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Differentiation matrices for univariate polynomials

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Abstract

Differentiation matrices are in wide use in numerical algorithms, although usually studied in an ad hoc manner. We collect here in this review paper elementary properties of differentiation matrices for univariate polynomials expressed in various bases, including orthogonal polynomial bases and non-degree-graded bases such as Bernstein bases and Lagrange and Hermite interpolational bases. We give new explicit formulations, and new explicit formulations for the pseudo-inverses which help to understand antidifferentiation, of many of these matrices. We also give the unique Jordan form for these (nilpotent) matrices and a new unified formula for the transformation matrix.

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References

  1. Amiraslani, A., Fassbender, H., Shayanfar, N.: Birkhoff polynomial basis. Springer Proc. Math. Stat. 192, 1–25 (2017)

    Article  MathSciNet  Google Scholar 

  2. Berrut, J.-P., Trefethen, L.N: Barycentric Lagrange interpolation. SIAM Rev. 46(3), 509–514 (2004)

    Article  MathSciNet  Google Scholar 

  3. Boyd, J.P: Chebyshev and Fourier Spectral Methods. Courier Corporation (2001)

  4. Bronstein, M.: Symbolic Integration I: Transcendental Functions, vol. 1. Springer Science & Business Media (2006)

  5. Butcher, J.C., Corless, R.M., Gonzalez-Vega, L., Shakoori, A.: Polynomial algebra for Birkhoff interpolants. Numer. Algor. 56(3), 319–347 (2011)

    Article  MathSciNet  Google Scholar 

  6. Carnicer, J.M., Khiar, Y., Peña, J.M.: Optimal stability of the Lagrange formula and conditioning of the Newton formula. Journal of Approximation Theory (2017)

  7. Chelyshkov, V.S.: Alternative orthogonal polynomials and quadratures. Electron. Trans. Numer. Anal. 25, 17–26 (2006)

    MathSciNet  MATH  Google Scholar 

  8. Corless, R.M., Fillion, N.: A Graduate Introduction to Numerical Methods. Springer (2013)

  9. Corless, R.M., Trivedi, J.A.: Levin integration using differentiation matrices. In preparation (2018)

  10. Corless, R.M., Watt, S.M.: Bernstein bases are optimal, but, sometimes, Lagrange bases are better. In: Proceedings of SYNASC, Timisoara, pp 141–153. MITRON Press (2004)

  11. Davis, P.J.: Interpolation and Approximation. Blaisdell (1963)

  12. Diekema, E., Koornwinder, T.H.: Differentiation by integration using orthogonal polynomials, a survey. J. Approx. Theory 164(5), 637–667 (2012)

    Article  MathSciNet  Google Scholar 

  13. Don, W.S., Solomonoff, A.: Accuracy enhancement for higher derivatives using Chebyshev collocation and a mapping technique. SIAM J. Sci. Comput. 18(4), 1040–1055 (1997)

    Article  MathSciNet  Google Scholar 

  14. Embree, M.: Pseudospectra. In: Hogben, L (ed.) Handbook of Linear Algebra, chapter 23. Chapman and Hall/CRC (2013)

  15. Farin, G.: Curves and Surfaces for Computer-Aided Geometric Design: A Practical Guide. Elsevier (2014)

  16. Farouki, R.T., Goodman, T.N.T.: On the optimal stability of the Bernstein basis. Math. Comput. 65(216), 1553–1566 (1996)

    Article  MathSciNet  Google Scholar 

  17. Fitt, A.D., Hoare, G.T.Q.: The closed-form integration of arbitrary functions. Math. Gaz. 77(479), 227–236 (1993)

    Article  Google Scholar 

  18. Gottlieb, D., Lustman, L.: The spectrum of the Chebyshev collocation operator for the heat equation. SIAM J. Numer. Anal. 20(5), 909–921 (1983)

    Article  MathSciNet  Google Scholar 

  19. Henrici, P.: Elements of Numerical Analysis. Wiley (1964)

  20. Iserles, A., Nørsett, S., Olver, S.: Highly oscillatory quadrature: The story so far. Numer. Math. Adv. Appl., 97–118 (2006)

  21. Knuth, D.E.: Two notes on notation. Am. Math. Mon. 99(5), 403–422 (1992)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  23. Lorentz, G., Jetter, K., Riemenschneider, S.D.: Birkhoff Interpolation. Addison Wesley Publishing Company (1983)

  24. Olver, S., Townsend, A.: A fast and well-conditioned spectral method. SIAM Rev. 55(3), 462–489 (2013)

    Article  MathSciNet  Google Scholar 

  25. Rebillard, L.: Rational approximation in the complex plane using a τ-method and computer algebra. Numer. Algor. 16(2), 187–208 (1997)

    Article  MathSciNet  Google Scholar 

  26. Reddy, S.C., Weideman, J.A.C.: The accuracy of the Chebyshev differencing method for analytic functions. SIAM J. Numer. Anal. 42(5), 2176–2187 (2005)

    Article  MathSciNet  Google Scholar 

  27. Sadiq, B., Viswanath, D.: Barycentric Hermite interpolation. SIAM J. Sci. Comput. 35(3), A1254–A1270 (2013)

    Article  MathSciNet  Google Scholar 

  28. Schneider, C., Werner, W.: Hermite interpolation: the barycentric approach. Computing 46(1), 35–51 (1991)

    Article  MathSciNet  Google Scholar 

  29. Trefethen, L.N.: Spectral Methods in MATLAB. SIAM (2000)

  30. Trefethen, L.N.: Approximation Theory and Approximation Practice. SIAM (2013)

  31. Weideman, J.A.C., Reddy, S.C.: A matlab differentiation matrix suite. ACM Trans. Math. Softw. (TOMS) 26(4), 466–470 (2000)

    Article  MathSciNet  Google Scholar 

  32. Weideman, J.A.C., Trefethen, Lloyd N: The eigenvalues of second-order spectral differentiation matrices. SIAM J. Numer. Anal. 25(6), 1279–1298 (1988)

    Article  MathSciNet  Google Scholar 

  33. Yang, C.: Modified Chebyshev collocation method for pantograph-type differential equations. Appl. Numer. Math. 134, 132–144 (2018)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

We thank André Weideman and a referee for very helpful comments on an earlier draft. We also thank ORCCA and the Rotman Institute of Philosophy.

Funding

This work was supported by a Summer Undergraduate NSERC Scholarship for the third author. The second author was supported by an NSERC Discovery Grant.

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Authors and Affiliations

  1. STEM Department, The University of Hawaii-Maui College, Kahului, HI, USA

    Amirhossein Amiraslani

  2. Faculty of Mathematics, K. N. Toosi University of Technology, Tehran, Iran

    Amirhossein Amiraslani

  3. The Ontario Research Center for Computer Algebra, The University of Western Ontario, London, Ontario, Canada

    Robert M. Corless & Madhusoodan Gunasingam

  4. The School of Mathematical and Statistical Sciences, The University of Western Ontario, London, Ontario, Canada

    Robert M. Corless & Madhusoodan Gunasingam

Authors
  1. Amirhossein Amiraslani
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  2. Robert M. Corless
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  3. Madhusoodan Gunasingam
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Correspondence to Robert M. Corless.

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Amiraslani, A., Corless, R.M. & Gunasingam, M. Differentiation matrices for univariate polynomials. Numer Algor 83, 1–31 (2020). https://doi.org/10.1007/s11075-019-00668-z

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