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CHEBINT: A MATLAB/Octave toolbox for fast multivariate integration and interpolation based on chebyshev approximations over hypercubes

Published: 03 October 2013 Publication History

Abstract

We present the fast approximation of multivariate functions based on Chebyshev series for two types of Chebyshev lattices and show how a fast Fourier transform (FFT) based discrete cosine transform (DCT) can be used to reduce the complexity of this operation. Approximating multivariate functions using rank-1 Chebyshev lattices can be seen as a one-dimensional DCT while a full-rank Chebyshev lattice leads to a multivariate DCT.
We also present a MATLAB/Octave toolbox which uses this fast algorithms to approximate functions on a axis aligned hyper-rectangle. Given a certain accuracy of this approximation, interpolation of the original function can be achieved by evaluating the approximation while the definite integral over the domain can be estimated based on this Chebyshev approximation. We conclude with an example for both operations and actual timings of the two methods presented.

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  1. CHEBINT: A MATLAB/Octave toolbox for fast multivariate integration and interpolation based on chebyshev approximations over hypercubes

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          cover image ACM Transactions on Mathematical Software
          ACM Transactions on Mathematical Software  Volume 40, Issue 1
          September 2013
          165 pages
          ISSN:0098-3500
          EISSN:1557-7295
          DOI:10.1145/2513109
          Issue’s Table of Contents
          Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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          Publication History

          Published: 03 October 2013
          Accepted: 01 February 2013
          Revised: 01 July 2012
          Received: 01 October 2011
          Published in TOMS Volume 40, Issue 1

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          Author Tags

          1. Chebyshev lattices
          2. Godzina blending formulae
          3. MATLAB/Octave toolbox
          4. fast Fourier transform
          5. multivariate integration
          6. multivariate interpolation

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          • Research-article
          • Research
          • Refereed

          Funding Sources

          • Interuniversity Attraction Poles Programme, initiated by the Belgian State, Science Policy Office
          • KU Leuven

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          • (2018)Algorithm 988ACM Transactions on Mathematical Software10.1145/315773544:3(1-19)Online publication date: 11-Apr-2018
          • (2018)Hyperinterpolation for Spectral Wave Propagation Models in Three DimensionsContemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan10.1007/978-3-319-72456-0_17(351-372)Online publication date: 23-May-2018
          • (2017)Chopping a Chebyshev SeriesACM Transactions on Mathematical Software10.1145/299844243:4(1-21)Online publication date: 9-Jan-2017
          • (2017)Multivariate polynomial interpolation on LissajousChebyshev nodesJournal of Approximation Theory10.1016/j.jat.2017.03.003219:C(15-45)Online publication date: 1-Jul-2017
          • (2016)Bivariate Lagrange interpolation at the node points of Lissajous curves - the degenerate caseApplied Mathematics and Computation10.1016/j.amc.2016.05.019289:C(409-425)Online publication date: 20-Oct-2016
          • (2015)Fast and exact reconstruction of arbitrary multivariate algebraic polynomials in Chebyshev form2015 International Conference on Sampling Theory and Applications (SampTA)10.1109/SAMPTA.2015.7148919(392-396)Online publication date: May-2015

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