Accurate error estimation in CG

0546795 - MÚ 2022 RIV NL eng J - Journal Article
Meurant, G. - Papež, Jan - Tichý, P.
Accurate error estimation in CG.
Numerical Algorithms. Roč. 88, č. 3 (2021), s. 1337-1359. ISSN 1017-1398. E-ISSN 1572-9265
R&D Projects: GA ČR(CZ) GA20-01074S
Institutional support: RVO:67985840
Keywords : accuracy of the estimate * conjugate gradients * error estimation
OECD category: Pure mathematics
Impact factor: 2.370, year: 2021
Method of publishing: Limited access
https://doi.org/10.1007/s11075-021-01078-w

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.
Permanent Link: http://hdl.handle.net/11104/0323175