To the problem of the interrelation of the matrix close to degeneracy and the stability of the solution of the system of linear algebraic equations describing a physical object.

Abstract

It is generally accepted that the proximity of the matrix to degeneracy uniquely determines the instability of the system of linear algebraic equations (SLAE). There is only an assumption that the instability of the numerical solution of the SLAE in the case of a matrix close to degeneracy is caused by the rounding error of the arithmetic operation of addition (subtraction). When determining the mechanism of the interrelation of the matrix close to degeneracy and the solvability of the SLAE, as a model example, all stages of the calculation of the electric circuit are analyzed, namely, the description of the graph of the linear electric circuit, the composition of the SLAE describing the circuit, and the solution of the complex system. For the first time, it was proposed to use the conductivity of the bipolar component located between the nodes of the electric circuit graph as an «indicator» of the matrix close to degeneracy. The dependence of the accuracy of the SLAE solution on the value of the conductivity of the indicator component was calculated for two variants of the voltage and conductivity of the indicator component in the SLAE. The analysis of the calculation results showed for the first time that the rounding error of the arithmetic operation of addition (subtraction) leads to the loss of accuracy or instability of the numerical solution of the SLAE with the matrix close to degeneracy. The influence of the error on the computational process of the numerical solution of the system was performed in two stages. For the first time, it was established that the interrelation between the matrix close to degeneracy and the stability of the SLAE solution depends on the selection of the voltage of the indicator component in the vector of SLAR variables at the stage of compiling the equations describing the electrical circuit. The interrelation exists if the voltage of the indicator component is not included in the vector of variables and does not exist if the voltage of the «indicator» component is included in the vector of variables because the second rounding error operation stage does not work. An algorithm for selecting the variables of the SLAE, which is composed, is proposed to obtain a stable solution for the system.