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arxiv: 2606.02744 · v1 · pith:UVJ5MZK3new · submitted 2026-06-01 · ✦ hep-ph · hep-th

IterInt: Evaluating iterated integrals via differential equations

Gideon Baur , Claude Duhr This is my paper

Pith reviewed 2026-06-28 13:17 UTC · model grok-4.3

classification ✦ hep-ph hep-th
keywords iterated integralsdifferential equationsnumerical evaluationmultiple polylogarithmsshuffle regularisationbanana integrals
0
0 comments X

The pith

IterInt converts iterated integrals into systems of first-order linear differential equations that standard solvers can evaluate numerically with high precision.

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

IterInt is a package in Mathematica and C++ that evaluates iterated integrals with arbitrary kernels. It transforms these integrals into a system of first-order linear differential equations. These equations are then solved using established numerical libraries for high precision. The package also automatically handles shuffle-regularisation to manage cases with poles at the integration start.

Core claim

The central claim is that by transforming iterated integrals into a system of first-order linear differential equations, IterInt enables efficient and high-precision numerical evaluation of integrals involving arbitrary integration kernels, including those requiring shuffle-regularisation.

What carries the argument

The transformation of the iterated integrals, defined by user-specified kernels, into a closed system of first-order linear differential equations.

If this is right

  • Comparison with GiNaC shows agreement for ordinary and elliptic multiple polylogarithms.
  • Validation against known results for banana integrals up to four loops.
  • Supports evaluation where the integrand has a pole at the starting point via automatic regularisation.

Where Pith is reading between the lines

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

  • This method could allow users to define custom kernels for integrals not covered by existing special functions.
  • Integration with other computational physics tools might enable broader applications in multi-loop calculations.

Load-bearing premise

Any user-defined integration kernels can be transformed into a finite closed system of first-order linear differential equations that remains numerically stable.

What would settle it

Finding an iterated integral with arbitrary kernels for which no finite system of first-order linear differential equations can be constructed or where the numerical solution becomes unstable.

Figures

Figures reproduced from arXiv: 2606.02744 by Claude Duhr, Gideon Baur.

Figure 1
Figure 1. Figure 1: The resulting tree graph representing the relations between the iterated in￾tegrals I(ω1, ω3, ω2;t), I(ω1, ω4, ω4;t), I(ω2, ω1;t) and I(ω2, ω3, ω4;t). Each arrow represents taking a derivative. The edges passing through the dashed line are removed for the computation (see the discussion in the main text). by combining the two systems into a single larger system, d dt   I(ω1, ω2;t) I(ω1, ω3;t) I(ω1;t) I… view at source ↗
Figure 1
Figure 1. Figure 1: figure 1. Afterwards the systems of differential equations are as small as possible without [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Numerical evaluation of the unequal-mass two-loop sunrise integral I111 (blue line). The red dots correspond to an independent evaluation of the integral by numerically integrating the Feynman parameter representation. Note that the range on the x-axis corresponds to the region below threshold, where the integral is real. The equal-mass sunrise integral. If all three masses are equal, there are only three … view at source ↗
Figure 3
Figure 3. Figure 3: Numerical evaluation using IterInt of the master integral I111 of the equal￾mass sunrise family (blue dots). The red line correspond to an independent evaluation of the integral using a series representation of the result. Note that the range on the x-axis corresponds to the region below threshold, where the integral is real. proceed in a different fashion. Indeed, we may exploit the fact that Mathematica … view at source ↗
Figure 4
Figure 4. Figure 4: Numerical evaluation using IterInt of the master integral I1111 of the equal￾mass banana family (blue dots and triangles). The red lines correspond to an independent evaluation of the integral using a truncated series expansion. Note that the range on the x-axis corresponds to the region above threshold, and both the real (solid lines) and imaginary (dashed line) parts are shown. Unlike for the three-loop … view at source ↗
Figure 5
Figure 5. Figure 5: Numerical evaluation using IterInt of the master integral I11111 of the equal￾mass 4-loop equal-mass banana family (blue markers). The red lines correspond to an independent evaluation of the integral. Note that the range on the x-axis corresponds to the region above threshold. Both the real (solid lines/circles) and imaginary (dashed line/triangles) parts are shown. ematica and in C++, and we have describ… view at source ↗
read the original abstract

