Fast polynomial transforms based on Toeplitz and Hankel matrices
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- by Alex Townsend, Marcus Webb and Sheehan Olver PDF
- Math. Comp. 87 (2018), 1913-1934 Request permission
Abstract:
Many standard conversion matrices between coefficients in classical orthogonal polynomial expansions can be decomposed using diagonally-scaled Hadamard products involving Toeplitz and Hankel matrices. This allows us to derive algorithms with an observed complexity of $\mathcal {O}(N\log ^2 \! N)$, based on the fast Fourier transform, for converting coefficients of a degree $N$ polynomial in one polynomial basis to coefficients in another. Numerical results show that this approach is competitive with state-of-the-art techniques, requires no precomputational cost, can be implemented in a handful of lines of code, and is easily adapted to extended precision arithmetic.References
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Additional Information
- Alex Townsend
- Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853
- Email: townsend@cornell.edu
- Marcus Webb
- Affiliation: Department of Computer Science, KU Leuven, 3001 Leuven, Belgium
- MR Author ID: 1004196
- Email: marcus.webb@cs.kuleuven.be
- Sheehan Olver
- Affiliation: Department of Mathematics, Imperial College, London SW7 2AZ, United Kingdom
- MR Author ID: 783322
- ORCID: 0000-0001-6920-0826
- Email: s.olver@imperial.ac.uk
- Received by editor(s): May 10, 2016
- Received by editor(s) in revised form: November 20, 2016, and March 9, 2017
- Published electronically: November 6, 2017
- Additional Notes: The work of the first author was supported by the National Science Foundation grant No. 1522577
The work of the second author was supported by the UK Engineering and Physical Sciences Research Council (EPSRC) grant EP/H023348/1 for the University of Cambridge Centre for Doctoral Training, the Cambridge Centre for Analysis, and the London Mathematical Society Cecil King Travel Scholarship 2015 - © Copyright 2017 American Mathematical Society
- Journal: Math. Comp. 87 (2018), 1913-1934
- MSC (2010): Primary 65T50, 65D05, 15B05
- DOI: https://doi.org/10.1090/mcom/3277
- MathSciNet review: 3787396