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arxiv: 2601.01007 · v2 · submitted 2026-01-03 · 🧮 math.NA · cs.NA

On solving nonlinear simultaneous equations arising from the double-exponential Sinc-collocation method for initial value problems

Yusaku Yamamoto , Ken'ichiro Tanaka This is my paper

Pith reviewed 2026-05-16 18:37 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords double-exponential Sinc-collocationGauss-Seidel iterationnonlinear simultaneous equationsglobal convergenceinitial value problemsordinary differential equationsconvergence analysisfixed-point iteration
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The pith

Gauss-Seidel fixed-point iteration converges globally for the nonlinear system from double-exponential Sinc-collocation on initial-value ODEs, with an explicit bound on its rate.

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

The double-exponential Sinc-collocation method produces nearly exponential error decay for initial-value ODE problems as the number of time points N grows, but it yields a coupled nonlinear system in nN unknowns when the ODE dimension is n. A simple Gauss-Seidel iteration has been observed to cut the residual by one or two orders of magnitude per step, yet no rigorous explanation existed. The paper supplies a sufficient condition on the nonlinear mapping that guarantees global convergence of the iteration from any starting vector and derives an upper bound on the contraction factor that directly accounts for the observed speed. Numerical examples confirm that the predicted rate matches the practical behavior on standard test problems.

Core claim

The Gauss-Seidel type fixed-point iteration applied to the nonlinear simultaneous equations that arise from the double-exponential Sinc-collocation discretization of initial-value ODE problems admits global convergence whenever the underlying nonlinear mapping satisfies a contraction or monotonicity condition, and the iteration possesses an explicit upper bound on its convergence factor that explains the rapid error reduction seen in practice.

What carries the argument

Gauss-Seidel fixed-point iteration applied to the nonlinear map obtained by substituting the double-exponential Sinc-collocation basis into the ODE initial-value problem.

If this is right

  • The iteration can be used with guaranteed global convergence for every ODE whose discretized map meets the stated contraction or monotonicity condition.
  • The number of iterations required stays small even as N increases, preserving the overall exponential accuracy of the Sinc-collocation method.
  • An a-priori bound on the convergence factor can be computed directly from the problem data to predict the work needed before the residual falls below a tolerance.
  • The same iteration and its convergence theory apply uniformly across the entire family of double-exponential Sinc-collocation schemes for first-order systems.

Where Pith is reading between the lines

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

  • The same contraction-mapping argument may extend to other collocation or spectral discretizations that produce similar nonlinear algebraic systems.
  • When the contraction factor bound is close to one, increasing N still improves accuracy faster than the added cost of extra iterations, suggesting the method remains competitive for very high precision.
  • The analysis supplies a concrete test that a practitioner can perform on a given ODE to decide in advance whether the iteration will converge reliably.

Load-bearing premise

The nonlinear mapping that defines the discretized equations must obey a contraction or monotonicity property that is not automatically satisfied by every ODE.

What would settle it

An ODE initial-value problem for which the Gauss-Seidel iteration either diverges from a standard initial guess or converges more slowly than the derived upper bound on the contraction factor.

read the original abstract

The double-exponential Sinc-collocation method is known as a super-accurate method for solving initial value problems of ordinary differential equations, for which the error decreases almost exponentially as a function of the number of sample points in the temporal direction, $N$. However, this method requires solving nonlinear simultaneous equations in $nN$ variables when the problem dimension is $n$. Recently, Ogata pointed out that Gauss-Seidel type fixed-point iteration works surprisingly well for solving these equations, typically reducing the error by one or two orders of magnitude at each iteration. In this paper, we analyze the convergence of this iteration and give a sufficient condition for its global convergence. We also provide an upper bound on its convergence factor, which explains the efficiency of this iteration. Some numerical examples that illustrate the validity of our analysis are also provided.

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 paper analyzes the convergence of a Gauss-Seidel-type fixed-point iteration for solving the nN-dimensional nonlinear system arising from the double-exponential Sinc-collocation discretization of initial-value ODE problems. It supplies a sufficient condition for global convergence of the iteration together with an upper bound on the convergence factor, and presents numerical examples to illustrate the analysis.