We introduce IterInt, a novel package implemented in both Mathematica and C++ for the numerical evaluation of iterated integrals involving arbitrary integration kernels. After the user has defined the integration kernels, IterInt transforms the iterated integrals into a system of first-order linear differential equations which can be solved efficiently and with high precision using well established libraries. IterInt is also able to automatically perform shuffle-regularisation. This makes it possible to evaluate also integrals where the integrand has a pole at the starting point of the integration path. As an illustration of our code, and also to validate it and gauge its performance, we compare the output of IterInt to the results obtained by GiNaC for ordinary and elliptic multiple polylogarithms, and also to existing results for the first few orders for banana integrals with up to four loops.

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

1 major / 2 minor

Summary. The paper introduces IterInt, a package implemented in both Mathematica and C++ for the numerical evaluation of iterated integrals with arbitrary user-defined kernels. The method transforms the integrals into a system of first-order linear differential equations solved using established libraries, with automatic shuffle regularisation to handle cases with poles at the integration starting point. Validation consists of direct numerical comparisons to GiNaC results for ordinary and elliptic multiple polylogarithms, as well as to published results for banana integrals up to four loops.

Significance. If the implementation is correct, IterInt supplies a general-purpose numerical tool for iterated integrals appearing in multi-loop calculations, extending beyond libraries limited to specific classes such as MPLs. The dual-language implementation and the reported agreement with independent codes constitute concrete strengths that support potential utility in the field.

major comments (1)
  1. [method description (paragraph following abstract)] The manuscript states that the transformation to a closed first-order linear DE system is always possible for arbitrary kernels, but provides no explicit algorithm, basis construction, or closure proof in the method description; this is load-bearing for the generality claim even though the finite-dimensionality argument is standard.
minor comments (2)
  1. [validation section] The validation paragraphs report agreement with GiNaC and banana results but do not tabulate achieved relative precision, CPU times, or the specific integration paths and weight values used; adding such a table would improve reproducibility.
  2. [implementation and solver paragraph] No discussion of numerical stability, step-size control, or error estimation in the DE solver is included, which would clarify the 'high precision' claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment of IterInt and the recommendation for minor revision. We address the single major comment below.

read point-by-point responses

  1. Referee: [method description (paragraph following abstract)] The manuscript states that the transformation to a closed first-order linear DE system is always possible for arbitrary kernels, but provides no explicit algorithm, basis construction, or closure proof in the method description; this is load-bearing for the generality claim even though the finite-dimensionality argument is standard.

    Authors: We agree that an expanded description of the basis construction and closure procedure would strengthen the presentation of the generality claim. In the revised manuscript we will add a concise subsection (or short appendix) outlining the algorithmic steps: (i) the finite-dimensional vector space spanned by the iterated integrals with the user-supplied kernels, (ii) the explicit construction of a basis via repeated differentiation and reduction using the shuffle product, and (iii) the resulting closed linear system of first-order DEs. This will make the standard finite-dimensionality argument fully explicit without altering the existing code or validation results. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper describes a software package that converts user-specified iterated integrals into a first-order linear DE system via the finite-dimensional vector space of weight-n integrals under differentiation. This is a direct algebraic construction, not a fitted or self-referential prediction. Validation is performed against independent external codes (GiNaC) and published banana-integral results, providing external benchmarks. No self-citations, ansatze, or renamings of known results serve as load-bearing steps in the central claim. The method is self-contained against external checks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The approach rests on standard existence and uniqueness theorems for linear ODEs and the algebraic properties of shuffle regularization; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • standard math Linear first-order ODE systems derived from iterated integrals admit unique solutions given appropriate boundary conditions.
    Invoked implicitly when stating that the integrals are transformed into solvable DE systems.
  • domain assumption Shuffle regularization preserves the value of the iterated integral when a pole is present at the lower limit.
    Stated as an automatic feature without further justification in the abstract.

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Forward citations

Cited by 1 Pith paper

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