Numerical Algorithms

, Volume 56, Issue 3, pp 319–347 | Cite as

Polynomial algebra for Birkhoff interpolants

  • John C. Butcher
  • Robert M. Corless
  • Laureano Gonzalez-Vega
  • Azar Shakoori
Original   Paper


We introduce a unifying formulation of a number of related problems which can all be solved using a contour integral formula. Each of these problems requires finding a non-trivial linear combination of possibly some of the values of a function f, and possibly some of its derivatives, at a number of data points. This linear combination is required to have zero value when f is a polynomial of up to a specific degree p. Examples of this type of problem include Lagrange, Hermite and Hermite–Birkhoff interpolation; fixed-denominator rational interpolation; and various numerical quadrature and differentiation formulae. Other applications include the estimation of missing data and root-finding.


Lagrange, Hermite, and Hermite–Birkhoff interpolation Contour integrals Barycentric form Fixed-denominator rational interpolation Root-finding 

Mathematics Subject Classifications (2010)

41A05 65D05 65D25 65D30 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Berriochoa, E., Cachafeiro, A.: Algorithms for solving Hermite interpolation problems using the Fast Fourier Transform. J. Comput. Appl. Math. (2009, in press)Google Scholar
  2. 2.
    Berrut, J.-P., Trefethen, L.N.: Barycentric Lagrange interpolation. SIAM Rev. 46(3), 501–517 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bini, D.A., Gemignani, L., Pan, V.Y.: Fast and stable QR eigenvalue algorithms for generalized companion matrices and secular equations. Numer. Math. 100(3), 373–408 (2005). doi:10.1007/s00211-005-0595-4 zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Birkhoff, G.D.: General mean value and remainder theorems with applications to mechanical differentiation and quadrature. Trans. Am. Math. Soc. 7(1), 107–136 (1906)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Brankin, R.W., Gladwell, I.: Shape-preserving local interpolation for plotting solutions of ODEs. IMA J. Numer. Anal. 9, 555–566 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bronstein, M., Salvy, B.: Full partial fraction decomposition of rational functions. In: ISSAC ’93: Proceedings of the 1993 International Symposium on Symbolic and Algebraic Computation, pp. 157–160. ACM, New York, NY, USA (1993)Google Scholar
  7. 7.
    Butcher, J.C.: A multistep generalization of Runge–Kutta methods with four or five stages. ACM J. 14(1), 84–99 (1967)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Celis, O.S.: Practical rational interpolation of exact and inexact data: theory and algorithms. Ph.D. thesis, University of Antwerp (2008)Google Scholar
  9. 9.
    Chin, F.Y.: The partial fraction expansion problem and its inverse. SIAM J. Comput. 6(3), 554–562 (1977)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Corless, R.M., Shakoori, A., Aruliah, D., Gonzalez-Vega, L.: Barycentric Hermite interpolants for event location in initial-value problems. JNAIAM 3(1–2), 1–18 (2008)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Corless, R.M., Watt, S.M.: Bernstein bases are optimal, but, sometimes, Lagrange bases are better. In: Proceedings of SYNASC, Timisoara, pp. 141–153. MIRTON Press (2004)Google Scholar
  12. 12.
    De Alba, L.M.: Handbook of linear algebra. In: Hogben, L., Brualdi, R., Greenbaum, A., Mathias, R. (eds.) Chapman & Hall/CRC, Boca Raton (2007)Google Scholar
  13. 13.
    Dyn, N., Lorentz, G.G., Riemenschneider, S.D.: Continuity of the Birkhoff interpolation. SIAM J. Numer. Anal. 19(3), 507–509 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Fiala, J.: An algorithm for Hermite–Birkhoff interpolation. Appl. Math. 18(3), 167–175 (1973)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Geddes, K.O., Czapor, S.R., Labahn, G.: Algorithms for Computer Algebra. Kluwer, Boston, MA (1992)Google Scholar
  16. 16.
    Gemignani, L.: Fast and stable computation of the barycentric representation of rational interpolants. Calcolo 33(3), 371–388 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Henrici, P.: Applied and Computational Complex Analysis. Wiley, New York (1974, 1986)Google Scholar
