论文标题
在福赛猜想上
On the Forsythe conjecture
论文作者
论文摘要
Forsythe在1968年提出了关于重新开始的共轭梯度法的渐近行为的猜想。我们将其几个结果转化为现代术语，并将猜想（最初仅用于对称正定矩阵）概括为对称和非对称矩阵。我们的概括是基于与给定矩阵及其转置的双面或跨迭代，该矩阵基于Arnoldi（或用于对称矩阵兰开斯）算法中使用的投影过程。我们证明了有关此迭代的限制行为的几个新结果，但是猜想仍然在很大程度上开放。
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.
