New error estimates for the conjugate gradient method

The conjugate gradient method is the default iterative method for the solution of linear systems of equations with a large symmetric positive definite matrix $A$. The development of techniques for estimating the norm of the error in iterates computed by this method has received considerable attention. Available methods for bracketing the $A$-norm of the error evaluate pairs of Gauss and Gauss–Radau quadrature rules to determine lower and upper bounds. The latter rule requires a user to allocate a node (the Radau node) between the origin and the smallest eigenvalue of the system matrix. The determination of such a node generally demands further computations to estimate the location of the smallest eigenvalue; see, e.g., Golub and Meurant (1997), Golub and Meurant (2010), Golub and Strakoš (1994), Meurant (1997), Meurant (1999). An approach that avoids the need to know a lower bound for the smallest eigenvalue is to replace the Gauss–Radau quadrature rule by an anti-Gauss rule as described by Calvetti et al. (2000). However, this approach may sometimes yield inaccurate error norm estimates. This paper proposes the use of pairs of Gauss and associated optimal averaged Gauss quadrature rules to estimate the $A$-norm of the error in iterates determined by the conjugate gradient method.