  Consider the problem of solving systems of linear algebraic equations $Ax=b$
with a real symmetric positive definite matrix $A$ using the conjugate gradient
(CG) method. To stop the algorithm at the appropriate moment, it is important
to monitor the quality of the approximate solution. One of the most relevant
quantities for measuring the quality of the approximate solution is the
$A$-norm of the error. This quantity cannot be easily computed, however, it can
be estimated. In this paper we discuss and analyze the behaviour of the
Gauss-Radau upper bound on the $A$-norm of the error, based on viewing CG as a
procedure for approximating a certain Riemann-Stieltjes integral. This upper
bound depends on a prescribed underestimate $\mu$ to the smallest eigenvalue of
$A$. We concentrate on explaining a phenomenon observed during computations
showing that, in later CG iterations, the upper bound loses its accuracy, and
is almost independent of $\mu$. We construct a model problem that is used to
demonstrate and study the behaviour of the upper bound in dependence of $\mu$,
and developed formulas that are helpful in understanding this behavior. We show
that the above mentioned phenomenon is closely related to the convergence of
the smallest Ritz value to the smallest eigenvalue of $A$. It occurs when the
smallest Ritz value is a better approximation to the smallest eigenvalue than
the prescribed underestimate $\mu$. We also suggest an adaptive strategy for
improving the accuracy of the upper bounds in the previous iterations.
