Consider the problem of solving systems of linear algebraic equations $Ax=b$ with a real symmetric positive definite matrix $A$ using the conjugate gradient (CG) method. To stop the algorithm at the appropriate moment, it is important to monitor the quality of the approximate solution. One of the most relevant quantities for measuring the quality of the approximate solution is the $A$-norm of the error. This quantity cannot be easily computed, however, it can be estimated. In this paper we discuss and analyze the behaviour of the Gauss-Radau upper bound on the $A$-norm of the error, based on viewing CG as a procedure for approximating a certain Riemann-Stieltjes integral. This upper bound depends on a prescribed underestimate $\mu$ to the smallest eigenvalue of $A$. We concentrate on explaining a phenomenon observed during computations showing that, in later CG iterations, the upper bound loses its accuracy, and is almost independent of $\mu$. We construct a model problem that is used to demonstrate and study the behaviour of the upper bound in dependence of $\mu$, and developed formulas that are helpful in understanding this behavior. We show that the above mentioned phenomenon is closely related to the convergence of the smallest Ritz value to the smallest eigenvalue of $A$. It occurs when the smallest Ritz value is a better approximation to the smallest eigenvalue than the prescribed underestimate $\mu$. We also suggest an adaptive strategy for improving the accuracy of the upper bounds in the previous iterations.

翻译：考虑使用共轭梯度法求解实对称正定矩阵$A$的线性代数方程组$Ax=b$。为在适当时刻终止算法，需监控近似解的质量。衡量近似解质量最相关的量之一是误差的$A$范数。该量虽不易直接计算，但可被估计。本文基于将共轭梯度法视为近似特定黎曼-斯蒂尔切斯积分的过程，讨论并分析了误差$A$范数的高斯-拉道上界行为。该上界依赖于预设的矩阵$A$最小特征值下界估计$\mu$。我们重点阐释计算过程中观察到的现象：在共轭梯度法后期迭代中，该上界精度下降且几乎与$\mu$无关。我们构造了一个模型问题以展示并研究上界关于$\mu$的依赖行为，同时推导出有助于理解该行为的公式。研究表明，上述现象与最小里兹值向$A$最小特征值的收敛密切相关。当最小里兹值对最小特征值的逼近精度优于预设下界$\mu$时，该现象即发生。我们还提出了一种自适应策略来提升先前迭代中上界的精度。