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PreprintCC BY 4.0arXiv

Performance Evaluation of an Adaptive Quadrature and a Double Exponential Formula Using Arbitrary-Precision Floating-Point Arithmetic

  • Tomonori Kouya
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Abstract

Using arbitrary-precision arithmetic provided by the GNU Multiple Precision Floating-Point Reliable Library, we implement AQE11D---that is, Ninomiya's adaptive 9-point Newton--Cotes rule extended with a sequence of higher-order rules---and Takahasi and Moris' double exponential (DE) formula. We evaluate them for Kahaner's 21 test problems. For both absolute tolerances 10−50 and 10−100, AQE11D attains target accuracy on all 21 problems; however, for strong endpoint singularity such as 1/√x, it requires about 5.4×107 function evaluations at 10−100, roughly 7×104 times as many as the DE formula. The formula converges on 18 problems at both tolerances, demonstrating its strength against endpoint singularities but also its failure, as it stands, on problems with a singularity inside the integration interval.

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Cite

BibTeXRIS
BibTeX
@misc{kouya2026performance,
  title = {{Performance Evaluation of an Adaptive Quadrature and a Double Exponential Formula Using Arbitrary-Precision Floating-Point Arithmetic}},
  author = {Tomonori Kouya},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.13187},
  eprint = {2608.13187},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.13187},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Performance Evaluation of an Adaptive Quadrature and a Double Exponential Formula Using Arbitrary-Precision Floating-Point Arithmetic
AU  - Tomonori Kouya
PY  - 2026
DA  - 2026/09/09
PB  - arXiv
DO  - 10.48550/arxiv.2608.13187
UR  - https://arxiv.org/abs/2608.13187
AB  - Using arbitrary-precision arithmetic provided by the GNU Multiple Precision Floating-Point Reliable Library, we implement AQE11D---that is, Ninomiya's adaptive 9-point Newton--Cotes rule extended with a sequence of higher-order rules---and Takahasi and Moris' double exponential (DE) formula. We evaluate them for Kahaner's 21 test problems. For both absolute tolerances 10^−50 and 10^−100, AQE11D attains target accuracy on all 21 problems; however, for strong endpoint singularity such as 1/√x, it requires about 5.4×10^7 function evaluations at 10^−100, roughly 7×10^4 times as many as the DE formula. The formula converges on 18 problems at both tolerances, demonstrating its strength against endpoint singularities but also its failure, as it stands, on problems with a singularity inside the integration interval.
ER  -