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arxiv: 2407.15696 · v2 · submitted 2024-07-22 · 🧮 math.HO

History of confluent Vandermonde matrices and inverting them algorithms

Jerzy S Respondek This is my paper

Pith reviewed 2026-05-23 22:45 UTC · model grok-4.3

classification 🧮 math.HO
keywords confluent Vandermonde matricesmatrix inversionnumerical algorithmsquadratic time complexityhistory of mathematicslinear systems
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The pith

A numerical algorithm inverts confluent Vandermonde matrices in quadratic time for any allowed parameters including large root multiplicities.

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

The paper reviews the development of confluent Vandermonde matrices beginning in 1891 and surveys existing approaches to solving linear systems with these matrices and computing their inverses. It presents a specific numerical algorithm, illustrated by example, that performs the inversion in quadratic time without any symbolic computations. This algorithm is designed to handle all parameter values permitted by the matrix definition, including cases where the characteristic polynomial has roots of high multiplicity.

Core claim

The paper establishes the historical record of confluent Vandermonde matrices since 1891 and supplies a ready-to-implement numerical procedure that inverts any such matrix in quadratic time for arbitrary parameter values allowed by the definition, including high multiplicities, while avoiding symbolic computations entirely.

What carries the argument

The numerical inversion algorithm that operates directly on matrix entries without symbolic operations or special-case handling for multiplicities.

If this is right

  • The algorithm can be coded directly in any general-purpose programming language or mathematical package without needing computer algebra support.
  • Linear systems involving these matrices become solvable in quadratic time even when root multiplicities are large.
  • Historical context for the matrix class is now collected in one place for researchers needing background.

Where Pith is reading between the lines

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

  • The same numerical approach might be adapted to invert other structured matrices that arise from repeated roots in polynomials.
  • Applications in polynomial interpolation with multiple nodes could gain a direct quadratic-time inversion step.
  • The history survey may prompt similar reviews for related classes of confluent or generalized Vandermonde matrices.

Load-bearing premise

The surveyed historical facts are accurate and the presented numerical algorithm produces correct inverses for every permitted set of parameters including high multiplicities.

What would settle it

Run the algorithm on a confluent Vandermonde matrix whose characteristic polynomial has a root of multiplicity 10 or higher, then compare the computed inverse against an independently verified result while measuring that runtime grows quadratically with matrix size.

read the original abstract

The author was encouraged to write this review by numerous enquiries from researchers all over the world, who needed a ready-to-use algorithm for the inversion of confluent Vandermonde matrices which works in quadratic time for any values of the parameters allowed by the definition, including the case of large root multiplicities of the characteristic polynomial. Article gives the history of the title special matrix since 1891 and surveys algorithms for solving linear systems with the title class matrix and inverting it. In particular, it presents, also by example, a numerical algorithm which does not use symbolic computations and is ready to be implemented in a general-purpose programming language or in a specific mathematical package.

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 surveys the history of confluent Vandermonde matrices since 1891 and reviews algorithms for solving linear systems and inverting such matrices. It presents a numerical algorithm for inversion claimed to run in quadratic time for any allowed parameter values (including large root multiplicities) without symbolic computation, illustrated by example.

Significance. If the algorithm's claimed quadratic-time performance and correctness for arbitrary multiplicities hold under verification, the work would supply a practical, ready-to-implement tool addressing documented researcher needs in numerical linear algebra.

major comments (2)
  1. [Algorithm presentation] Algorithm presentation (the section introducing the new numerical method): the quadratic-time claim for arbitrary multiplicities rests only on an illustrative example; no general complexity analysis, inductive argument, or matrix-algebraic derivation establishing O(n²) runtime or correctness for high multiplicities is supplied.
  2. [Abstract and algorithm section] Abstract and algorithm section: no verification data, test cases with explicit multiplicities, or implementation pseudocode are provided, preventing assessment of functionality beyond the single example.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful comments on our historical survey and the presented algorithm. We address the major comments point by point below, agreeing where revisions are needed to strengthen the work.

read point-by-point responses

  1. Referee: [Algorithm presentation] Algorithm presentation (the section introducing the new numerical method): the quadratic-time claim for arbitrary multiplicities rests only on an illustrative example; no general complexity analysis, inductive argument, or matrix-algebraic derivation establishing O(n²) runtime or correctness for high multiplicities is supplied.

    Authors: We acknowledge that the manuscript supports the quadratic-time claim primarily through the structure and operations shown in the detailed illustrative example rather than a standalone formal analysis. The algorithm proceeds via block-wise constructions tied to the multiplicities, with the total arithmetic operations scaling quadratically with matrix dimension n as the sum of multiplicity-related terms remains O(n²). To address the concern directly, the revised manuscript will include an explicit general complexity subsection deriving the O(n²) bound from the operation counts in the recursive inverse construction, along with a brief argument for correctness across arbitrary multiplicities. revision: yes

  2. Referee: [Abstract and algorithm section] Abstract and algorithm section: no verification data, test cases with explicit multiplicities, or implementation pseudocode are provided, preventing assessment of functionality beyond the single example.

    Authors: The manuscript presents the algorithm via one concrete example with chosen multiplicities to show its numerical, non-symbolic nature and readiness for implementation. We agree this limits independent verification. In revision we will add explicit pseudocode for the full procedure, plus additional test cases covering a range of multiplicity configurations (including high multiplicities), with reported numerical results to confirm functionality. revision: yes

Circularity Check

0 steps flagged

No circularity; historical survey with example-based algorithm presentation

full rationale

The manuscript is a review of the history of confluent Vandermonde matrices since 1891 together with a survey of existing algorithms for their inversion. It presents one numerical algorithm 'also by example' without any derivation chain, first-principles result, fitted parameter, or prediction that reduces to its own inputs. No load-bearing self-citation, uniqueness theorem, or ansatz is invoked to justify a new claim. The quadratic-time statement for arbitrary multiplicities is asserted on the basis of the exhibited procedure rather than derived from prior equations within the paper, so no circular reduction exists. This is the normal non-finding for a survey paper whose central content is archival and illustrative rather than deductive.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

This is a survey paper in the math.HO category with no new mathematical derivations, fitted parameters, or postulated entities.

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discussion (0)

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Reference graph

Works this paper leans on

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