Cubature formulas of degree nine for symmetric planar regions

Authors:
Robert Piessens and Ann Haegemans

Journal:
Math. Comp. **29** (1975), 810-815

MSC:
Primary 65D30

DOI:
https://doi.org/10.1090/S0025-5718-1975-0368393-5

MathSciNet review:
0368393

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Abstract | References | Similar Articles | Additional Information

Abstract: A method of constructing 19-point cubature formulas with degree of exactness 9 is given for two-dimensional regions and weight functions which are symmetric in each variable. For some regions, e.g., the square and the circle, these formulas can be reduced to 18-point formulas.

**[1]**Philip Rabinowitz and Nira Richter,*Perfectly symmetric two-dimensional integration formulas with minimal numbers of points*, Math. Comp.**23**(1969), 765–779. MR**0258281**, https://doi.org/10.1090/S0025-5718-1969-0258281-4**[2]**I. P. Mysovskih,*On the construction of cubature formulas with the smallest number of nodes*, Dokl. Akad. Nauk SSSR**178**(1968), 1252–1254 (Russian). MR**0224284****[3]**Richard Franke,*Minimal point cubatures of precision seven for symmetric planar regions*, SIAM J. Numer. Anal.**10**(1973), 849–862. MR**0343544**, https://doi.org/10.1137/0710070**[4]**Seymour Haber,*Numerical evaluation of multiple integrals*, SIAM Rev.**12**(1970), 481–526. MR**0285119**, https://doi.org/10.1137/1012102**[5]**A. H. Stroud,*Approximate calculation of multiple integrals*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971. Prentice-Hall Series in Automatic Computation. MR**0327006****[6]**A. H. Stroud,*Integration formulas and orthogonal polynomials for two variables*, SIAM J. Numer. Anal.**6**(1969), 222–229. MR**0261788**, https://doi.org/10.1137/0706020**[7]**J. Albrecht,*Formeln zur numerischen Integration über Kreisbereiche*, Z. Angew. Math. Mech.**40**(1960), 514–517 (German). MR**0120765**, https://doi.org/10.1002/zamm.19600401014**[8]**A. HAEGEMANS & R. PIESSENS,*Tables of Cubature Formulas of Degree Nine for Symmetric Planar Regions*. (Report to be published.)

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1975-0368393-5

Keywords:
Approximate integration,
cubature formula,
degree of exactness,
planar region,
orthogonal polynomials

Article copyright:
© Copyright 1975
American Mathematical Society