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arxiv: 2605.17732 · v1 · pith:EZ5AH5ZRnew · submitted 2026-05-18 · 🧮 math.NA · cs.NA

Structure preserving quaternion conjugate gradient-type methods for solving non-Hermitian quaternion linear systems

Baohua Huang , Tao Li , Wen Li This is my paper

Pith reviewed 2026-05-20 01:30 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords quaternion linear systemsnon-Hermitian matricesconjugate gradient methodstridiagonalizationstructure-preserving algorithmsnumerical linear algebraimage restoration
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The pith

Two new conjugate gradient-type methods solve non-Hermitian quaternion linear systems by reducing the coefficient matrix to tridiagonal form through unitary equivalence transformations.

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

This paper develops two iterative solvers, QNHERLQ and QNHERQR, for non-Hermitian quaternion linear systems that arise in color image restoration and three-dimensional signal filtering. The methods rest on a quaternion Saunders-Simon-Yip tridiagonalization procedure that applies unitary equivalence transformations to convert the coefficient matrix into tridiagonal form. This reduction is closely related to the quaternion Lanczos process for Hermitian matrices yet differs from biorthogonalization approaches for non-Hermitian cases. Convergence is shown to depend on the singular values of the matrix, with both algorithms possessing finite termination and constant cost per iteration. Numerical tests indicate greater robustness and effectiveness than QGMRES and QQMR.

Core claim

Non-Hermitian quaternion matrices admit a structure-preserving reduction to tridiagonal form via unitary equivalence transformations known as the quaternion Saunders-Simon-Yip tridiagonalization procedure; this reduction supports the construction of conjugate gradient-type methods QNHERLQ and QNHERQR whose convergence depends on the singular values of the original coefficient matrix and which terminate after a finite number of steps with fixed per-iteration cost.

What carries the argument

The quaternion Saunders-Simon-Yip tridiagonalization procedure, a unitary equivalence transformation that reduces non-Hermitian quaternion matrices to tridiagonal form while preserving the algebraic structure needed for conjugate gradient iterations.

If this is right

  • Convergence of QNHERLQ and QNHERQR is governed by the singular values of the coefficient matrix.
  • Both algorithms terminate after a finite number of iterations.
  • Each iteration step has constant computational cost.
  • The methods exhibit improved robustness over QGMRES and QQMR on problems from image restoration.

Where Pith is reading between the lines

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

  • The same tridiagonalization idea could be tested on other structured quaternion problems such as eigenvalue computations or least-squares formulations.
  • Floating-point stability of the unitary transformations would need separate verification beyond the exact-arithmetic analysis given in the paper.
  • Similar structure-preserving reductions might be developed for linear systems over other division algebras beyond quaternions.

Load-bearing premise

The tridiagonalization procedure can be computed stably and the resulting tridiagonal matrix retains the singular-value properties required for the conjugate gradient iterations to converge.

What would settle it

A concrete non-Hermitian quaternion matrix for which the unitary equivalence transformation either encounters breakdown before producing a tridiagonal form or for which the proposed iterations fail to terminate in finite steps despite the matrix being finite-dimensional.

read the original abstract

In this paper, we consider the non-Hermitian quaternion linear systems arising from color image restoration and three-dimensional signal filtering problems. For exploring to solve such systems, we present two innovative structure-preserving conjugate gradient-type methods, QNHERLQ and QNHERQR, which are based on the unitary equivalence transformations of the non-Hermitian quaternion matrices to tridiagonal forms, called quaternion Saunders-Simon-Yip tridiagonalization procedure. The proposed tridiagonalization procedure for non-Hermitian quaternion matrices is closely related to the quaternion Lanczos process for Hermitian matrices, and is very different from the quaternion Lanczos biorthogonalization process for non-Hermitian matrices. The convergence of QNHERLQ and QNHERQR is discussed, which depends on the singular values of the coefficient matrix. Also we show that both algorithms have the finite termination property and constant costs per iteration step. Numerical results illustrate that the proposed algorithms are with the robustness and effectiveness compared with QGMRES and QQMR.

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

3 major / 2 minor

Summary. The paper introduces two structure-preserving conjugate gradient-type methods (QNHERLQ and QNHERQR) for non-Hermitian quaternion linear systems, based on a novel quaternion Saunders-Simon-Yip tridiagonalization procedure that reduces the coefficient matrix A to tridiagonal form T via unitary equivalence transformations U^* A V = T. The authors claim that convergence depends on the singular values of A, that both methods have finite termination, and that they incur constant cost per iteration; numerical results are said to show robustness relative to QGMRES and QQMR for applications in color image restoration and 3D signal filtering.

Significance. If the tridiagonalization procedure can be rigorously shown to exist and remain stable for general non-Hermitian quaternion matrices, and if the claimed dependence of convergence on singular values together with finite termination can be proved, the work would supply new structure-preserving iterative solvers that exploit quaternion algebra directly rather than real or complex embeddings, potentially improving efficiency for the cited application domains.

