The influence of isolated largest eigenvalues on the numerical convergence of the CG method

This paper considers the dependence of the convergence history of the CG method on largest eigenvalues of a symmetric positive definite matrix. It is demonstrated that, in solving ill-conditioned linear systems, the reproduction of largest eigenvalues can be so intensive that large eigenvalues cannot be treated as isolated. On the other hand, since the moment at which the smallest isolated eigenvalues start to govern the numerical convergence of the CG method, the character of convergence mainly depends on the smallest Ritz values.