We consider the solution of systems of linear algebraic equations (SLAEs) with an ill- conditioned or degenerate exact matrix and an approximate right-hand side. An approach to solving such a problem is proposed and justified, which makes it possible to improve the conditionality of the SLAE matrix and, as a result, obtain an approximate solution that is stable to perturbations of the right hand side with higher accuracy than using other methods. The approach is implemented by an algorithm that uses so-called minimal pseudoinverse matrices. The results of numerical experiments are presented that confirm the theoretical provisions of the article.

翻译：我们考虑求解具有病态或退化精确矩阵及近似右端项的线性代数方程组。提出并论证了一种解决此类问题的方法，该方法能够改善方程组矩阵的条件数，从而获得比使用其他方法更稳定且更精确的近似解（对右端项扰动具有更高鲁棒性）。该方法通过使用所谓的最小伪逆矩阵算法实现。文中给出了数值实验结果，验证了文章的理论结论。