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The demands of accuracy in measurements and engineering models today, rendersthe condition number of problems larger. While a corresponding increase in theprecision of floating point numbers ensured a stable computing, the uncertaintyin convergence when using residue as a stopping criterion has increased. Wepresent an analysis of the uncertainty in convergence when using relativeresidue as a stopping criterion for iterative solution of linear systems, andthe resulting over/under computation for a given tolerance in error. This showsthat error estimation is significant for an efficient or accurate solution evenwhen the condition number of the matrix is not large. An $\mathcal{O}(1)$ errorestimator for iterations of the CG algorithm was proposed more than two decadesago. Recently, an $\mathcal{O}(k^2)$ error estimator was described for theGMRES algorithm which allows for non-symmetric linear systems as well, where$k$ is the iteration number. We suggest a minor modification in this GMRESerror 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 errorvector in Bi-CG algorithm. The robust performance of these estimates as astopping criterion results in increased savings and accuracy in computation, ascondition number and size of problems increase.

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