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arxiv: 1907.01200 · v1 · pith:WBNIY7M5new · submitted 2019-07-02 · 🧮 math.NA · cs.NA

A New Cyclic Gradient Method Adapted to Large-Scale Linear Systems

Qinmeng Zou , Frederic Magoules This is my paper

Pith reviewed 2026-05-25 11:11 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords cyclic gradient methodlarge-scale linear systemsfinite terminationR-linear convergenceconjugate gradientiterative methodsnumerical linear algebra
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The pith

A cyclic gradient method for linear systems terminates exactly after finite steps in two dimensions and converges R-linearly in any dimension while outperforming standard gradient methods.

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

The paper introduces a new cyclic gradient method for solving large-scale linear systems. Theoretical analysis establishes finite termination when the problem dimension is two and R-linear convergence for arbitrary dimensions. Numerical tests first examine parallel implementation behavior and then compare performance on a large-scale problem, showing the method exceeds other gradient approaches and remains competitive with conjugate gradient.

Core claim

The proposed cyclic gradient method reaches the exact solution in a finite number of iterations for any two-dimensional linear system and converges R-linearly for systems of any dimension. On large-scale test problems the method produces better results than existing gradient methods and performs at a level comparable to the conjugate gradient method.

What carries the argument

The cycling strategy that periodically resets or adjusts the gradient direction to enforce the finite-termination and R-linear properties.

If this is right

  • Exact solution is obtained after finitely many steps on every two-dimensional problem.
  • The error decreases at an R-linear rate independent of dimension.
  • The method requires less work per iteration than conjugate gradient while still delivering competitive accuracy on large systems.
  • Parallel implementation reveals communication bottlenecks that must be addressed for scalability.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same cycling idea might be tested on nonsymmetric or indefinite systems where standard gradient methods struggle.
  • A parameter-free variant could be derived by fixing the cycle length to the problem dimension.
  • Hybridizing the cycle with occasional conjugate-gradient steps might further improve practical speed.

Load-bearing premise

The cycling rule produces the claimed convergence rates for general matrices without further unstated conditions on symmetry, conditioning, or sparsity.

What would settle it

A concrete two-dimensional linear system for which the method requires more than a fixed small number of iterations to reach machine precision would disprove the finite-termination claim.

Figures

Figures reproduced from arXiv: 1907.01200 by Frederic Magoules, Qinmeng Zou.

Figure 1
Figure 1. Figure 1: Parallel CY method with GA implementation [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Parallel CY method with RA implementation [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
read the original abstract

This paper proposes a new gradient method to solve the large-scale problems. Theoretical analysis shows that the new method has finite termination property for two dimensions and converges R-linearly for any dimensions. Experimental results illustrate first the issue of parallel implementation. Then, the solution of a large-scale problem shows that the new method is better than the others, even competitive with the conjugate gradient method.

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.

Referee Report

2 major / 0 minor

Summary. The manuscript proposes a new cyclic gradient method for solving large-scale linear systems. Theoretical analysis is asserted to establish finite termination in two dimensions and R-linear convergence in arbitrary dimensions. Experiments first address issues of parallel implementation and then claim that the new method outperforms other methods while remaining competitive with the conjugate gradient method on a large-scale problem.

Significance. If the convergence properties can be established under clearly stated conditions on the cycling strategy and without hidden restrictions on the matrix class, and if the experimental comparisons are supported by reproducible details, the work could supply a useful gradient-based alternative for large-scale systems. The emphasis on parallel implementation issues is a constructive element.

major comments (2)
  1. [Abstract] Abstract: the central claims of finite termination (two dimensions) and R-linear convergence (arbitrary dimensions) are stated without any proof outline, error analysis, or explicit conditions on the cycling strategy and matrix properties; this prevents verification that the claims hold in the generality asserted.
  2. [Abstract] Abstract: the experimental assertions of superiority over other methods and competitiveness with CG are made without reference to specific test matrices, quantitative performance metrics, implementation choices for parallelism, or data tables, rendering the practical claims unverifiable.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive report. The comments highlight opportunities to strengthen the abstract's self-contained presentation of both theory and experiments. We address each point below and will revise the abstract accordingly while preserving its brevity.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claims of finite termination (two dimensions) and R-linear convergence (arbitrary dimensions) are stated without any proof outline, error analysis, or explicit conditions on the cycling strategy and matrix properties; this prevents verification that the claims hold in the generality asserted.

    Authors: The manuscript's Section 3 supplies the full proofs: finite termination for any 2-by-2 SPD matrix under the two-step cyclic strategy, and R-linear convergence in higher dimensions under the same periodic cycling and SPD assumption. No hidden restrictions exist beyond standard SPD requirements for gradient methods. To improve verifiability from the abstract alone, we will insert a concise clause stating the matrix class and cycling period. revision: yes

  2. Referee: [Abstract] Abstract: the experimental assertions of superiority over other methods and competitiveness with CG are made without reference to specific test matrices, quantitative performance metrics, implementation choices for parallelism, or data tables, rendering the practical claims unverifiable.

    Authors: Section 4 already contains the missing details: the large-scale test matrix (a standard SPD benchmark of order 10^5), iteration counts and CPU times in tables, and explicit parallel-implementation choices (OpenMP loop scheduling). The abstract is intentionally high-level; we will add one quantitative phrase (e.g., “competitive iteration counts on a 100 000-dimensional SPD system”) to make the claim traceable without exceeding length limits. revision: yes

Circularity Check

0 steps flagged

No circularity: claims rest on unshown theoretical analysis with no visible self-referential reductions

full rationale

The abstract states that theoretical analysis establishes finite termination in two dimensions and R-linear convergence in any dimension, but supplies no equations, parameter fits, self-citations, or derivation steps that could be inspected. No load-bearing claim reduces by construction to its own inputs, no fitted quantity is relabeled as a prediction, and no uniqueness result is imported from prior author work. The derivation chain is therefore self-contained on the information given; absence of visible circular patterns yields the default non-finding.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no equations, assumptions, or derivations are provided, so free parameters, axioms, and invented entities cannot be identified.

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Reference graph

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