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]**P. RABINOWITZ & N. RICHTER, "Perfectly symmetric two-dimensional integration formulas with minimal number of points,"*Math. Comp.*, v. 23, 1969, pp. 765-779. MR**41**#2928. MR**0258281 (41:2928)****[2]**I. P. MYSOVSKIH, "On the construction of cubature formulas with fewest nodes,"*Dokl. Akad. Nauk SSSR*, v. 178, 1968, pp. 1252-1254 =*Soviet Math. Dokl.*, v. 9, 1968, pp. 277-280. MR**36**#7328. MR**0224284 (36:7328)****[3]**R. FRANKE, "Minimal point cubatures of precision seven for symmetric regions,"*SIAM J. Numer. Anal.*, v. 10, 1971, pp. 849-882. MR**0343544 (49:8285)****[4]**S. HABER, "Numerical evaluation of multiple integrals,"*SIAM Rev.*, v. 12, 1970, pp. 481-526. MR**44**#2342. MR**0285119 (44:2342)****[5]**A. H. STROUD,*Approximate Calculation of Multiple Integrals*, Prentice-Hall Ser. in Automatic Computation, Prentice-Hall, Englewood Cliffs, N. J., 1971. MR**48**#5348. MR**0327006 (48:5348)****[6]**A. H. STROUD, "Integration formulas and orthogonal polynomials for two variables,"*SIAM J. Numer. Anal.*, v. 6, 1969, pp. 222-229. MR**41**#6400. MR**0261788 (41:6400)****[7]**J. ALBRECHT, "Formeln zur numerischen Integration über Kreisbereiche,"*Z. Angew. Math. Mech.*, v. 40, 1960, pp. 514-517. MR**22**#11514. MR**0120765 (22:11514)****[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