Forming a methodology of basic matrices in the study of poorly conditioned linear systems

Authors

DOI:

https://doi.org/10.15587/1729-4061.2019.184530

Keywords:

basic matrix method, rectangular constraint matrix, poorly conditioned SLAE

Abstract

Algorithm of the basic matrix method for analysis of properties of the system of linear arithmetic equation (SLAE) in various changes introduced in the model, in particular, when including-excluding a group of rows and columns (based on "framing") without re-solving the problem from beginning has been improved. Conditions of compatibility (incompatibility) of restrictions were established and vectors of the fundamental solution system in a case of compatibility were established. Influence of accuracy of representing the model elements (mantis length, order value, thresholds of machine zero and overflow) and variants of computation organization on solution properties was studied. Specifically, effect of magnitude and completeness of rank was studied on an example of a SLAE with a poorly conditioned constraint matrix. A program was developed for implementation of conducting calculations using the basic matrix methods (BMM) and Gauss method, that is, long arithmetic was used for models with rational elements. Algorithms and computer-aided implementation of Gaussian methods and artificial basic matrices (as a variant of the basic matrix method) in MATLAB and Visual C++ environments with the use of the technology of exact calculation of the method elements, first of all, for poorly conditioned systems with different dimensions were proposed.

Using as an example Hilbert matrices, which are characterized as "inconvenient" matrices, an experiment was conducted to analyze properties of a linear system at different dimensions, accuracy of the input data and computation scenarios. Formats ("exact" and "inexact") of representation of model elements (mantis length, order value, thresholds of machine zero and overflow) as well as variants of organization of basic computation operations during calculation and their influence on solution properties have been developed. In particular, influence of rank magnitude and completeness was traced on an example of an SLAE with a poorly conditioned constraint matrix

Author Biographies

Volodymyr Kudin, Taras Shevchenko National University of Kyiv Volodymyrska str., 60, Kyiv, Ukraine, 01033

Doctor of Technical Sciences, Professor

Department of Intelligent and Information Systems

Viacheslav Onotskyi, Taras Shevchenko National University of Kyiv Volodymyrska str., 60, Kyiv, Ukraine, 01033

PhD, Assistant

Department of Computational Mathematics

Andriy Onyshchenko, Taras Shevchenko National University of Kyiv Volodymyrska str., 60, Kyiv, Ukraine, 01033

Doctor of Economic Sciences, Professor

Department of Information Systems and Technologies

Yurii Stupak, National Metallurgical Academy of Ukraine Gagarina ave., 4, Dnipro, Ukraine, 49600

PhD

Department of Theory, Technology and Automation of Metallurgical Processes

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Published

2019-11-21

How to Cite

Kudin, V., Onotskyi, V., Onyshchenko, A., & Stupak, Y. (2019). Forming a methodology of basic matrices in the study of poorly conditioned linear systems. Eastern-European Journal of Enterprise Technologies, 6(4 (102), 57–67. https://doi.org/10.15587/1729-4061.2019.184530

Issue

Vol. 6 No. 4 (102) (2019): Mathematics and Cybernetics - applied aspects

Section

Mathematics and Cybernetics - applied aspects

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Copyright (c) 2019 Volodymyr Kudin, Viacheslav Onotskyi, Andriy Onyshchenko, Yurii Stupak

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