Differentiation matrices for univariate polynomials
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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.
KeywordsDifferentiation matrices Polynomial bases Lagrange interpolational bases Hermite interpolational bases Bernstein bases Orthogonal polynomial bases Newton bases
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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.
This work was supported by a Summer Undergraduate NSERC Scholarship for the third author. The second author was supported by an NSERC Discovery Grant.
- 3.Boyd, J.P: Chebyshev and Fourier Spectral Methods. Courier Corporation (2001)Google Scholar
- 4.Bronstein, M.: Symbolic Integration I: Transcendental Functions, vol. 1. Springer Science & Business Media (2006)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)Google Scholar
- 8.Corless, R.M., Fillion, N.: A Graduate Introduction to Numerical Methods. Springer (2013)Google Scholar
- 9.Corless, R.M., Trivedi, J.A.: Levin integration using differentiation matrices. In preparation (2018)Google Scholar
- 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)Google Scholar
- 11.Davis, P.J.: Interpolation and Approximation. Blaisdell (1963)Google Scholar
- 14.Embree, M.: Pseudospectra. In: Hogben, L (ed.) Handbook of Linear Algebra, chapter 23. Chapman and Hall/CRC (2013)Google Scholar
- 15.Farin, G.: Curves and Surfaces for Computer-Aided Geometric Design: A Practical Guide. Elsevier (2014)Google Scholar
- 19.Henrici, P.: Elements of Numerical Analysis. Wiley (1964)Google Scholar
- 20.Iserles, A., Nørsett, S., Olver, S.: Highly oscillatory quadrature: The story so far. Numer. Math. Adv. Appl., 97–118 (2006)Google Scholar
- 23.Lorentz, G., Jetter, K., Riemenschneider, S.D.: Birkhoff Interpolation. Addison Wesley Publishing Company (1983)Google Scholar
- 29.Trefethen, L.N.: Spectral Methods in MATLAB. SIAM (2000)Google Scholar
- 30.Trefethen, L.N.: Approximation Theory and Approximation Practice. SIAM (2013)Google Scholar