When the CG method for solving linear algebraic systems was formulated about 70 years ago by Lanczos, Hestenes, and Stiefel, it was considered an iterative process possessing a mathematical finite termination property. CG was placed into a rich mathematical context, including links with Gauss quadrature and continued fractions. The optimality property of CG was described via a normalized weighted polynomial least squares approximation to zero. This highly nonlinear problem explains the adaptation of CG iterates to the given data. Karush and Hayes immediately considered CG in infinite dimensional Hilbert spaces and investigated its superlinear convergence. Since then, the view of CG and other Krylov subspace methods has changed. Today these methods are primarily used as computational tools, and their behavior is typically characterized using linear upper bounds or heuristics based on clustering of eigenvalues. Such simplifications limit the mathematical understanding and also negatively affect their practical application. This paper offers a different perspective. Focusing on CG and GMRES, it presents mathematically important and practically relevant phenomena that uncover their behavior through a discussion of computed examples. These examples provide an easily accessible approach that enables understanding of the methods, while pointers to more detailed analyses in the literature are given. This approach allows readers to choose the level of depth and thoroughness appropriate for their intentions. Some of the points made in this paper illustrate well known facts. Others challenge mainstream views and explain existing misunderstandings. Several points refer to recent results leading to open problems. We consider CG and GMRES crucially important for the mathematical understanding, further development, and practical applications also of other Krylov subspace methods.

翻译：大约70年前，当兰佐斯、赫斯滕斯和斯蒂费尔提出用于求解线性代数方程组的共轭梯度法（CG）时，它被视为一种具备数学有限终止性质的迭代过程。CG被置于丰富的数学背景之中，包括与高斯求积法和连分数的联系。CG的最优性质通过归一化加权多项式对零的最小二乘逼近进行描述。这一高度非线性的问题解释了CG迭代如何适应给定数据。卡鲁什和海耶斯立即在无限维希尔伯特空间中研究了CG方法，并探讨了其超线性收敛性。此后，人们对CG及其他Krylov子空间方法的认知发生了变化。如今，这些方法主要作为计算工具使用，其行为特征通常采用线性上界或基于特征值聚类的启发式方法进行描述。此类简化不仅限制了数学理解，也对其实际应用产生了负面影响。本文提出了一种不同的视角。以CG和广义最小残差法（GMRES）为核心，通过讨论计算实例来揭示其行为中既具数学重要性又具实际意义的现象。这些实例提供了一种易于理解的方法，帮助读者掌握算法原理，同时附有文献中更详细分析的索引。这种框架使读者能够根据自身需求选择合适的研究深度和完整度。文中部分观点阐明了广为人知的事实，另一些则挑战了主流观点并解释了现有误解，还有若干观点涉及近期研究成果所引发的开放性问题。我们认为，CG和GMRES对于数学理解、方法发展及其他Krylov子空间方法的实际应用均具有至关重要的价值。