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Polynomial algebra for Birkhoff interpolants

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Abstract

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.

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References

  1. Berriochoa, E., Cachafeiro, A.: Algorithms for solving Hermite interpolation problems using the Fast Fourier Transform. J. Comput. Appl. Math. (2009, in press)

  2. Berrut, J.-P., Trefethen, L.N.: Barycentric Lagrange interpolation. SIAM Rev. 46(3), 501–517 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  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

    Article  MATH  MathSciNet  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  5. Brankin, R.W., Gladwell, I.: Shape-preserving local interpolation for plotting solutions of ODEs. IMA J. Numer. Anal. 9, 555–566 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  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)

  7. Butcher, J.C.: A multistep generalization of Runge–Kutta methods with four or five stages. ACM J. 14(1), 84–99 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  8. Celis, O.S.: Practical rational interpolation of exact and inexact data: theory and algorithms. Ph.D. thesis, University of Antwerp (2008)

  9. Chin, F.Y.: The partial fraction expansion problem and its inverse. SIAM J. Comput. 6(3), 554–562 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  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)

  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)

  13. Dyn, N., Lorentz, G.G., Riemenschneider, S.D.: Continuity of the Birkhoff interpolation. SIAM J. Numer. Anal. 19(3), 507–509 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  14. Fiala, J.: An algorithm for Hermite–Birkhoff interpolation. Appl. Math. 18(3), 167–175 (1973)

    MATH  MathSciNet  Google Scholar 

  15. Geddes, K.O., Czapor, S.R., Labahn, G.: Algorithms for Computer Algebra. Kluwer, Boston, MA (1992)

  16. Gemignani, L.: Fast and stable computation of the barycentric representation of rational interpolants. Calcolo 33(3), 371–388 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  17. Henrici, P.: Applied and Computational Complex Analysis. Wiley, New York (1974, 1986)

  18. Higham, D.J.: Runge–Kutta defect control using Hermite–Birkhoff interpolation. SIAM J. Sci. Statist. Comput. 12(5), 991–999 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  19. Higham, N.J.: Accuracy and Stability of Numerical Algorithms, 2nd edn. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA (2002)

  20. Higham, N.J.: The numerical stability of barycentric Lagrange interpolation. IMA J. Numer. Anal. 24, 547–556 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  21. Higham, N.J.: Functions of Matrices : Theory and Computation. Society for Industrial and Applied Mathematics, Philadelphia (2008)

  22. Hou, S.-H., Pang, W.-K.: Inversion of confluent Vandermonde matrices. Comput. Math. Appl. 43, 1539–1547 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  23. Kung, H.T., Tong, D.M.: Fast algorithms for partial fraction decomposition. SIAM J. Comput. 6(3), 582–593 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  24. Lele, S.K.: Compact finite difference schemes with spectral-like resolution. J. Comput. Phys. 103(1), 16–42 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  25. Levinson, N., Redheffer, R.M.: Complex variables. Holden-Day, San Francisco (1970)

  26. Lorentz, G.G., Jetter, K., Riemenschneider, S.D.: Birkhoff Interpolation, vol. 19. Addison-Wesley, Reading, MA; Don Mills, ON (1983)

  27. Luther, U., Rost, K.: Matrix exponentials and inversion of confluent Vandermonde matrices. Electron. Trans. Numer. Anal. 18, 91–100 (2004)

    MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  29. Micchelli, C.A., Rivlin, T.J.: Quadrature formulae and Hermite–Birkhoff interpolation. Adv. Math. 11, 93–112 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  30. Milne-Thomson, L.M.: The Calculus of Finite Differences. Macmillan, London (1933)

    Google Scholar 

  31. Mühlbach, G.: An algorithmic approach to Hermite–Birkhoff interpolation. Numer. Math. 37, 339–347 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  32. Schneider, C., Werner, W.: Hermite interpolation: the barycentric approach. Computing 46, 35–51 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  33. Stoutemyer, D.R.: Multivariate partial fraction expansion. ACM Commun. Comput. Algebra 42(4), 206–210 (2008)

    Google Scholar 

  34. Trefethen, L.N.: Spectral Methods in Matlab. SIAM, Philadelphia, PA (2000)

  35. Tsitourras, C.: Runge–Kutta interpolants for high-order precision computations. Numer. Algorithms 44(3), 291–307 (2007)

    Article  MathSciNet  Google Scholar 

  36. Turnbull, H.W.: A note on partial fractions and determinants. Proc. Edinb. Math. Soc. 1, 49–54 (1927)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  38. von zur Gathen, J., Gerhard, J.: Modern Computer Algebra. Cambridge University Press, Cambridge, New York (1999)

  39. Zhao, J., Corless, R.M.: Compact finite difference method for integro-differential equations. Appl. Math. Comput. 177(1), 271–288 (2006)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Auckland, Auckland, New Zealand

    John C. Butcher

  2. Department of Applied Mathematics, University of Western Ontario, London, ON, N6A 5B7, Canada

    Robert M. Corless

  3. Departamento de Matematicas, Estadistica y Computacion, Universidad de Cantabria, Santander, 39071, Spain

    Laureano Gonzalez-Vega & Azar Shakoori

Authors
  1. John C. Butcher
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  2. Robert M. Corless
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  3. Laureano Gonzalez-Vega
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  4. Azar Shakoori
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Corresponding author

Correspondence to Robert M. Corless.

Additional information

This work was partially supported by the Natural Sciences & Engineering Research Council of Canada.

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Butcher, J.C., Corless, R.M., Gonzalez-Vega, L. et al. Polynomial algebra for Birkhoff interpolants. Numer Algor 56, 319–347 (2011). https://doi.org/10.1007/s11075-010-9385-x

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  • DOI: https://doi.org/10.1007/s11075-010-9385-x

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