method of artificial basis matrices. Cybernetics and Systems Analysis, 43(4), 563–570. https://doi.org/10.1007/s10559-007-0081-3
Kudin, V., Onotskyi, V., Al-Ammouri, A., & Shkvarchuk, L. (2019a). Advancement of a long arithmetic technology in the construction of algorithms for studying linear systems. Eastern-European Journal of Enterprise Technologies, 1(4), 14–21. https://doi.org/10.15587/1729-4061.2019.157521
Kudin, V., Onotskyi, V., Onyshchenko, A., & Stupak, Y. (2019b). Forming a methodology of basic matrices in the study of poorly conditioned linear systems. Eastern-European Journal of Enterprise Technologies, 6(4), 57–67. https://doi.org/10.15587/1729-4061.2019.184530
Kudin, V., Onyshchenko, A., & Onyshchenko, I. (2019). Algorithmizing the methods of basis matrices in the study of balace intersectoral ecological and economic models. Eastern-European Journal of Enterprise Technologies, 3(4), 45–55. https://doi.org/10.15587/1729-4061.2019.170516
Mrachkovskyi, D. (2023). Development of a mathematical model of the operator in a human-machine control system for a quarry excavator. Electromechanical Systems, Optimization Methods and Modeling, 62(3), 41–48. https://doi.org/10.32782/2072-2052.2023.3.62.4
Strauch, B. (2017). Investigating Human Error: Incidents, Accidents, and Complex Systems. Taylor & Francis Group.
Tan, Y., Feng, D., & Shen, H. (2017). Research for Unmanned Aerial Vehicle components reliability evaluation model considering the influences of human factors. In B. Xu, & Y. Chen (Eds.), 3rd International Conference on Mechanical, Electronic and Information Technology Engineering, Vol. 139. EDP Sciences. https://doi.org/10.1051/matecconf/201713900221
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