On numerical solution of systems of linear algebraic equations with ill-conditioned matrices

The results of the numerical solution of systems of linear algebraic equations (SLAE) with symmetric and asymmetrical ill-conditioned matrices by the regularization method are presented in the paper. Positive-definite and oscillatory matrices are considered. The article shows that in order to regularize the computational process according to the Tikhonov method, it is enough to replace the system matrix $A_n$ with the matrix $ A_n+\alpha E_n $ where $E_n$ is the identity matrix, and $\alpha$ is some positive number (regularization parameter) that tends to zero.