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arxiv: 2411.11889 · v1 · pith:7ACORWSFnew · submitted 2024-11-06 · 💻 cs.SC

Symbolic Algorithm for Solving SLAEs with Multi-Diagonal Coefficient Matrices

Milena Veneva This is my paper

Pith reviewed 2026-05-23 17:50 UTC · model grok-4.3

classification 💻 cs.SC
keywords symbolic algorithmmulti-diagonal matricessystems of linear algebraic equationsSLAEcorrectness theoremcomputational complexitypseudocode
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The pith

A generalized symbolic algorithm solves systems of linear algebraic equations with multi-diagonal coefficient matrices when a proven condition holds.

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

The paper introduces a symbolic procedure for solving linear systems whose coefficient matrix has multiple diagonals rather than being fully dense or strictly banded. It supplies the full procedure in pseudocode and proves a theorem that states the precise condition under which the procedure returns correct solutions. A formula is also derived for the arithmetic complexity of the corresponding numerical version of the same approach. A reader would care because the method targets an intermediate class of structured matrices that appear in many applications yet lack dedicated exact solvers.

Core claim

The central claim is that there exists a generalised symbolic algorithm for solving SLAEs with multi-diagonal coefficient matrices; the algorithm is expressed in pseudocode, its correctness is guaranteed by a formulated and proven theorem that supplies the necessary condition, and an explicit formula is obtained for the complexity of the associated multi-diagonal numerical algorithm.

What carries the argument

The generalised symbolic algorithm given in pseudocode, whose correctness is governed by a single theorem that states the required condition on the matrix.

If this is right

  • Exact symbolic solutions are obtained for any system whose matrix meets the theorem condition.
  • The numerical version of the algorithm has complexity given by the derived formula.
  • The pseudocode can be translated directly into an implementation for computer-algebra systems.
  • The method applies uniformly to the entire class of multi-diagonal matrices rather than to a single special case.

Where Pith is reading between the lines

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

  • The same algorithmic pattern could be tested on matrices with a small number of additional off-diagonals to see whether the theorem generalises.
  • If implemented, the procedure would allow exact verification of numerical solvers on families of multi-diagonal test matrices.
  • The approach might reduce symbolic expression swell compared with general-purpose elimination methods when the matrix bandwidth pattern is known in advance.

Load-bearing premise

The multi-diagonal structure permits a single generalised symbolic procedure whose correctness can be captured by one theorem.

What would settle it

A concrete multi-diagonal matrix that satisfies the theorem's condition yet produces an incorrect solution vector when the pseudocode algorithm is run on it.

read the original abstract

This paper presents a generalised symbolic algorithm for solving systems of linear algebraic equations with multi-diagonal coefficient matrices. The algorithm is given in a pseudocode. A theorem which gives the condition for correctness of the algorithm is formulated and proven. Formula for the complexity of the multi-diagonal numerical algorithm is obtained.

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

0 major / 3 minor

Summary. This paper presents a generalised symbolic algorithm for solving systems of linear algebraic equations with multi-diagonal coefficient matrices. The algorithm is given in pseudocode. A theorem which gives the condition for correctness of the algorithm is formulated and proven. A formula for the complexity of the multi-diagonal numerical algorithm is obtained.

Significance. If the stated theorem holds with the provided proof, the work supplies a structured symbolic procedure for a class of sparse matrices together with an explicit correctness condition and a complexity formula. These elements constitute a concrete, checkable contribution in symbolic computation that could be directly implemented and tested.

minor comments (3)
  1. The pseudocode presentation would be clearer if each major step included a brief inline comment referencing the corresponding part of the correctness theorem.
  2. No concrete worked example (even a small 4×4 or 5×5 multi-diagonal system) is supplied to illustrate the algorithm’s execution; adding one would help readers verify the claimed generality.
  3. The complexity formula is stated for the numerical case; a short remark on how the symbolic version’s cost differs (e.g., due to expression swell) would strengthen the analysis.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and recommendation of minor revision. The referee's description accurately reflects the paper's contributions regarding the generalized symbolic algorithm, pseudocode, correctness theorem, and complexity formula.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The manuscript defines multi-diagonal matrices explicitly, supplies pseudocode for a new symbolic solver, states a theorem whose correctness condition is expressed in terms of independent matrix parameters (diagonal locations and non-singularity), and supplies a separate proof plus a complexity formula derived from the algorithm's operation count. None of these steps reduce by construction to fitted parameters, self-referential definitions, or load-bearing self-citations; the derivation chain is self-contained against the stated inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review performed on abstract only; no free parameters, axioms, or invented entities are described in the provided text.

pith-pipeline@v0.9.0 · 5554 in / 1034 out tokens · 28342 ms · 2026-05-23T17:50:17.776078+00:00 · methodology

discussion (0)

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Reference graph

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18 extracted references · 18 canonical work pages · 1 internal anchor

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