Year: 2026
Type: preprint
Abstract: We develop Toeplitz infinite GMRES for solving large sparse analytic parameterized systems $A(s)x(s)=z$ at many parameter values. Our 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 this basic recurrence. Moreover, we show that the Arnoldi basis can be simply represented using one block row and a nilpotent matrix. For $A(s)$ admitting a split form, we further use this structure to develop an implicitly rebased method. A residual lower bound for square-summable Taylor coefficients explains the effect of scaling outside the unit disk. Theoretical analysis and numerical experiments illustrate the computational efficiency of the proposed methods.
Source: arXiv (Cornell University)
Authors: Gao, Weiguo, Jiang, Feiyang
Institution: Fudan University
Language: English
Topic: Matrix Theory and Algorithms
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7216174508