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
Highlights
It was historically predetermined that a considerable body of scientific research is largely focused on the study of properties of linear systems, in particular, systems of linear algebraic equations (SLAE) [1,2,3]
It is not difficult to be convinced that typical SLAE were implemented for a rectangular constraint matrix on an assumption of rank completeness [1,2,3]
When constructing a fundamental system of solutions (FSS), rank magnitude remains a fundamental problem of establishing the system properties [3]
Summary
It was historically predetermined that a considerable body of scientific research is largely focused on the study of properties of linear systems, in particular, systems of linear algebraic equations (SLAE) [1,2,3]. Compatibility of SLAE with a rectangular constraint matrix is studied, for example, by application of the well-known Kronecker-Capelli theorem In this case, solvability of the problem is reduced to comparison of rank magnitudes of matrices of main and extended constraint systems. Quantitative inaccuracies in representation of model elements can cause qualitative distinctions in geometric structure (multifaceted set), in particular, dimension of the set, minimal face, etc These properties can be of significance in conditions of poor conditionality. It seems appropriate to develop new and improve existing methods and algorithms of analyzing the impact of changes in presentation of SLAE elements whose values are in the “zone” close to the machine zero, in particular, when solving systems of equations in different variants of model representation (mantis length, order value, etc.) taking into account peculiarities of the constraint matrix structure
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