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Error Estimators and Their Analysis for CG, Bi-CG, and GMRES

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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 the Bi-CG algorithm. The robust performance of these estimates as a stopping criterion results in increased savings and accuracy in computation, as the condition number and size of problems increase.

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Notes

  1. https://math.nist.gov/MatrixMarket/

  2. Code freely available at https://github.com/CSPL-IISc/Error-Linsolve.

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Correspondence to P. Jain, K. Manglani or M. Venkatapathi.

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Translated from Sibirskii Zhurnal Vychislitel’noi Matematiki, 2023, Vol. 26, No. 2, pp. 161-181. https://doi.org/10.15372/SJNM20230204.

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Jain, P., Manglani, K. & Venkatapathi, M. Error Estimators and Their Analysis for CG, Bi-CG, and GMRES. Numer. Analys. Appl. 16, 135–153 (2023). https://doi.org/10.1134/S1995423923020040

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  • DOI: https://doi.org/10.1134/S1995423923020040

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