The worst-case root-convergence factor of GMRES(1)

TL;DR

This work studies the worst-case asymptotic convergence of MRI/GMRES(1) for Ax=b, deriving explicit worst-case root-convergence factors in two key structural settings: A symmetric and M=I−A skew-symmetric. By formulating GMRES(1) iterations as vector-dependent nonlinear eigenvalue problems (I_2 and Π) and analyzing eigenpairs that are combinations of A’s eigenvectors, the authors obtain closed-form bounds: for symmetric definite A, ρ^* ≤ |(λ_max−λ_min)/(λ_max+λ_min)|, while for symmetric indefinite A, ρ^*=1; for skew-symmetric M, GMRES(1) converges unconditionally with ρ^* = m^*/√(1+(m^*)^2). The q-linear convergence factor is shown to coincide with the worst-case factor. The paper also extends the analysis to restarted Anderson acceleration rAA(1), providing counterexamples to a conjecture and comparing performance with GMRES(1) under skew-symmetric M, along with numerical experiments validating the theory and highlighting the influence of the initial guess. These results offer theoretical guidance for choosing restart parameters and preconditioning to accelerate convergence. All mathematical expressions are presented with precise notation to support reproducibility and SEO relevance.

Abstract

In this work, we analyze the asymptotic convergence factor of minimal residual iteration (MRI) (or GMRES(1)) for solving linear systems $Ax=b$ based on vector-dependent nonlinear eigenvalue problems. The worst-case root-convergence factor is derived for linear systems with $A$ being symmetric or $I-A$ being skew-symmetric. When $A$ is symmetric, the asymptotic convergence factor highly depends on the initial guess. While $M=I-A$ is skew-symmetric, GMRES(1) converges unconditionally and the worst-case root-convergence factor relies solely on the spectral radius of $M$. We also derive the q-linear convergence factor, which is the same as the worst-case root-convergence factor. Numerical experiments are presented to validate our theoretical results.

Figures (4)

• Figure 1: Root-convergence factor $\varrho_k(r_0)=\|r_k\|^{1/k}$ (solid lines) as a function of the iteration index $k$ with 1000 random initial guesses for $A_1$ (left) and $A_2$ (right). The dashed line is the theoretical worst-case root convergence factor, denoted as $\varrho^*$.
• Figure 2: Root-convergence factor $\varrho_k(r_0)=\|r_k\|^{1/k}$ (solid lines) as a function of the iteration index $k$ with 1000 random initial guesses for $A_3$ (left) and for $A_3$ (right) with the first elements of the initial guesses being zero. The dashed line is the theoretical worst-case root-convergence factor, denoted as $\varrho^*$.
• Figure 3: Root-convergence factor $\varrho_k(r_0)=\|r_k\|^{1/k}$ (solid lines) as a function of the iteration index $k$ with two different initial guesses for $A_2$. The dashed line is the theoretical worst-case root-convergence factor. Cross markers corresponds to $x_0=[1, 2\sqrt{2}, 0,0,0]^T$, where $\varrho^*=0.3333$ . Circle markers are for $x_0=[1, 0, 8,0,0]^T$, where $\varrho^*=0.6000$.
• Figure 4: Root-convergence factor $\varrho_k(r_0)=\|r_k\|^{1/k}$ (solid lines) as a function of the iteration index $k$ with 1000 random initial $x_0$ for $A_4$ (left). The dashed line is the theoretical worst-case root convergence factor, denoted as $\varrho^*$. Cross markers are $\varrho_k(r_0)=\|r_k\|^{1/k}$ as a function of the iteration index $k$ with $x_0=Q(:,3)+Q(:,5)+Q(:,7)$ for $A_4$ (right).

Theorems & Definitions (47)

• Definition 1.1: Root convergence
• Definition 1.2: q-linear convergence,toth2015convergence
• Lemma 2.1
• Proof 1
• Lemma 2.2
• Lemma 2.3
• Proof 2
• Theorem 2.4
• Proof 3
• Theorem 2.5