发表机构
University of Cambridge; California Institute of Technology; Cornell University（剑桥大学; 加州理工学院; 康奈尔大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究解决了Forsythe猜想，明确仅当重启长度s为1、2、3时猜想成立，s≥4时存在反例，通过双正交恒等式、Sturm序列等完成分类与反例构造。
AI 中文摘要
1968年提出的Forsythe猜想断言：对于每个重启长度$s$，在实对称正定问题上的每一个精确算术重启共轭梯度迭代要么终止，要么其归一化残差会沿偶、奇重启子序列分别收敛。除经典最速下降情形外，该渐近问题近六十年来未得到完全一般性解决。我们在原有限维情形下按重启长度给出完全分类并确定了一个尖锐阈值：当$s=2$和$s=3$时，每个问题要么终止，要么具有收敛的偶、奇残差方向；对于所有$s\boldsymbol{\text{≥}}4$，存在一个维数为$s+4$的对角正定反例，其永不终止且偶残差方向不收敛。结合Akaike关于$s=1$的定理，这表明猜想的普适结论仅对$s∈\{1,2,3\}$成立，对所有$s\boldsymbol{\text{≥}}4$不成立。正结果源于与次数无关的双正交恒等式及低次数不动点集分析；在重启长度为4时，有理区间算术和Sturm序列验证了重标化平方权重映射的主导向量场的横截Hopf点，解析周期轨道与影射论证得到了重启长度为4的反例，次数提升将该构造扩展至所有更大的重启长度。
英文摘要
Forsythe's conjecture, published in 1968, asserts that for each restart length $s$, every exact-arithmetic restarted conjugate-gradient iteration on a real symmetric positive definite problem either terminates or has normalised residuals that converge separately along the even and odd restart subsequences. Apart from the classical steepest-descent case, this asymptotic question remained unresolved in full generality for nearly six decades. We give a complete classification by restart length in the original finite-dimensional setting and identify a sharp threshold. For $s=2$ and $s=3$, every problem either terminates or has convergent even and odd residual directions. For every $s\ge4$, there is a diagonal positive definite counterexample of dimension $s+4$ which never terminates and whose even residual directions do not converge. Together with Akaike's theorem for $s=1$, this shows that the conjectured universal conclusion is true precisely for $s\in\{1,2,3\}$ and false for every $s\ge4$. The positive results follow from a degree-independent double-orthogonality identity and an analysis of the low-degree fixed-point sets. At restart length four, rational interval arithmetic and Sturm sequences certify a transverse Hopf point of the leading vector field of the rescaled squared-weight map. Analytic periodic-orbit and shadowing arguments yield the counterexample at restart length four, and degree elevation extends the construction to every larger restart length. The classification for all $s\ge2$ is also formally verified in Lean.
Comments151 pages, 3 figures
