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

Jain, P and Manglani, K and Venkatapathi, M (2023) Error Estimators and Their Analysis for CG, Bi-CG, and GMRES. In: Numerical Analysis and Applications, 16 (2). pp. 135-153.

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Abstract

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 O(1) error estimator for iterations of the CG algorithm was proposed more than two decades ago. Recently, an O(k2) 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 O(n) error estimator for A-norm and l2 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.

Item Type:	Journal Article
Publication:	Numerical Analysis and Applications
Publisher:	Pleiades Publishing
Additional Information:	The copyright for this article belongs to the Pleiades Publishing.
Keywords:	Bi-CG; condition number; Conjugate Gradients; error; GMRES; stopping criteria
Department/Centre:	Division of Interdisciplinary Sciences > Computational and Data Sciences
Date Deposited:	01 Aug 2023 05:10
Last Modified:	01 Aug 2023 05:10
URI:	https://eprints.iisc.ac.in/id/eprint/82740

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