## Fast algorithms for discrete polynomial transforms

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- by Daniel Potts, Gabriele Steidl and Manfred Tasche PDF
- Math. Comp.
**67**(1998), 1577-1590 Request permission

## Abstract:

Consider the Vandermonde-like matrix ${\mathbf {P}}:=(P_k(\cos \frac {j\pi }{N}))_{j,k=0}^N$, where the polynomials $P_k$ satisfy a three-term recurrence relation. If $P_k$ are the Chebyshev polynomials $T_k$, then ${\mathbf {P}}$ coincides with ${\mathbf {C}}_{N+1}:= (\cos \frac {jk\pi }{N})_{j,k=0}^N$. This paper presents a new fast algorithm for the computation of the matrix-vector product ${\mathbf {Pa}}$ in $O(N \log ^2N)$ arithmetical operations. The algorithm divides into a fast transform which replaces ${\mathbf {Pa}}$ with ${\mathbf {C}}_{N+1} {\mathbf {\tilde a}}$ and a subsequent fast cosine transform. The first and central part of the algorithm is realized by a straightforward cascade summation based on properties of associated polynomials and by fast polynomial multiplications. Numerical tests demonstrate that our fast polynomial transform realizes ${\mathbf {Pa}}$ with almost the same precision as the Clenshaw algorithm, but is much faster for $N\ge 128$.## References

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## Additional Information

**Daniel Potts**- Affiliation: Fachbereich Mathematik, Universität Rostock, D–18051 Rostock
- Email: daniel.potts@stud.uni-rostock.de
**Gabriele Steidl**- Affiliation: Fakultät für Mathematik und Informatik, Universität Mannheim, D–68131 Mannheim
- Email: steidl@kiwi.math.uni-mannheim.de
**Manfred Tasche**- Affiliation: Fachbereich Mathematik, Universität Rostock, D–18051 Rostock
- Email: manfred.tasche@mathematik.uni-rostock.de
- Received by editor(s): March 15, 1996
- Received by editor(s) in revised form: March 13, 1997
- © Copyright 1998 American Mathematical Society
- Journal: Math. Comp.
**67**(1998), 1577-1590 - MSC (1991): Primary 65T99, 42C10, 33C25
- DOI: https://doi.org/10.1090/S0025-5718-98-00975-2
- MathSciNet review: 1474655

Dedicated: Dedicated to Professor G. Maess on the occasion of his 60th birthday