Numerische Mathematik

, Volume 141, Issue 1, pp 1–19 | Cite as

An improved Gregory-like method for 1-D quadrature

  • Bengt FornbergEmail author
  • Jonah A. Reeger


The quadrature formulas described by James Gregory (1638–1675) improve the accuracy of the trapezoidal rule by adjusting the weights near the ends of the integration interval. In contrast to the Newton–Cotes formulas, their weights are constant across the main part of the interval. However, for both of these approaches, the polynomial Runge phenomenon limits the orders of accuracy that are practical. For the algorithm presented here, this limitation is greatly reduced. In particular, quadrature formulas on equispaced 1-D node sets can be of high order (tested here up through order 20) without featuring any negative weights.

Mathematics Subject Classification

Primary: 65D30 65D32 Secondary 65B15 


  1. 1.
    Advanpix: Multiprecision computing toolbox for MATLAB. Advanpix LLC, Yokohama, Japan. Accessed 8 Aug 2018
  2. 2.
    Bailey, D.H., Borwein, J.M.: High-precision numerical integration: progress and challenges. J. Symb. Comput. 46, 741–754 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bayona, V., Flyer, N., Fornberg, B., Barnett, G.A.: On the role of polynomials in RBF-FD approximations: II. Numerical solution of elliptic PDEs. J. Comput. Phys. 332, 257–273 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bocher, P., De Meyer, H., Berghe, G.: On Gregory- and modified Gregory-type corrections to Newton–Cotes quadrature. J. Comput. Appl. Math. 50, 145–158 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Brunner, H., van der Houwen, P.J.: The Numerical Solution of Volterra equations. CWI Monographs. Elsevier, Amsterdam (1986)Google Scholar
  6. 6.
    De Swardt, S.A., De Villiers, J.M.: Gregory type quadrature based on quadratic nodal spline interpolation. Numer. Math. 85, 129–153 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    De Villiers, J.: Mathematics of Approximation. Atlantis Press, Amsterdam (2012)CrossRefzbMATHGoogle Scholar
  8. 8.
    De Villiers, J.M.: A nodal spline interpolant for the Gregory rule of even order. Numer. Math. 66, 123–137 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Fornberg, B.: A Practical Guide to Pseudospectral Methods. Cambridge University Press, Cambridge (1996)CrossRefzbMATHGoogle Scholar
  10. 10.
    Fornberg, B., Flyer, N.: A Primer on Radial Basis Functions with Applications to the Geosciences. SIAM, Philadelphia (2015)CrossRefzbMATHGoogle Scholar
  11. 11.
    Fornberg, B., Flyer, N.: Solving PDEs with radial basis functions. Acta Numer. 24, 215–258 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Fraser, D.C.: Newton’s interpolation formulas. Further notes. J. Inst. Actuar. 52(274), 117–135 (1920)CrossRefGoogle Scholar
  13. 13.
    Gregory, J.: Letter to J. Collins, 23 November 1670, pp. 203–212. In: Rigaud: Correspondence of Scientific Men. Oxford University Press (1841)Google Scholar
  14. 14.
    Javed, M., Trefethen, L.N.: Euler–Maclaurin and Gregory interpolants. Numer. Math. 132, 201–216 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Jordan, C.: Calculus of Finite Differences, 2nd edn. Chelsea, New York (1950)zbMATHGoogle Scholar
  16. 16.
    Phillips, G.M.: Gregory’s method for numerical integration. Am. Math. Mon. 79(3), 270–274 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Pólya, G.: Über die Konvergenz von Quadraturverfahren. Math. Zeitschrift. 37, 264–286 (1933)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Reeger, J.A., Fornberg, B.: Numerical quadrature over smooth surfaces with boundaries. J. Comput. Phys. 355, 176–190 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Romberg, W.: Vereinfachte numerische Integration. Det Kongelige Norske Videnskabers Selskab Forhandlinger 28(7), 30–36 (1955)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Takahasi, H., Mori, M.: Double exponential formulas for numerical integration. Res. Inst. Math. Sci. 9, 721–741 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Trefethen, L.N.: Approximation Theory and Approximation Practice. SIAM, Philadelphia (2013)zbMATHGoogle Scholar
  22. 22.
    Trefethen, L.N., Weideman, J.A.C.: The exponentially convergent trapezoidal rule. SIAM Rev. 56, 384–458 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Applied MathematicsUniversity of ColoradoBoulderUSA
  2. 2.Department of MathematicsUnited States Naval AcademyAnnapolisUSA

Personalised recommendations