On a solving ill-conditioned linear systems by iterative methods

The question of application of iterative conjugate gradients methods to solve ill-conditioned linear algebraic equations is discussed in this work. It is shown that for the arbitrary decisions these methods do not improve the precision in comparison with direct methods. Nevertheless for one main class that imitated non-orthogonal system expansion of function the concerned methods may get 5-6$^{\mathrm{th}}$ order of accuracy while direct ones can't get at least 1$^{\mathrm{th}}$.