Error estimators and their analysis for CG, Bi-CG and GMRES

The demands of accuracy in measurements and engineering models today render 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.