Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.16492

On the cyclic limiting behavior of the restarted Arnoldi algorithm for real symmetric matrices with restart length $s=2$

Vance Faber·Jarek Liesen·Jörg Liesen·Petr Tichý
View original source ↗Read source document ↗Export metadata ↓

Abstract

We prove that the restarted Arnoldi algorithm for real symmetric matrices with restart length $s=2$ has a cyclic limiting behavior. In the terminology established in (arXiv:2209.14579), this is a proof of the ACI($s$) conjecture for $A=A^T\in \mathbb{R}^{n\times n}$ and $s=2$. As explained in (arXiv:2209.14579), this also proves the Forsythe conjecture (Numerische Mathematik, 11 (1968), pp. 57-76) about the limiting behavior of the conjugate gradient method and positive definite $A=A^T\in\mathbb{R}^{n\times n}$ for $s=2$. The proof we give is an extensively rewritten and simplified version of a proof found using Codex with GPT-5.6 Sol. At the end of this paper, we discuss its origin and evolution, as well as related recent developments around the Forsythe conjecture. Our motivation is to present a proof that is instructive and transparent, so that it can enhance the knowledge of interested human readers.

Explore related subjects

math.NAcs.NA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Vance Faber, Jarek Liesen, Jörg Liesen, Petr Tichý. 2026-10-12. On the cyclic limiting behavior of the restarted Arnoldi algorithm for real symmetric matrices with restart length $s=2$. https://arxiv.org/abs/2610.16492

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Viscosity-uniform error estimates for a semi-implicit Fourier method in Besov spaces

We prove error estimates for a semi-implicit Fourier--Galerkin method for two-dimensional incompressible flow on the torus, with viscosity $0\leν\le1$. The temporal error is first order in $H^s$ for $H^{s+2}$ initial data and integer $s\ge1$. For data in $B^m_{2,1}$, with integer $m\ge3$, the fully discrete error is of order $τ+N^{-(m-2)}$ in $B^1_{2,1}$ and $B^0_{\infty,1}$. These estimates hold on a common local interval, with constants and time-step thresholds independent of viscosity and Fourier resolution. For data in $B^{2+\varepsilon}_{2,1}$, $0<\varepsilon<1$, the error relative to Euler flow is bounded by $C(τ^{(1+\varepsilon)/2}+N^{-\varepsilon} +ν^{(1+\varepsilon)/2})$. The consistency bounds use viscosity-weighted regularity. Orthogonality of the Leray and Fourier projectors in the block energies gives the endpoint estimates, and stability with respect to the initial data yields the fractional rate. An $L^2$ residual tolerance $δ$ contributes $CNδ$ to the endpoint error of computed iterates. We report refinement studies for smooth and fixed multiscale data, together with qualitative flow comparisons. An exactly solvable rough shear has fractional errors, up to logarithms, in $B^1_{2,1}$ and first-order errors in $B^0_{\infty,1}$.

math.NA↗
arXiv2026

Quantum circuits for the advection-diffusion equation with boundary conditions based on LCHS

This paper proposes a systematic and explicit quantum circuit framework for solving advection-diffusion equations with boundary conditions, based on the Linear Combination of Hamiltonian Simulations (LCHS) method.By employing the Finite Volume Method (FVM) combined with various flux construction schemes, we elaborate the design of quantum circuits tailored explicitly for Robin boundary conditions (including Dirichlet and Neumann boundary conditions as special cases) and periodic boundary conditions. The Robin boundary elimination is encoded by newly constructed endpoint-projector circuits, and all physical and boundary-induced inhomogeneous terms are incorporated through an outer linear combination of unitaries (LCU). The error analysis adopts an LCU-weighted form, preserving the decay of the LCHS kernel in the product-formula bounds. Complementing this analysis, a comprehensive gate complexity analysis demonstrates the quantum advantages over classical computing in high-dimensional scenarios. We simulate the proposed circuits on a fault-tolerant emulator, and numerical results validate the effectiveness of the proposed framework across homogeneous, inhomogeneous, and high-dimensional cases.The proposed framework is compatible with numerous spatial discretization methods and numerical schemes, extends naturally to other linear PDEs, and establishes a practical foundation for solving large-scale PDE problems on future fault-tolerant quantum computers.

math.NA↗
arXiv2026

Generalized Modulo Hysteresis Encoding

Unlimited sensing and its extension via modulo hysteresis provide an efficient encoding scheme for high dynamic range signals by folding the signal's amplitude into the range of the analog-to-digital converter prior to sampling. Motivated by shortcomings of the modulo hysteresis model, in this work we introduce generalized modulo encoders that flexibly design the folding by incorporating a general transition function and allowing the folding times to depend on the nonideality of the folding transition. We rigorously study mathematical properties of the respective operators and derive conditions under which a bandlimited signal is uniquely determined by its generalized modulo samples. Finally, we propose a recovery algorithm backed by mathematical guarantees and illustrate our theoretical results through numerical experiments.

math.NA↗