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@article{Olver:2020:10.1017/s0962492920000045,
author = {Olver, S and Slevinsky, RM and Townsend, A},
doi = {10.1017/s0962492920000045},
journal = {Acta Numerica},
pages = {573--699},
title = {Fast algorithms using orthogonal polynomials},
url = {http://dx.doi.org/10.1017/s0962492920000045},
volume = {29},
year = {2020}
}

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TY  - JOUR
AB  - 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.
AU  - Olver,S
AU  - Slevinsky,RM
AU  - Townsend,A
DO  - 10.1017/s0962492920000045
EP  - 699
PY  - 2020///
SN  - 0962-4929
SP  - 573
TI  - Fast algorithms using orthogonal polynomials
T2  - Acta Numerica
UR  - http://dx.doi.org/10.1017/s0962492920000045
UR  - https://www.cambridge.org/core/journals/acta-numerica/article/fast-algorithms-using-orthogonal-polynomials/4FAD8C7C28EC20EE7583465C1A89AA3D
UR  - http://hdl.handle.net/10044/1/85269
VL  - 29
ER  -

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