Significance. If the sufficient condition can be shown to hold for the intended class of problems, the result would provide a theoretical explanation for the observed rapid convergence of the iteration and strengthen the practical utility of the double-exponential Sinc-collocation method. The work is a targeted contribution to the numerical analysis of high-order collocation schemes.

major comments (2)
  1. [§3] The sufficient condition for global convergence is stated in terms of the nonlinear mapping satisfying a contraction (or monotonicity) property with Lipschitz constant strictly less than one in a suitable norm. No general analytic verification of this hypothesis is supplied for arbitrary right-hand sides; the numerical examples only demonstrate rapid convergence on specific test problems and therefore do not secure the hypothesis for the method's claimed scope.
  2. [§3] The upper bound on the convergence factor is derived under the same contraction assumption. Because the assumption is not established independently of the data, the bound remains conditional and does not yet constitute an unconditional explanation of the iteration's efficiency across the target class of ODEs.
minor comments (2)
  1. The notation distinguishing the continuous ODE solution, the Sinc-collocation approximant, and the fixed-point iterates should be introduced with a single consolidated table or list of symbols.
  2. Figure captions for the convergence plots should explicitly state the norm in which the error is measured and the stopping tolerance used.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The points raised help clarify the scope of our convergence analysis. We respond to each major comment below and indicate planned revisions.

read point-by-point responses

  1. Referee: [§3] The sufficient condition for global convergence is stated in terms of the nonlinear mapping satisfying a contraction (or monotonicity) property with Lipschitz constant strictly less than one in a suitable norm. No general analytic verification of this hypothesis is supplied for arbitrary right-hand sides; the numerical examples only demonstrate rapid convergence on specific test problems and therefore do not secure the hypothesis for the method's claimed scope.

    Authors: We agree that the paper presents a sufficient condition for global convergence based on the nonlinear mapping being contractive with Lipschitz constant strictly less than one. No general analytic verification is provided that this condition holds for arbitrary right-hand sides, as such a verification would require additional assumptions on the nonlinearity and lies outside the paper's scope. The analysis instead derives the condition as a general criterion guaranteeing convergence when satisfied, and the numerical examples illustrate that rapid convergence occurs for the tested problems, consistent with the condition holding in those cases. We will revise §3 to explicitly state that the condition is sufficient but its verification for a given ODE may need to be checked separately. revision: partial

  2. Referee: [§3] The upper bound on the convergence factor is derived under the same contraction assumption. Because the assumption is not established independently of the data, the bound remains conditional and does not yet constitute an unconditional explanation of the iteration's efficiency across the target class of ODEs.

    Authors: The upper bound is derived under the contraction assumption and is therefore conditional. The manuscript's aim is to supply a theoretical explanation for the observed rapid convergence of the Gauss-Seidel iteration when the sufficient condition holds, which matches the numerical behavior. An unconditional bound valid for all ODEs would require proving the contraction property in general, which we do not claim. We will add a clarifying remark in the revised manuscript to emphasize the conditional nature of the bound. revision: partial

standing simulated objections not resolved
  • Providing a general analytic verification that the contraction property holds for arbitrary right-hand sides of the ODE IVPs without further assumptions on the nonlinearity.

Circularity Check

0 steps flagged

Standard fixed-point analysis of Gauss-Seidel iteration; no reduction to fitted inputs or self-definition

full rationale

The paper derives a sufficient condition for global convergence and an upper bound on the convergence factor by applying standard contraction-mapping or monotonicity arguments from fixed-point theory to the nonlinear algebraic system produced by double-exponential Sinc-collocation. These quantities are expressed directly in terms of the Lipschitz constant or monotonicity modulus of the iteration map; they are not obtained by fitting parameters to the same data the bound is intended to explain, nor are they defined in terms of the target result itself. Numerical examples illustrate rapid convergence but are not used to construct the bound. No load-bearing self-citation chain or ansatz smuggling is present; the central claim therefore retains independent mathematical content.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard contraction-mapping or monotonicity assumptions from fixed-point theory together with the known discretization properties of the double-exponential Sinc-collocation method; no new free parameters or invented entities are introduced.

axioms (1)
  • domain assumption The nonlinear mapping induced by the Sinc-collocation discretization satisfies a contraction or monotonicity condition sufficient for global convergence of Gauss-Seidel iteration.
    Invoked to guarantee the iteration converges from any initial guess; appears in the statement of the sufficient condition.

pith-pipeline@v0.9.0 · 5444 in / 1258 out tokens · 84307 ms · 2026-05-16T18:37:29.059285+00:00 · methodology

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Reference graph

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