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 x_k so that the process could be stopped whenever x_k is accurate enough. One of the most relevant quantities for monitoring the quality of x_k is the squared A-norm of the error vector x − x_k. 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.

在实际计算中，（预处理）共轭梯度（P）CG方法是求解带有实对称正定矩阵A的线性代数方程A x = b的系统的迭代选择方法。在迭代过程中，监视近似解x _k的质量很重要，这样只要x _k足够准确，就可以停止该过程。监视x _k质量最相关的量之一是误差向量x − x _k的平方A-范_数。此数量不容易评估；但是，可以估计。许多现有的估算技术都受到CG的启发，因为CG是逼近某个Riemann-Stieltjes积分的过程。最自然的技术是基于高斯正交逼近，并为感兴趣的量提供了一个下限。可以使用必须在即将到来的CG迭代中计算出的项来廉价地评估边界。如果平方A误差向量的-norm迅速减小，则下限表示严格的估计。在本文中，我们提出了一种启发式策略，旨在回答需要多少次CG迭代才能获得具有规定精度的估计值的问题。数值实验表明，所提出的策略是有效且鲁棒的。