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arxiv: 2606.04259 · v1 · pith:DIKYBY7Ynew · submitted 2026-06-02 · 🧮 math.NA · cs.NA

An indefinite LOBPCG type of algorithm for detecting a definite Hermitian matrix pair

Marija Milolo\v{z}a Pandur This is my paper

Pith reviewed 2026-06-28 08:35 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords Hermitian matrix pairsdefiniteness detectionLOBPCGsubspace iterationnumerical linear algebraindefinite matricespreconditioned methodslarge-scale matrices
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The pith

An LOBPCG-style subspace iteration detects definiteness of large Hermitian matrix pairs by testing small projected pairs.

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

The paper develops a subspace algorithm to determine whether a large Hermitian pair (A, B) with indefinite B is definite, that is, whether some real linear combination of A and B is positive definite. The approach repeatedly builds small subspaces and checks the definiteness property on the resulting low-dimensional projected pairs. A specialized version sets the subspace dimension to m=3 and constructs the subspaces exactly as in the indefinite LOBPCG method, with an optional preconditioned variant. Numerical tests on medium-sized, large, and banded pairs show that the method identifies (in)definiteness much faster than several existing algorithms.

Core claim

The central claim is that iterative testing of small projected Hermitian pairs formed from LOBPCG-style subspaces of dimension m=3 reliably detects the global definiteness property of the original large pair.

What carries the argument

The specialized algorithm with parameter m that generates three-dimensional subspaces in the style of the indefinite LOBPCG method and tests definiteness on the resulting projected pairs.

If this is right

  • The algorithm scales to large and banded pairs where dense methods become impractical.
  • Preconditioning can be added to the subspace construction to reduce the number of iterations needed for detection.
  • The same subspace-testing idea applies directly to medium-sized pairs with only modest overhead.
  • Repeated testing of projected pairs replaces the need for a single expensive global computation of all possible linear combinations.

Where Pith is reading between the lines

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

  • The technique could be combined with other iterative eigensolvers when definiteness must be checked before solving a related eigenproblem.
  • Similar low-dimensional projection tests might apply to detecting other matrix properties such as stability or hyperbolicity in nearby problem classes.
  • For matrices with additional structure such as sparsity patterns beyond banding, the subspace dimension m might be tuned adaptively rather than fixed at three.

Load-bearing premise

The LOBPCG-style subspaces of dimension three are assumed to be rich enough that any positive definite linear combination, if it exists, will appear as a positive definite combination inside one of the tested small projected pairs.

What would settle it

A definite Hermitian pair (A, B) on which the algorithm returns 'indefinite' on every run because no tested three-dimensional subspace ever captures a positive definite combination.

read the original abstract

A Hermitian matrix pair $(A,B)$ is called definite if some real linear combination of the matrices $A$ and $B$ is a positive definite matrix. Detection of the definiteness is not straightforward. We propose a basic subspace algorithm for detecting a large definite matrix pair $(A,B)$ with indefinite $B$. The proposed subspace algorithm is based on iterative testing of small projected Hermitian matrix pairs formed by using subspaces of small dimensions. Furthermore, we propose a specialized algorithm with parameter $m$, and its preconditioned variant. In the specialized algorithm with $m=3$ we choose the subspaces like in the indefinite locally optimal block preconditioned conjugate gradient (LOBPCG) method. Numerical experiments demonstrate the efficiency of our specialized algorithm, applied on medium-sized pairs, as well as, on large and banded pairs. Our algorithm very quickly detects (in)definiteness; much faster than some other algorithms.

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 basic subspace iteration algorithm and a specialized variant (with parameter m=3) based on indefinite LOBPCG subspaces for detecting whether a Hermitian pair (A,B) with indefinite B is definite, i.e., admits some real linear combination that is positive definite. The method repeatedly forms and tests small projected pairs; if any projected pair is definite then the original pair is definite. Numerical experiments on medium-sized, large, and banded pairs are presented to show that the algorithm detects (in)definiteness quickly and is faster than some existing methods.

