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On the Petras algorithm for verified integration of piecewise analytic functions

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Ma{l}gorzata Moczurad, Piotr Zgliczyński

Numerical Analysis · Submitted 12 Mar 2016

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We present an estimation of the complexity (measured by a number of evaluations of an integrand) of the algorithm, both upper and lower bounds, for a wider class of functions.

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arXiv ID1603.03945 (version 1)
Pith IDFGF4VK32, Pith's permanent record of this paper and its review
CitationMa{\l}gorzata Moczurad, Piotr Zgliczy\'nski. "On the Petras algorithm for verified integration of piecewise analytic functions." arXiv:1603.03945 (2016). https://pith.science/paper/FGF4VK32
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The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,596 input tokens.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.15 · human participants 0.01 · controlled comparison 0.09 · randomized assignment 0.01 · states quantitative result 0.97 · compares with prior results 0.93 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We consider the algorithm for verified integration of piecewise analytic functions given by Petras. The analysis of the algorithm contained in Patras' paper is limited to a narrow class of functions and gives upper bounds only. We present an estimation of the complexity (measured by a number of evaluations of an integrand) of the algorithm, both upper and lower bounds, for a wider class of functions. We show examples with complexity $\Theta(|\ln\eps|/\eps^{p-1})$, for any $p >1$, where $\eps$ is the desired accuracy of the computed integral.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We consider the algorithm for verified integration of piecewise analytic functions given by Petras. background 0.99 0.09 0.03 asserted 0.83
s1 The analysis of the algorithm contained in Patras' paper is limited to a narrow class of functions and gives upper bounds only. gap 0.95 0.08 0.05 asserted 1.00
s2 We present an estimation of the complexity (measured by a number of evaluations of an integrand) of the algorithm, both upper and lower bounds, for a wider class of functions. contribution 0.99 0.95 0.59 asserted 1.00
s3 We show examples with complexity $\Theta(|\ln\eps|/\eps^{p-1})$, for any $p >1$, where $\eps$ is the desired accuracy of the computed integral. result 1.00 0.92 0.55 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

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  • references and citing sentences
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  • P abstract
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  • S sections
  • T body sentences
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  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

We consider the algorithm for verified integration of piecewise analytic functions given by Petras. The analysis of the algorithm contained in Patras' paper is limited to a narrow class of functions and gives upper bounds only. We present an estimation of the complexity (measured by a number of evaluations of an integrand) of the algorithm, both upper and lower bounds, for a wider class of functions. We show examples with complexity $\Theta(|\ln\eps|/\eps^{p-1})$, for any $p >1$, where $\eps$ is the desired accuracy of the computed integral.

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