Toeplitz Infinite GMRES for Parameterized Linear Systems
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 𝒪(np2+p3) arithmetic and 𝒪(np+p2) 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.
Related
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BibTeX
@misc{gao2026toeplitz,
title = {{Toeplitz Infinite GMRES for Parameterized Linear Systems}},
author = {Weiguo Gao and Feiyang Jiang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01740},
eprint = {2610.01740},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01740},
}
RIS
TY - UNPB
TI - Toeplitz Infinite GMRES for Parameterized Linear Systems
AU - Weiguo Gao
AU - Feiyang Jiang
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.01740
UR - https://arxiv.org/abs/2610.01740
AB - 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 𝒪(np^2+p^3) arithmetic and 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.
ER -