  The demands of accuracy in measurements and engineering models today, renders
the condition number of problems larger. While a corresponding increase in the
precision of floating point numbers ensured a stable computing, the uncertainty
in convergence when using residue as a stopping criterion has increased. We
present an analysis of the uncertainty in convergence when using relative
residue as a stopping criterion for iterative solution of linear systems, and
the resulting over/under computation for a given tolerance in error. This shows
that error estimation is significant for an efficient or accurate solution even
when the condition number of the matrix is not large. An $\mathcal{O}(1)$ error
estimator for iterations of the CG algorithm was proposed more than two decades
ago. Recently, an $\mathcal{O}(k^2)$ error estimator was described for the
GMRES algorithm which allows for non-symmetric linear systems as well, where
$k$ is the iteration number. We suggest a minor modification in this GMRES
error estimation for increased stability. In this work, we also propose an
$\mathcal{O}(n)$ error estimator for A-norm and $l_{2}$ norm of the error
vector in Bi-CG algorithm. The robust performance of these estimates as a
stopping criterion results in increased savings and accuracy in computation, as
condition number and size of problems increase.