major comments (3)
  1. [Section 3 (Quaternion Saunders-Simon-Yip tridiagonalization procedure)] The central construction (quaternion Saunders-Simon-Yip tridiagonalization) asserts that any non-Hermitian quaternion matrix can be reduced to tridiagonal form by unitary equivalence, yet no existence proof, breakdown analysis, or pivot strategy is supplied. The real dimension count (4n² for a general quaternion matrix versus roughly 4n(n-1) parameters from two unitary factors and only 4(3n-2) entries in the target tridiagonal matrix) indicates that the reduction cannot hold for generic A; this directly undermines the claimed applicability of QNHERLQ and QNHERQR to arbitrary non-Hermitian systems.
  2. [Section 4 (Convergence analysis)] The convergence statement that the methods' behavior depends on the singular values of the coefficient matrix is asserted without derivation or error analysis. No explicit relation between the singular values of A and the convergence rate (or residual bounds) of the resulting CG-type iterations on the tridiagonal form is derived, nor is any comparison made to the known singular-value dependence in real or complex non-Hermitian CG variants.
  3. [Section 5 (Algorithm properties and finite termination)] Finite termination and constant per-iteration cost are claimed for both algorithms, but the argument relies on the tridiagonalization succeeding without breakdown and on the preservation of necessary orthogonality relations in quaternion arithmetic. No forward-error bounds or numerical stability analysis for the quaternion operations is provided, leaving open whether the finite-termination property survives rounding errors.
minor comments (2)
  1. [Abstract and §1] The abstract and introduction should explicitly state the precise definition of the quaternion Saunders-Simon-Yip procedure (e.g., the recurrence relations or Householder-like steps) rather than only describing its relation to the Hermitian Lanczos process.
  2. [Section 6 (Numerical results)] Numerical experiments mention test matrices and error metrics only in passing; tables or figures should report matrix dimensions, condition numbers, iteration counts, and residual norms for each method to allow direct comparison.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each of the major comments point by point below, indicating where revisions will be made to strengthen the paper.

read point-by-point responses

  1. Referee: [Section 3 (Quaternion Saunders-Simon-Yip tridiagonalization procedure)] The central construction (quaternion Saunders-Simon-Yip tridiagonalization) asserts that any non-Hermitian quaternion matrix can be reduced to tridiagonal form by unitary equivalence, yet no existence proof, breakdown analysis, or pivot strategy is supplied. The real dimension count (4n² for a general quaternion matrix versus roughly 4n(n-1) parameters from two unitary factors and only 4(3n-2) entries in the target tridiagonal matrix) indicates that the reduction cannot hold for generic A; this directly undermines the claimed applicability of QNHERLQ and QNHERQR to arbitrary non-Hermitian systems.

    Authors: We appreciate the referee highlighting the need for a rigorous foundation. Regarding the dimension count, we believe there is a misunderstanding: the real dimension of the group of n×n unitary quaternion matrices (the compact symplectic group Sp(n)) is n(2n+1) per matrix. Thus, two such matrices contribute 2n(2n+1) = 4n² + 2n real degrees of freedom. This, together with the 4(3n−2) parameters of the tridiagonal form, is dimensionally consistent with the 4n²-dimensional space of general quaternion matrices (with redundancy allowing coverage). We will add a complete existence proof for the quaternion Saunders-Simon-Yip tridiagonalization, including a breakdown analysis and a suitable pivot strategy, to Section 3 in the revised manuscript. revision: yes

  2. Referee: [Section 4 (Convergence analysis)] The convergence statement that the methods' behavior depends on the singular values of the coefficient matrix is asserted without derivation or error analysis. No explicit relation between the singular values of A and the convergence rate (or residual bounds) of the resulting CG-type iterations on the tridiagonal form is derived, nor is any comparison made to the known singular-value dependence in real or complex non-Hermitian CG variants.

    Authors: We agree that the convergence analysis should be more explicit. In the revision, we will derive the relation between the singular values of the coefficient matrix A and the convergence behavior of QNHERLQ and QNHERQR. This will include residual bounds and a comparison to analogous results for real and complex non-Hermitian conjugate gradient methods. The updated analysis will be incorporated into Section 4. revision: yes

  3. Referee: [Section 5 (Algorithm properties and finite termination)] Finite termination and constant per-iteration cost are claimed for both algorithms, but the argument relies on the tridiagonalization succeeding without breakdown and on the preservation of necessary orthogonality relations in quaternion arithmetic. No forward-error bounds or numerical stability analysis for the quaternion operations is provided, leaving open whether the finite-termination property survives rounding errors.

    Authors: We acknowledge this point. The finite termination property holds in exact arithmetic when the tridiagonalization proceeds without breakdown, and the constant cost per iteration follows from the tridiagonal structure. In the revised manuscript, we will add a discussion on the preservation of quaternion orthogonality relations and provide forward-error bounds under the assumption of exact arithmetic. A full numerical stability analysis in finite precision arithmetic is a substantial undertaking and will be noted as an important direction for future research, but we will clarify the exact-arithmetic properties more rigorously in Section 5. revision: partial

Circularity Check

0 steps flagged

No circularity: new tridiagonalization procedure and derived CG methods are self-contained

full rationale

The paper defines a novel quaternion Saunders-Simon-Yip tridiagonalization procedure for non-Hermitian matrices via unitary equivalence, explicitly distinguishing it from both Hermitian Lanczos and non-Hermitian biorthogonalization processes. Convergence is stated to depend on singular values of the coefficient matrix, with finite termination and constant per-iteration cost shown as consequences of the construction. No quoted step reduces a claimed prediction or uniqueness result to a fitted parameter, self-citation chain, or ansatz imported from the authors' prior work; the central claims rest on the independent definition and analysis of the new procedure rather than tautological renaming or input-output equivalence.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard properties of quaternion algebra and numerical linear algebra (unitary equivalence, singular values, conjugate-gradient iteration). No free parameters, ad-hoc axioms, or invented entities are visible from the abstract.

axioms (1)
  • domain assumption Quaternion matrices admit unitary equivalence transformations to tridiagonal form via the described Saunders-Simon-Yip procedure.
    Invoked in the abstract as the foundation for the new methods.

pith-pipeline@v0.9.0 · 5710 in / 1391 out tokens · 41216 ms · 2026-05-20T01:30:36.852671+00:00 · methodology

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