70 years of Krylov subspace methods: The journey continues

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.