When the conjugate gradient (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. With the deep insight of the original authors, CG was placed into a very 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, as well as other Krylov subspace methods developed in the meantime, has changed. Today these methods are considered primarily 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 of Krylov subspace methods, and also negatively affect their practical application.

中文翻译：

---

通过示例理解 CG 和 GMRES

大约 70 年前，当 Lanczos、Hestenes 和 Stiefel 提出用于求解线性代数系统的共轭梯度 (CG) 方法时，它被认为是具有数学有限终止性质的迭代过程。凭借原作者的深刻见解，CG 被置于非常丰富的数学背景中，包括与高斯求积和连分数的联系。 CG 的最优性属性通过归一化加权多项式最小二乘近似为零来描述。这个高度非线性的问题解释了 CG 迭代对给定数据的适应。 Karush 和 Hayes 立即考虑无限维希尔伯特空间中的 CG 并研究其超线性收敛性。从那时起，CG 以及同时发展的其他 Krylov 子空间方法的观点发生了变化。如今，这些方法主要被视为计算工具，它们的行为通常使用线性上限或基于特征值聚类的启发法来表征。这种简化限制了对克雷洛夫子空间方法的数学理解，也对其实际应用产生了负面影响。