Methods for solving ill-conditioned systems of linear algebraic equations that improve the conditionality

The problem of solving systems of linear algebraic equations (SLAE) with an ill-conditioned or degenerate exact matrix and an approximate right-hand side is considered. A scheme for solving such a problem is proposed and justified, which makes it possible to improve the conditionality of the SLAE matrix. As a result, an approximate solution that is stable to perturbations of the right-hand side is obtained with a higher accuracy than when using some other methods. The scheme is implemented by an algorithm that uses minimal pseudoinverse matrices. The results of numerical experiments are presented, confirming the theoretical provisions of the article.