Differentiation matrices for univariate polynomials

  • Amirhossein Amiraslani
  • Robert M. CorlessEmail author
  • Madhusoodan Gunasingam
Original Paper


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.


Differentiation 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.

Funding information

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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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.STEM DepartmentThe University of Hawaii-Maui CollegeKahuluiUSA
  2. 2.Faculty of MathematicsK. N. Toosi University of TechnologyTehranIran
  3. 3.The Ontario Research Center for Computer AlgebraThe University of Western OntarioLondonCanada
  4. 4.The School of Mathematical and Statistical SciencesThe University of Western OntarioLondonCanada

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