Forsythe formulated a conjecture about the asymptotic behavior of the restarted conjugate gradient method in 1968. We translate several of his results into modern terms, and generalize the conjecture (originally formulated only for symmetric positive definite matrices) to symmetric and nonsymmetric matrices. Our generalization is based on a two-sided or cross iteration with the given matrix and its transpose, which is based on the projection process used in the Arnoldi (or for symmetric matrices the Lanczos) algorithm. We prove several new results about the limiting behavior of this iteration, but the conjecture still remains largely open.

翻译：福瑟(Forsythe)对1968年重新启动的共振梯度法的无症状行为的推测。 我们把他的几项结果翻译成现代术语,并将推测(最初只为正对正确定矩阵而拟订)概括为对称和非对称矩阵。 我们的概括是基于对给定矩阵及其转换的双向或交叉迭代,它基于Arnoldi(或对称矩阵兰佐斯)算法所使用的预测过程。 我们证明了关于这种迭代行为限制的几项新结果,但推测仍然基本开放。