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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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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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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DOI: https://doi.org/10.1007/s11075-019-00668-z