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

G. W. Forbes

doi:10.1016/j.cam.2023.115643

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

G. W. Forbes

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Abstract

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.

Original language	English
Article number	115643
Pages (from-to)	1-8
Number of pages	8
Journal	Journal of Computational and Applied Mathematics
Volume	441
DOIs	https://doi.org/10.1016/j.cam.2023.115643
Publication status	Published - 15 May 2024

Bibliographical note

© 2023 The Author(s). Published by Elsevier B.V. Version archived for private and non-commercial use with the permission of the author/s and according to publisher conditions. For further rights please contact the publisher.

Keywords

Cubature formulas
Numerical integration
Optimization
Unit disk

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10.1016/j.cam.2023.115643Licence: CC BY

Publisher version (open access)Final published version, 1.28 MBLicence: CC BY

Cite this

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title = "Comment re: Schemes for cubature over the unit disk found via numerical optimization",

abstract = "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.",

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AB - 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.

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

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