IMPROVED METHOD FOR STABILITY RESEARCH OF SOLUTIONS OF SYSTEMS OF LINEAR ALGEBRAIC EQUATIONS

Abstract

A review of existing methods for stability research of solutions of systems of linear algebraic equations (SLAE) depending on the input data, that is, parameter variations, have been carried out. Methods for stability research of solving systems of linear algebraic equations, such as condition numbers, modular determinants and the construction of a table of signs using the original and improved construction methods, were considered. Software for stability research of systems of linear algebraic equations have been developed. The software is using condition numbers, modular determinants, and construction of table of signs to find accurate estimates of the variations of solutions of SLAEs that depend on parameters variations.It is shown that stability research using condition numbers gives a very rough estimate of possible errors in solutions, but that research is simple to implement, and for SLAEs it can immediately show that some systems are ill-conditioned, which saves research time, especially if SLAEs have a very large dimensionality. The stability research using modular determinants requires large calculations, but they give a fairly reliable upper estimate with respect to possible variations of individual components of solutions of systems of linear algebraic equations. This is a very important feature of the method because the individual components of solutions may experience significant variations that are not taken into account in the research using condition numbers. The study of stability by constructing a table of signs makes it possible to find the maximum variations of individual components of solutions of a system of linear algebraic equations, which in fact can be significantly less than the upper estimate of possible variations found by method of modular determinants. The paper proposes an improved method for constructing a table of signs, which finds a more accurate range of possible variations of solutions of a system of linear algebraic equations.A comparative analysis was conducted between the traditional method of constructing a table of signs using individual determinants and an improved method of constructing a table of signs based on the derivatives of the division of determinants using the Cramer formula. According to the analysis, the improved method in 30% of cases finds variations that are 1.3 times greater than the variations that the previous method finds, and in 5% of cases these variations are 2 or more times greater than the previous ones. This suggests that the traditional method in some cases underestimated the possible deviations of solutions that depend on variations of the input data.