发表机构
Fudan University; Shanghai Key Laboratory of Contemporary Applied Mathematics（复旦大学; 上海市当代应用数学重点实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出Toeplitz无限GMRES方法，利用块上三角Toeplitz结构高效求解多参数线性系统，通过增量递推和动态刷新策略降低计算与存储开销，并扩展至矩阵函数场景。
AI 中文摘要
我们开发了Toeplitz无限GMRES方法，用于在多个参数值下求解大型稀疏解析参数化系统$A(s)x(s)=z$。该方法利用伴随Krylov序列中的块上三角Toeplitz结构来构造Arnoldi过程，并在不存储完整Arnoldi向量的情况下恢复解的近似。我们推导了增量递推关系，对于$p$步Arnoldi过程，需要$\mathcal{O}(np^2+p^3)$的算术运算和$\mathcal{O}(np+p^2)$的存储空间，不包括分解设置，并假设线性成本的系数作用和三角求解。一种动态生成器刷新策略解决了基本递推中的抵消问题。对于允许固定分离表示的矩阵函数，我们进一步开发了一种基于紧凑、压缩幂零矩阵的隐式重基方法。两种变体在精确算术下都保持Arnoldi过程，其中重基方法需要精确的矩阵函数作用。对于平方可求和Taylor系数，残差下界解释了归一化单位圆外的缩放障碍。理论分析和数值实验说明了所提出方法的计算效率。
英文摘要
We develop Toeplitz infinite GMRES for solving large sparse analytic parameterized systems $A(s)x(s)=z$ at many parameter values. The method exploits a block upper triangular Toeplitz structure in the companion Krylov sequence to construct the Arnoldi process and recover solution approximations without storing full Arnoldi vectors. We derive incremental recurrences requiring $\mathcal{O}(np^2+p^3)$ arithmetic and $\mathcal{O}(np+p^2)$ storage for $p$ Arnoldi steps, excluding factorization setup and assuming linear-cost coefficient actions and triangular solves. A dynamic generator-refreshing strategy addresses cancellation in the basic recurrence. For matrix functions admitting a fixed separated representation, we further develop an implicitly rebased method based on a compact, contractive nilpotent matrix. Both variants preserve the Arnoldi process in exact arithmetic, with exact matrix-function actions required for the rebased method. A residual lower bound for square-summable Taylor coefficients explains a scaling obstruction outside the normalized unit disk. Theoretical analysis and numerical experiments illustrate the computational efficiency of the proposed methods.
