Iteration number for the conjugate gradient method

When solving linear systems and, in particular when solving large scale ill-conditioned problems it is important to understand the behaviour of the conjugate gradient method. The conjugate gradient method converges typically in three phases, an initial phase of rapid convergence but short duration, which depends essentially only on the initial error, a fairly linearly convergent phase, which depends on the spectral condition number and finally a superlinearly convergent phase, which depends on how the smallest eigenvalues are distributed. In the paper, this is explained by proper estimates of the rate of convergence.