Advertisement

BIT Numerical Mathematics

, Volume 47, Issue 2, pp 441–453 | Cite as

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

  • A. Sommariva
  • M. VianelloEmail author
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 Nmn 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.

Key words

Gauss-like cubature polygons Green’s formula 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    T. M. Apostol, Calculus, vol. II, 2nd edn., Blaisdell, Waltham, MA, 1969.zbMATHGoogle 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.zbMATHCrossRefGoogle Scholar
  3. 3.
    R. Cools, An encyclopaedia of cubature formulas, J. Complexity, 19(3) (2003), pp. 445–453.zbMATHCrossRefMathSciNetGoogle 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.zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    C. F. Dunkl, Orthogonal polynomials on the hexagon, SIAM J. Appl. Math., 47(2) (1987), pp. 343–351.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    C. F. Dunkl and Y. Xu, Orthogonal Polynomials of Several Variables, Encycl. Math. Appl., vol. 81, Cambridge University Press, Cambridge, 2001.zbMATHGoogle 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).zbMATHGoogle Scholar
  8. 8.
    W. Gautschi, Orthogonal polynomials (in Matlab), J. Comput. Appl. Math., 178 (2005), pp. 215–234.zbMATHCrossRefMathSciNetGoogle 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.zbMATHGoogle Scholar
  10. 10.
    G. Green, An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism, Nottingham, 1828.Google Scholar
  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.Google Scholar
  13. 13.
    M. Held, FIST: Fast industrial-strength triangulation of polygons, Algorithmica, 30(4) (2001), pp. 563–596.zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    V. I. Krylov, Approximate Calculation of Integrals, The Macmillan Co., New York London, 1962.zbMATHGoogle 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.zbMATHCrossRefMathSciNetGoogle 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.Google Scholar
  17. 17.
    The MathWorks, MATLAB documentation set, 2006 version (available online at http://www.mathworks.com).Google Scholar
  18. 18.
    A. Narkhede and D. Manocha, Graphics Gems 5, A. Paeth, ed., Academic Press, Boston, MA, 1995.Google Scholar
  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.CrossRefMathSciNetGoogle Scholar
  20. 20.
    W. Pleśniak, Remarks on Jackson’s theorem in ℝN, East J. Approx., 2(3) (1996), pp. 301–308.MathSciNetzbMATHGoogle 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.zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    A. Sommariva and M. Vianello, Numerical cubature on scattered data by radial basis functions, Computing, 76 (2006), pp. 295–310.zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    A. Sommariva and M. Vianello, Meshless cubature by Green’s formula, Appl. Math. Comput., 183 (2006), pp. 1098–1107.zbMATHCrossRefMathSciNetGoogle 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.Google Scholar
  25. 25.
    A. H. Stroud, Approximate calculation of multiple integrals, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971.Google Scholar

Copyright information

© Springer Science + Business Media B.V. 2007

Authors and Affiliations

  1. 1.Department of Pure and Applied MathematicsUniversity of PadovaPadovaItaly

Personalised recommendations