  18. 18.
    Higham, D.J.: Runge–Kutta defect control using Hermite–Birkhoff interpolation. SIAM J. Sci. Statist. Comput. 12(5), 991–999 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Higham, N.J.: Accuracy and Stability of Numerical Algorithms, 2nd edn. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA (2002)Google Scholar
  20. 20.
    Higham, N.J.: The numerical stability of barycentric Lagrange interpolation. IMA J. Numer. Anal. 24, 547–556 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Higham, N.J.: Functions of Matrices : Theory and Computation. Society for Industrial and Applied Mathematics, Philadelphia (2008)Google Scholar
  22. 22.
    Hou, S.-H., Pang, W.-K.: Inversion of confluent Vandermonde matrices. Comput. Math. Appl. 43, 1539–1547 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Kung, H.T., Tong, D.M.: Fast algorithms for partial fraction decomposition. SIAM J. Comput. 6(3), 582–593 (1977)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Lele, S.K.: Compact finite difference schemes with spectral-like resolution. J. Comput. Phys. 103(1), 16–42 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Levinson, N., Redheffer, R.M.: Complex variables. Holden-Day, San Francisco (1970)Google Scholar
  26. 26.
    Lorentz, G.G., Jetter, K., Riemenschneider, S.D.: Birkhoff Interpolation, vol. 19. Addison-Wesley, Reading, MA; Don Mills, ON (1983)Google Scholar
  27. 27.
    Luther, U., Rost, K.: Matrix exponentials and inversion of confluent Vandermonde matrices. Electron. Trans. Numer. Anal. 18, 91–100 (2004)zbMATHMathSciNetGoogle Scholar
  28. 28.
    Mahoney, J.F., Sivazlian, B.D.: Partial fractions expansion: a review of computational methodology and efficiency. J. Comput. Appl. Math. 9(3), 247–269 (1983)zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Micchelli, C.A., Rivlin, T.J.: Quadrature formulae and Hermite–Birkhoff interpolation. Adv. Math. 11, 93–112 (1973)zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Milne-Thomson, L.M.: The Calculus of Finite Differences. Macmillan, London (1933)Google Scholar
  31. 31.
    Mühlbach, G.: An algorithmic approach to Hermite–Birkhoff interpolation. Numer. Math. 37, 339–347 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Schneider, C., Werner, W.: Hermite interpolation: the barycentric approach. Computing 46, 35–51 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Stoutemyer, D.R.: Multivariate partial fraction expansion. ACM Commun. Comput. Algebra 42(4), 206–210 (2008)Google Scholar
  34. 34.
    Trefethen, L.N.: Spectral Methods in Matlab. SIAM, Philadelphia, PA (2000)Google Scholar
  35. 35.
    Tsitourras, C.: Runge–Kutta interpolants for high-order precision computations. Numer. Algorithms 44(3), 291–307 (2007)CrossRefMathSciNetGoogle Scholar
  36. 36.
    Turnbull, H.W.: A note on partial fractions and determinants. Proc. Edinb. Math. Soc. 1, 49–54 (1927)CrossRefGoogle Scholar
  37. 37.
    van Deun, J., Deckers, K., Bultheel, A., Weideman, J.: Algorithm 882: near-best fixed pole rational interpolation with applications in spectral methods. ACM Trans. Math. Softw. 35(2), 1–21 (2008)CrossRefGoogle Scholar
  38. 38.
    von zur Gathen, J., Gerhard, J.: Modern Computer Algebra. Cambridge University Press, Cambridge, New York (1999)Google Scholar
  39. 39.
    Zhao, J., Corless, R.M.: Compact finite difference method for integro-differential equations. Appl. Math. Comput. 177(1), 271–288 (2006)CrossRefMathSciNetGoogle Scholar
  40. 40.
    Zhao, J., Davison, M., Corless, R.M.: Compact finite difference method for american option pricing. J. Comput. Appl. Math. 206(1), 306–321 (2007)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2010

Authors and Affiliations

  • John C. Butcher
    • 1
  • Robert M. Corless
    • 2
  • Laureano Gonzalez-Vega
    • 3
  • Azar Shakoori
    • 3
  1. 1.Department of MathematicsUniversity of AucklandAucklandNew Zealand
  2. 2.Department of Applied MathematicsUniversity of Western OntarioLondonCanada
  3. 3.Departamento de Matematicas, Estadistica y ComputacionUniversidad de CantabriaSantanderSpain

Personalised recommendations