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Fast algorithms using orthogonal polynomials

Published online by Cambridge University Press:  30 November 2020

Sheehan Olver ,
Richard Mikaël Slevinsky  and
Alex Townsend
Sheehan Olver
Affiliation:
Department of Mathematics, Imperial College, London SW7 2AZ, UK E-mail: s.olver@imperial.ac.uk
Richard Mikaël Slevinsky
Affiliation:
Department of Mathematics, University of Manitoba, Winnipeg, Manitoba, Canada R3T 2M8 E-mail: Richard.Slevinsky@umanitoba.ca
Alex Townsend
Affiliation:
Department of Mathematics, Cornell University, Ithaca, NY 14853, USA E-mail: townsend@cornell.edu
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Abstract

We review recent advances in algorithms for quadrature, transforms, differential equations and singular integral equations using orthogonal polynomials. Quadrature based on asymptotics has facilitated optimal complexity quadrature rules, allowing for efficient computation of quadrature rules with millions of nodes. Transforms based on rank structures in change-of-basis operators allow for quasi-optimal complexity, including in multivariate settings such as on triangles and for spherical harmonics. Ordinary and partial differential equations can be solved via sparse linear algebra when set up using orthogonal polynomials as a basis, provided that care is taken with the weights of orthogonality. A similar idea, together with low-rank approximation, gives an efficient method for solving singular integral equations. These techniques can be combined to produce high-performance codes for a wide range of problems that appear in applications.

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Research Article
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Acta Numerica , Volume 29 , May 2020 , pp. 573 - 699
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© The Author(s), 2020. Published by Cambridge University Press

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References

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