Comment re: Schemes for cubature over the unit disk found via numerical optimization

A recent paper in this journal contained speculation about the potential for improving some of its novel high-order cubature schemes. Those schemes are for approximating uniformly weighted integrals over the unit disk. The two highest order rules were (i) a degree-65 solution with 733 cubature point...

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Bibliographic Details
Published in:Journal of computational and applied mathematics 2024-05, Vol.441, p.115643, Article 115643
Main Author: Forbes, G.W.
Format: Article
Language:English
Subjects:
Cubature formulas
Numerical integration
Optimization
Unit disk
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Summary:A recent paper in this journal contained speculation about the potential for improving some of its novel high-order cubature schemes. Those schemes are for approximating uniformly weighted integrals over the unit disk. The two highest order rules were (i) a degree-65 solution with 733 cubature points and (ii) a degree-77 solution with 1021 cubature points. Both schemes possessed six-fold rotational symmetry and it is reported here that, as speculated, one point can indeed be dropped from each segment so that the end results have just 727 and 1015 cubature points, respectively. The details for these new schemes are now presented along with further discussion of this type of solution and additional characterization of related options of even higher orders.
ISSN:0377-0427
1879-1778
DOI:10.1016/j.cam.2023.115643