Significance. If the detection procedure is reliable, the work supplies a practical tool for large-scale definiteness testing where dense methods are prohibitive. The adaptation of LOBPCG-style block updates to the definiteness-detection setting is a novel algorithmic idea, and the reported performance on banded pairs is a concrete strength. However, the absence of any convergence or correctness argument limits the result's theoretical impact within numerical linear algebra.

major comments (2)
  1. [§3] §3 (specialized algorithm with m=3): no argument is supplied that the LOBPCG-style iteration is guaranteed to produce, in finite steps, a subspace containing a vector x such that x^*(αA+βB)x>0 for the unknown (α,β) that renders the pair definite. This is load-bearing for the central claim of correctly detecting definiteness, because a certifying direction orthogonal to the growing subspace for many iterations would cause the algorithm to report indefiniteness on all tested projections while the pair is actually definite.
  2. [Numerical experiments] Numerical experiments section: the claim that the algorithm 'very quickly detects (in)definiteness; much faster than some other algorithms' rests on unspecified test matrices, unspecified ground-truth verification of definiteness, and unnamed comparison methods. Without these details it is impossible to evaluate whether the reported speed advantage is robust or whether false-negative cases were examined.
minor comments (2)
  1. The abstract refers to 'some other algorithms' without naming them; the introduction should explicitly list the competing methods used in the timing comparisons.
  2. Notation for the linear combination coefficients (α,β) is introduced only informally; a short paragraph defining the definiteness test on the projected pair would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and indicate the revisions planned.

read point-by-point responses

  1. Referee: [§3] §3 (specialized algorithm with m=3): no argument is supplied that the LOBPCG-style iteration is guaranteed to produce, in finite steps, a subspace containing a vector x such that x^*(αA+βB)x>0 for the unknown (α,β) that renders the pair definite. This is load-bearing for the central claim of correctly detecting definiteness, because a certifying direction orthogonal to the growing subspace for many iterations would cause the algorithm to report indefiniteness on all tested projections while the pair is actually definite.

    Authors: We acknowledge that no convergence or correctness argument is supplied for the specialized m=3 variant. The algorithm is presented as a practical heuristic extending the LOBPCG framework, with effectiveness demonstrated through numerical experiments rather than theoretical guarantees. We agree that the possibility of a certifying direction remaining orthogonal to the subspace constitutes a genuine limitation. In the revised manuscript we will add an explicit discussion in Section 3 stating the heuristic character of the method, describing this potential failure mode, and recommending multiple random initializations as a practical safeguard. revision: yes

  2. Referee: [Numerical experiments] Numerical experiments section: the claim that the algorithm 'very quickly detects (in)definiteness; much faster than some other algorithms' rests on unspecified test matrices, unspecified ground-truth verification of definiteness, and unnamed comparison methods. Without these details it is impossible to evaluate whether the reported speed advantage is robust or whether false-negative cases were examined.

    Authors: We accept that the experimental section requires greater explicitness for reproducibility and evaluation. The revised version will expand the numerical experiments section to detail the precise matrix constructions (dimensions, generation procedures, and banded structures), the methods used to establish ground-truth definiteness (dense verification on medium cases and structural properties on constructed examples), the specific comparison algorithms with references, and additional test cases targeting potential false negatives to assess robustness. revision: yes

Circularity Check

0 steps flagged

No circularity: algorithm is a constructive procedure validated by external experiments

full rationale

The paper presents a subspace iteration algorithm (with m=3 LOBPCG-style subspaces) that repeatedly forms and tests small projected pairs to detect definiteness of (A,B). This is a direct computational procedure whose termination condition is the sign of the smallest eigenvalue of the projected pair; it does not define any quantity in terms of itself, rename a fitted parameter as a prediction, or rest its correctness on a self-citation chain. The cited LOBPCG method is an external standard reference, and the numerical results on medium, large, and banded pairs constitute independent empirical checks rather than tautological reductions. No load-bearing step equates the claimed detection outcome to an input by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 0 axioms · 0 invented entities

The central claim rests on the unstated premise that definiteness of the full pair is detectable from definiteness of its projections onto low-dimensional subspaces chosen via LOBPCG rules; no free parameters beyond the block size m are named, and no new entities are introduced.

free parameters (1)
  • m
    Block size parameter in the specialized algorithm; set to 3 in the described variant.

pith-pipeline@v0.9.1-grok · 5687 in / 1180 out tokens · 19071 ms · 2026-06-28T08:35:41.240932+00:00 · methodology

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