Using computed examples for the Conjugate Gradient method and GMRES, we recall important building blocks in the understanding of Krylov subspace methods over the last 70 years. Each example consists of a description of the setup and the numerical observations, followed by an explanation of the observed phenomena, where we keep technical details as small as possible. Our goal is to show the mathematical beauty and hidden intricacies of the methods, and to point out some persistent misunderstandings as well as important open problems. We hope that this work initiates further investigations of Krylov subspace methods, which are efficient computational tools and exciting mathematical objects that are far from being fully understood.

翻译：使用“共融渐进法”和“GMERES”的计算实例,我们回顾过去70年来了解Krylov子空间方法的重要基石,每个例子都包含对设置的描述和数字观测,然后是所观察到现象的解释,我们在此之后尽可能少地保留技术细节。我们的目标是展示这些方法的数学美和隐藏的复杂之处,并指出一些长期存在的误解和重要的未决问题。我们希望这项工作能够对Krylov子空间方法展开进一步的调查,这些方法都是高效的计算工具和远未被完全理解的令人兴奋的数学对象。