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Fast algorithms for discrete polynomial transforms


Authors: Daniel Potts, Gabriele Steidl and Manfred Tasche
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
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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 $O(N \log^2\!N)$ 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 $N\ge 128$.


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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

DOI: https://doi.org/10.1090/S0025-5718-98-00975-2
Keywords: Discrete polynomial transform, Vandermonde-like matrix, fast cosine transform, fast polynomial transform, Chebyshev knots, cascade summation
Received by editor(s): March 15, 1996
Received by editor(s) in revised form: March 13, 1997
Dedicated: Dedicated to Professor G. Maess on the occasion of his 60th birthday
Article copyright: © Copyright 1998 American Mathematical Society

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