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arxiv: 2605.05551 · v1 · submitted 2026-05-07 · 🧮 math.NA · cs.NA

Recognition: unknown

The double splitting iteration method for solving the large indefinite least squares problem

Jun Li, Lingsheng Meng

Pith reviewed 2026-05-08 07:26 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords indefinite least squaresdouble splitting iterationnormal equationsiterative methodsnumerical linear algebralarge-scale problemsconvergence analysis
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The pith

A double splitting of the normal equations produces an iterative solver for large indefinite least squares problems that is faster and more robust than single-splitting schemes.

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

The paper develops an iterative framework for large indefinite least squares problems by splitting the associated normal equations twice instead of once. Specific implementations of this double-splitting strategy are described and tested on numerical examples. The authors report that the new approach reduces iteration counts and improves convergence reliability compared with conventional single-splitting methods. A sympathetic reader would care because indefinite least squares problems arise in applications with large, ill-conditioned matrices where single-splitting iterations can stall or diverge, and any method that reliably shortens run time without extra matrix assumptions would lower the practical cost of solving them.

Core claim

The proposed double splitting iterative paradigm outperforms conventional single splitting approaches in both computational efficiency and convergence robustness for the large indefinite least squares problem.

What carries the argument

Double splitting of the normal equations, which partitions the coefficient matrix into two auxiliary matrices to define the iteration operator and to control convergence behavior for indefinite systems.

If this is right

  • The double-splitting iteration converges in fewer steps than single-splitting iterations on large indefinite least squares instances.
  • Total computational time decreases because fewer matrix-vector products are needed.
  • Convergence remains stable across varied problem sizes and conditioning without requiring extra constraints on the coefficient matrix.
  • The concrete splitting constructions serve as templates for designing further double-splitting solvers.

Where Pith is reading between the lines

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

  • If the double-splitting construction extends beyond least squares, the same partitioning idea might accelerate iterative solvers for other indefinite linear systems.
  • Optimal parameter selection within each split could be studied to further shorten iteration counts on matrices with known spectral properties.
  • Direct comparison against Krylov-subspace methods on the same test set would clarify whether the efficiency gain is specific to splitting or holds more broadly.

Load-bearing premise

The specific double splitting implementations will reliably improve convergence and efficiency for general large indefinite least squares problems without additional restrictions on matrix properties or parameter choices.

What would settle it

Numerical tests on a collection of large indefinite least squares problems in which the double-splitting method requires more iterations or fails to converge on cases where a single-splitting method succeeds would falsify the claimed superiority.

Figures

Figures reproduced from arXiv: 2605.05551 by Jun Li, Lingsheng Meng.

Figure 1
Figure 1. Figure 1: α vs CPU times of DS iteration method in Example 4.1. Example 3.1. [18] we construct the ILS problem (1.1) using random large scale dense matrix A1 = rand(p, n), and A2 = 7 ∗ eye(q, n), b1 = rand(p, 1), b2 = rand(q, 1), where p = 40000, n = q = [11000 : 1000 : 14000] and m = p+q view at source ↗
Figure 2
Figure 2. Figure 2: The testing parameters vs CPU times of DS iteration method for Example 4.2 view at source ↗
read the original abstract

Addressing large-scale indefinite least squares (ILS) problem poses notable computational bottlenecks in the field of numerical linear algebra. State-of-the-art iterative schemes for such problems are predominantly constructed upon the single splitting of the coefficient matrix derived from the corresponding normal equation. In this work, we put forward an innovative iterative framework grounded in the double splitting of normal equations tailored for ILS problem. Specifically, we elaborate on a distinct implementations of the double splitting strategy, which offer constructive insights and methodological references for subsequent research on double splitting-based iterative methods. Two numerical experiments further corroborate that the proposed double splitting iterative paradigm outperforms conventional single splitting approaches in both computational efficiency and convergence robustness.

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 / 2 minor

Summary. The manuscript proposes a double-splitting iterative framework for large-scale indefinite least-squares (ILS) problems, obtained by applying two distinct double-splitting strategies to the normal equations. It presents two concrete implementations and reports two numerical experiments claiming that the new paradigm outperforms conventional single-splitting methods in both computational efficiency and convergence robustness.

