Abstract:
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.