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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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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 $N\ge 128$.
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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

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