Solution of ill-conditioned system of linear algebraic equations

The paper deals with the numerical solution for a system of linear algebraic equations which are ill-conditioned for some values of the problem parameter. For example, the parameter may be time. The solution of such system according to Cramer's rule or the Gauss method, for example, is impossible in the vicinity of singularity of the system matrix. An offered algorithm allows to pass successfully the vicinity of the singularity and own singular point, where the system matrix degenerates. This algorithm involves the method of solution continuation with respect to the best parameter.