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Accurate error estimation in CG

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Abstract

In practical computations, the (preconditioned) conjugate gradient (P)CG method is the iterative method of choice for solving systems of linear algebraic equations Ax = b with a real symmetric positive definite matrix A. During the iterations, it is important to monitor the quality of the approximate solution xk so that the process could be stopped whenever xk is accurate enough. One of the most relevant quantities for monitoring the quality of xk is the squared A-norm of the error vector x − xk. This quantity cannot be easily evaluated; however, it can be estimated. Many of the existing estimation techniques are inspired by the view of CG as a procedure for approximating a certain Riemann–Stieltjes integral. The most natural technique is based on the Gauss quadrature approximation and provides a lower bound on the quantity of interest. The bound can be cheaply evaluated using terms that have to be computed anyway in the forthcoming CG iterations. If the squared A-norm of the error vector decreases rapidly, then the lower bound represents a tight estimate. In this paper, we suggest a heuristic strategy aiming to answer the question of how many forthcoming CG iterations are needed to get an estimate with the prescribed accuracy. Numerical experiments demonstrate that the suggested strategy is efficient and robust.

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Funding

The work of J. Papež and P. Tichý was supported by the Grant Agency of the Czech Republic under grant no. 20-01074S.

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Authors and Affiliations

  1. Paris, France

    Gérard Meurant

  2. Institute of Mathematics, Czech Academy of Sciences, Prague, Czech Republic

    Jan Papež

  3. Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic

    Petr Tichý

Authors
  1. Gérard Meurant
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  2. Jan Papež
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  3. Petr Tichý
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Correspondence to Petr Tichý.

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Appendices

Appendix

Algorithm 2 (MATLAB code, preconditioned version)

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Meurant, G., Papež, J. & Tichý, P. Accurate error estimation in CG. Numer Algor 88, 1337–1359 (2021). https://doi.org/10.1007/s11075-021-01078-w

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