Product Gauss cubature over polygons based on Green’s integration formula

BIT Numerical Mathematics volume 47, pages441–453(2007)Cite this article

Abstract

We have implemented in Matlab a Gauss-like cubature formula over convex, nonconvex or even multiply connected polygons. The formula is exact for polynomials of degree at most 2n-1 using N∼mn 2 nodes, m being the number of sides that are not orthogonal to a given line, and not lying on it. It does not need any preprocessing like triangulation of the domain, but relies directly on univariate Gauss–Legendre quadrature via Green’s integral formula. Several numerical tests are presented.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    T. M. Apostol, Calculus, vol. II, 2nd edn., Blaisdell, Waltham, MA, 1969.

    Google Scholar 

  2. 2.

    J. Berntsen and T. O. Espelid, Algorithm 706: DCUTRI: An algorithm for adaptive cubature over a collection of triangles, ACM Trans. Math. Softw., 18(3) (1992), pp. 329–342.

    MATH  Article  Google Scholar 

  3. 3.

    R. Cools, An encyclopaedia of cubature formulas, J. Complexity, 19(3) (2003), pp. 445–453.

    MATH  Article  MathSciNet  Google Scholar 

  4. 4.

    A. R. DiDonato and R. K. Hageman, A method for computing the integral of the bivariate normal distribution over an arbitrary polygon, SIAM J. Sci. Stat. Comput., 3(4) (1982), pp. 434–446.

    MATH  Article  MathSciNet  Google Scholar 

  5. 5.

    C. F. Dunkl, Orthogonal polynomials on the hexagon, SIAM J. Appl. Math., 47(2) (1987), pp. 343–351.

    MATH  Article  MathSciNet  Google Scholar 

  6. 6.

    C. F. Dunkl and Y. Xu, Orthogonal Polynomials of Several Variables, Encycl. Math. Appl., vol. 81, Cambridge University Press, Cambridge, 2001.

    Google Scholar 

  7. 7.

    W. Gautschi, Orthogonal polynomials: computation and approximation, in Numerical Mathematics and Scientific Computation, Oxford Science Publications, Oxford University Press, New York, 2004 (software available at http://www.cs.purdue.edu/archives/2002/wxg/codes).

    Google Scholar 

  8. 8.

    W. Gautschi, Orthogonal polynomials (in Matlab), J. Comput. Appl. Math., 178 (2005), pp. 215–234.

    MATH  Article  MathSciNet  Google Scholar 

  9. 9.

    G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd edn., Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, MD, 1996.

    Google Scholar 

  10. 10.

    G. Green, An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism, Nottingham, 1828.

  11. 11.

    A. Guessab, On the approximate calculation of integrals on a polygon in R 2, in Numerical Methods and Approximation Theory, III (Niš, 1987), pp. 225–239 Univ. Niš, Niš, 1988.

    Google Scholar 

  12. 12.

    A. I. Ivanova, Certain cases of L. A. Lyusternik’s cubature formula for regular polygons (Russian), Vyčisl. Mat. Vyčisl. Tehn. 1, (1953), pp. 27–36.

  13. 13.

    M. Held, FIST: Fast industrial-strength triangulation of polygons, Algorithmica, 30(4) (2001), pp. 563–596.

    MATH  Article  MathSciNet  Google Scholar 

  14. 14.

    V. I. Krylov, Approximate Calculation of Integrals, The Macmillan Co., New York London, 1962.

    Google Scholar 

  15. 15.

    J. A. Liggett, Exact formulae for areas, volumes and moments of polygons and polyhedra, Comm. Appl. Numer. Methods, 4(6) (1988), pp. 815–820.

    MATH  Article  MathSciNet  Google Scholar 

  16. 16.

    J. N. Lyness and R. Cools, A survey of numerical cubature over triangles, in Mathematics of Computation 1943–1993: a half-century of computational mathematics, Vancouver, BC, 1993, Proc. Sympos. Appl. Math., vol. 48, pp. 127–150, Am. Math. Soc., Providence, RI, 1994.

  17. 17.

    The MathWorks, MATLAB documentation set, 2006 version (available online at http://www.mathworks.com).

  18. 18.

    A. Narkhede and D. Manocha, Graphics Gems 5, A. Paeth, ed., Academic Press, Boston, MA, 1995.

  19. 19.

    M. Nooijen, G. te Velde, and E. J. Baerends, Symmetric numerical integration formulas for regular polygons, SIAM J. Numer. Anal., 27(1) (1990), pp. 98–218.

    Article  MathSciNet  Google Scholar 

  20. 20.

    W. Pleśniak, Remarks on Jackson’s theorem in ℝN, East J. Approx., 2(3) (1996), pp. 301–308.

    MathSciNet  MATH  Google Scholar 

  21. 21.

    H. T. Rathod and H. S. Govinda Rao, Integration of trivariate polynomials over linear polyhedra in Euclidean three-dimensional space, J. Aust. Math. Soc., Ser. B, 39(3) (1998), pp. 355–385.

    MATH  MathSciNet  Article  Google Scholar 

  22. 22.

    A. Sommariva and M. Vianello, Numerical cubature on scattered data by radial basis functions, Computing, 76 (2006), pp. 295–310.

    MATH  Article  MathSciNet  Google Scholar 

  23. 23.

    A. Sommariva and M. Vianello, Meshless cubature by Green’s formula, Appl. Math. Comput., 183 (2006), pp. 1098–1107.

    MATH  Article  MathSciNet  Google Scholar 

  24. 24.

    A. Sommariva and M. Vianello, Polygauss: Matlab code for Gauss-like cubature over polygons, software downloadable from: www.math.unipd.it/∼marcov/software.html.

  25. 25.

    A. H. Stroud, Approximate calculation of multiple integrals, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971.

Download references

Author information

Affiliations

  1. Department of Pure and Applied Mathematics, University of Padova, via Belzoni 7, Padova, 35131, Italy

    A. Sommariva & M. Vianello

Authors
  1. A. Sommariva
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Vianello
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Vianello.

Additional information

AMS subject classification (2000)

65F20

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Sommariva, A., Vianello, M. Product Gauss cubature over polygons based on Green’s integration formula . Bit Numer Math 47, 441–453 (2007). https://doi.org/10.1007/s10543-007-0131-2

Download citation

Key words

  • Gauss-like cubature
  • polygons
  • Green’s formula