Gauss method application to solution of ill-conditioned systems of linear algebraic equations

The paper deals with the numerical solution of the system of linear algebraic equations that is singular under certain values of the problem parameter, which can be, for example, time. The solution of such system by the Kramer's rule or using Gaussian elimination method is impossible in the neighborhood of singularity of the system matrix. The algorithm is proposed which can successfully overcome the neighborhood of the singularity and singular points where the system matrix degenerates. The algorithm implies the application of the method of solution continuation with respect to the best parameter and Gaussian elimination method for linear algebraic equations’ system.