Obtaining cubatures for rectangles and other planar regions by using orthogonal polynomials

Author:
Richard Franke

Journal:
Math. Comp. **25** (1971), 803-817

MSC:
Primary 65D30

DOI:
https://doi.org/10.1090/S0025-5718-1971-0300440-5

MathSciNet review:
0300440

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Abstract | References | Similar Articles | Additional Information

Abstract: A. H. Stroud has recently shown the existence of cubature formulas for planar regions which use ${m^2}$ points and have polynomial precision $2m - 1$. In this paper, the author gives sufficient conditions for the existence of formulaa using fewer than ${m^2}$ points, and having polynomial precision $2m - 1$. An algorithm is given for computing such formulas, and is shown to be useful in a more general setting than given in the theorem. Numerical examples are given, both in terms of previously known and new cubature formulas.

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Additional Information

Keywords:
Approximate integration,
cubature formula,
orthogonal polynomials,
algebraic function,
common zeros,
positive weights,
polynomial precision,
rectangles,
planar regions

Article copyright:
© Copyright 1971
American Mathematical Society