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Mathematics of Computation

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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
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 $O(N \log ^2N)$ 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 $N\ge 128$.

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Additional Information

Daniel Potts
Affiliation: Fachbereich Mathematik, Universität Rostock, D–18051 Rostock

Gabriele Steidl
Affiliation: Fakultät für Mathematik und Informatik, Universität Mannheim, D–68131 Mannheim

Manfred Tasche
Affiliation: Fachbereich Mathematik, Universität Rostock, D–18051 Rostock

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