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

Abstract

We present an analysis of the uncertainty in the convergence of iterative linear solvers when using relative residue as a stopping criterion, and the resulting over/under computation for a given tolerance in error. This shows that error estimation is indispensable for efficient and accurate solution of moderate to high conditioned linear systems ($\kappa>100$), where $\kappa$ is the condition number of the matrix. An $\mathcal{O}(1)$ error estimator for iterations of the CG (Conjugate Gradient) algorithm was proposed more than two decades ago. Recently, an $\mathcal{O}(k^2)$ error estimator was described for the GMRES (Generalized Minimal Residual) 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 (Bi-Conjugate Gradient) 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