On the cyclic limiting behavior of the restarted Arnoldi algorithm for real symmetric matrices with restart length $s=2$

Abstract

We prove that the restarted Arnoldi algorithm for real symmetric matrices with restart length $s=2$ has a cyclic limiting behavior. In the terminology established in (arXiv:2209.14579), this is a proof of the ACI($s$) conjecture for $A=A^T\in \mathbb{R}^{n\times n}$ and $s=2$. As explained in (arXiv:2209.14579), this also proves the Forsythe conjecture (Numerische Mathematik, 11 (1968), pp. 57-76) about the limiting behavior of the conjugate gradient method and positive definite $A=A^T\in\mathbb{R}^{n\times n}$ for $s=2$. The proof we give is an extensively rewritten and simplified version of a proof found using Codex with GPT-5.6 Sol. At the end of this paper, we discuss its origin and evolution, as well as related recent developments around the Forsythe conjecture. Our motivation is to present a proof that is instructive and transparent, so that it can enhance the knowledge of interested human readers.