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

Volodymyr Kudin, Viacheslav Onotskyi, Andriy Onyshchenko, Yurii Stupak


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


basic matrix method; rectangular constraint matrix; poorly conditioned SLAE


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