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arxiv: 2604.15766 · v1 · submitted 2026-04-17 · 🧮 math.NA · cs.NA

Efficient Solution of Generalized Sylvester Equations via Preconditioned Alternating Anderson Acceleration

Hongjia Chen , Chun-Hua Zhang , Zhongming Teng , Lei Du This is my paper

Pith reviewed 2026-05-10 08:46 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords generalized Sylvester equationsAnderson accelerationpreconditioningNeumann seriesiterative methodsmatrix equationsnumerical linear algebra
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The pith

A preconditioned alternating Anderson acceleration method solves generalized Sylvester equations more efficiently by accelerating fixed-point iterations with a Neumann series approximation.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces the preconditioned-alternating Anderson acceleration (P-aAA) method for generalized Sylvester matrix equations that arise in scientific applications. It alternates between preconditioned fixed-point iterations and Anderson acceleration updates using a first-order Neumann series approximation for the preconditioner. This targets cases where the spectral radius of the associated operator is large but at most 1, which slows standard iterations. The authors prove that the preconditioner improves the convergence rate without raising overall computational complexity. Numerical tests on medium- and large-scale problems show the method reduces iteration counts and run times compared with existing approaches.

Core claim

The central claim is that the P-aAA method, which combines a matrix-oriented variant of Anderson acceleration with a novel preconditioning operator based on a first-order Neumann series approximation, efficiently solves generalized Sylvester equations. By alternating between preconditioned fixed-point iterations and acceleration updates, the scheme reduces both iteration count and computational cost. Theoretical analysis establishes that the preconditioning accelerates convergence without increasing overall complexity, and extensive experiments confirm consistent outperformance over state-of-the-art methods on medium- and large-scale instances.

What carries the argument

The key machinery is the preconditioning operator derived from a first-order Neumann series approximation to the inverse of the fixed-point operator, which enhances the convergence rate of the alternating Anderson acceleration scheme for generalized Sylvester equations.

Load-bearing premise

The spectral radius of the associated operator is large but not greater than 1, allowing the fixed-point iteration base to converge and the Neumann preconditioner to be effective for general coefficient matrices.

What would settle it

A concrete counterexample would be a test problem with spectral radius at most 1 on which the preconditioned method requires the same or more iterations and time than the non-preconditioned alternating Anderson acceleration version.

Figures

Figures reproduced from arXiv: 2604.15766 by Chun-Hua Zhang, Hongjia Chen, Lei Du, Zhongming Teng.

Figure 1
Figure 1. Figure 1: The iteration numbers and relative residual error of the pro [PITH_FULL_IMAGE:figures/full_fig_p015_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The iteration numbers and relative residual error of the pro [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The iteration numbers and relative residual error of the pro [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
read the original abstract

This paper considers the numerical solution of generalized Sylvester matrix equations, which arise in many scientific and engineering applications but remain challenging to solve efficiently, particularly when the coefficient matrices are general and the spectral radius of the associated operator is large but not greater than $1$. We propose a new iterative method, termed preconditioned-alternating Anderson acceleration (P-aAA), which combines a matrix-oriented variant of Anderson acceleration (AA) with a novel preconditioning strategy. The method alternates between preconditioned fixed-point iterations and Anderson acceleration updates, thereby reducing both computational cost and iteration count. A key contribution is the development of an efficient preconditioning operator based on a first-order Neumann series approximation, which avoids expensive operator inversions while enhancing convergence. We theoretically prove that the proposed preconditioning operator accelerates the convergence rate without increasing the overall computational complexity. Extensive numerical experiments further demonstrate that the proposed approach consistently outperforms existing state-of-the-art methods for both medium- and large-scale problems, achieving substantial reductions in computation time and iteration number.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The ledger captures the key domain assumption on spectral radius and the tunable parameters implied by the method description; no new entities are postulated.

free parameters (2)
  • Anderson acceleration depth m
    Number of previous iterates retained for the acceleration update; chosen by the user or tuned for performance.
  • Neumann series truncation order
    Order at which the infinite series for the preconditioner is cut off; balances accuracy and cost.
axioms (1)
  • domain assumption Spectral radius of the fixed-point operator is at most 1
    Invoked to guarantee convergence of the base iteration and effectiveness of the preconditioner.

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