  In practical conjugate gradient (CG) computations it is important to monitor
the quality of the approximate solution to $Ax=b$ so that the CG algorithm can
be stopped when the required accuracy is reached. The relevant convergence
characteristics, like the $A$-norm of the error or the normwise backward error,
cannot be easily computed. However, they can be estimated. Such estimates often
depend on approximations of the smallest or largest eigenvalue of~$A$.
  In the paper we introduce a new upper bound for the $A$-norm of the error,
which is closely related to the Gauss-Radau upper bound, and discuss the
problem of choosing the parameter $\mu$ which should represent a lower bound
for the smallest eigenvalue of $A$.The new bound has several practical
advantages, the most important one is that it can be used as an approximation
to the $A$-norm of the error even if $\mu$ is not exactly a lower bound for the
smallest eigenvalue of $A$. In this case, $\mu$ can be chosen, e.g., as the
smallest Ritz value or its approximation. We also describe a very cheap
algorithm, based on the incremental norm estimation technique, which allows to
estimate the smallest and largest Ritz values during the CG computations. An
improvement of the accuracy of these estimates of extreme Ritz values is
possible, at the cost of storing the CG coefficients and solving a linear
system with a tridiagonal matrix at each CG iteration. Finally, we discuss how
to cheaply approximate the normwise backward error. The numerical experiments
demonstrate the efficiency of the estimates of the extreme Ritz values, and
show their practical use in error estimation in CG.