Significance. If the empirical superiority can be placed on a firm theoretical footing, the double-splitting approach would constitute a useful methodological extension for ILS problems, where the normal-equation operator can possess negative eigenvalues. The work supplies no machine-checked proofs, reproducible code, or parameter-free derivations, so its significance remains tied to the two reported experiments.

major comments (2)
  1. [Numerical Experiments] The central claim of improved efficiency and robustness rests entirely on two numerical experiments (described in the abstract and the final section). No information is given on matrix dimensions, the spectrum of the normal-equation operator (in particular the location and magnitude of negative eigenvalues), the choice of splitting parameters, or the precise single-splitting baselines used for comparison. Without these details the experiments cannot be verified to cover the indefinite regime where the method is claimed to be advantageous.
  2. [Theoretical Analysis] No convergence analysis or theorem is supplied for either double-splitting iteration. The claim that the methods converge faster and more reliably than single-splitting schemes on arbitrary large ILS problems therefore lacks the necessary supporting derivation; the admissibility of the splitting parameters for indefinite matrices is not established.
minor comments (2)
  1. [Abstract] The abstract states that the double-splitting strategy 'offer constructive insights' but does not indicate which concrete algorithmic choices (e.g., how the two splittings are combined or how parameters are selected) constitute the novelty.
  2. [Method Description] Notation for the two distinct implementations is introduced without an explicit comparison table or pseudocode, making it difficult to see at a glance how they differ from each other and from standard single-splitting schemes.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript proposing a double-splitting iterative framework for large-scale indefinite least-squares problems. We address each major comment point by point below and will revise the manuscript accordingly to improve clarity and reproducibility while maintaining the focus on the proposed methodology and its empirical validation.

read point-by-point responses

  1. Referee: [Numerical Experiments] The central claim of improved efficiency and robustness rests entirely on two numerical experiments (described in the abstract and the final section). No information is given on matrix dimensions, the spectrum of the normal-equation operator (in particular the location and magnitude of negative eigenvalues), the choice of splitting parameters, or the precise single-splitting baselines used for comparison. Without these details the experiments cannot be verified to cover the indefinite regime where the method is claimed to be advantageous.

    Authors: We agree that these details are necessary for full verification and reproducibility. In the revised manuscript we will expand the numerical experiments section to report the dimensions of all test matrices, describe the eigenvalue spectrum of the normal-equation operators (including the location and magnitude of negative eigenvalues), specify the splitting parameters employed, and identify the exact single-splitting baselines used for comparison. These additions will confirm that the experiments address the indefinite regime. revision: yes

  2. Referee: [Theoretical Analysis] No convergence analysis or theorem is supplied for either double-splitting iteration. The claim that the methods converge faster and more reliably than single-splitting schemes on arbitrary large ILS problems therefore lacks the necessary supporting derivation; the admissibility of the splitting parameters for indefinite matrices is not established.

    Authors: The manuscript introduces the double-splitting framework and supports its advantages through two numerical experiments rather than providing general convergence theorems. The abstract and text do not claim convergence for arbitrary large ILS problems; the stated conclusion is limited to the reported experiments. We will revise the manuscript to clarify the scope of the claims, add a brief discussion of the empirical admissibility of the chosen splitting parameters, and explicitly note that a full theoretical convergence analysis for indefinite matrices is an important topic for future work. revision: yes

Circularity Check

0 steps flagged

No significant circularity; new method proposal with independent empirical validation

full rationale

The paper introduces a double-splitting iterative framework for the indefinite least squares problem, describes specific implementations, and validates performance claims solely via two numerical experiments. No derivation chain, equations, or theoretical steps are presented that reduce any claim to a self-definitional fit, a fitted input renamed as prediction, or a load-bearing self-citation. The central assertion of outperformance over single-splitting methods rests on external numerical evidence rather than any construction that is equivalent to its own inputs by definition. This is the expected non-finding for a method-proposal paper whose support is empirical.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The abstract invokes standard assumptions from numerical linear algebra for iterative convergence but introduces no explicit free parameters, new axioms, or invented entities.

axioms (1)
  • domain assumption Standard convergence assumptions for splitting-based iterative methods applied to normal equations of indefinite least squares problems
    Any iterative solver for this problem class requires background results on matrix splittings and spectral properties to guarantee behavior.

pith-pipeline@v0.9.0 · 5400 in / 1105 out tokens · 51652 ms · 2026-05-08T07:26:58.285136+00:00 · methodology

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Reference graph

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