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 查看全文>>