Hostname: page-component-6766d58669-zlvph Total loading time: 0 Render date: 2026-05-19T10:17:15.993Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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.

Information

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

References 13

Alpert, B. K. and Rokhlin, V. (1991), ‘A fast algorithm for the evaluation of Legendre expansions’, SIAM J. Sci. Statist. Comput. 12, 158–179.CrossRefGoogle Scholar
Andrews, G. E., Askey, R. and Roy, R. (1999), Special Functions, Cambridge University Press.CrossRefGoogle Scholar
Atkinson, K. and Han, W. (2012), Spherical Harmonics and Approximations on the Unit Sphere: An Introduction , Vol. 2044 of Lecture Notes in Mathematics, Springer.Google Scholar
Aurentz, J. L. and Slevinsky, R. M. (2020), ‘On symmetrizing the ultraspherical spectral method for self-adjoint problems’, J. Comput. Phys. 410, 109383.CrossRefGoogle Scholar
Aurentz, J. L. and Trefethen, L. N. (2017), ‘Chopping a Chebyshev series’, ACM Trans. Math. Software 43, 33:1–33:21.CrossRefGoogle Scholar
Bailey, D. H., Jeyabalan, K. and Li, X. S. (2004), ‘A comparison of three high-precision quadrature schemes’, Exp. Math. 14, 317–329.CrossRefGoogle Scholar
Berry, M. (2007), ‘Focused tsunami waves’, Proc. Roy. Soc. A Math. Phys. Engrg Sci. 463(2087), 3055–3071.Google Scholar
Beuchler, S. and Schöberl, J. (2006), ‘New shape functions for triangular $p$ -FEM using integrated Jacobi polynomials’, Numer. Math. 103, 339–366.CrossRefGoogle Scholar
Bochner, S. (1929), ‘Über Sturm–Liouvillesche Polynomsysteme’, Math. Z. 29, 730–736.CrossRefGoogle Scholar
Bogaert, I. (2014), ‘Iteration-free computation of Gauss–Legendre quadrature nodes and weights’, SIAM J. Sci. Comput. 36, A1008–A1026.CrossRefGoogle Scholar
Bogaert, I., Michiels, B. and Fostier, J. (2012), ‘ $\!\!\!\!\mathbf{\mathcal{O}}(1)$ computation of Legendre polynomials and Gauss–Legendre nodes and weights for parallel computing’, SIAM J. Sci. Comput. 34, C83–C101.Google Scholar
Borges, C. F. and Gragg, W. B. (1993), A parallel divide and conquer algorithm for the generalized real symmetric definite tridiagonal eigenproblem. In Numerical Linear Algebra and Scientific Computing (Reichel, L., Ruttan, A. and Varga, R. S., eds), de Gruyter, pp. 11–29.Google Scholar
Boyd, J. P. (2013), ‘Finding the zeros of a univariate equation: Proxy rootfinders, Chebyshev interpolation, and the companion matrix’, SIAM Rev. 55, 375–396.CrossRefGoogle Scholar
Bremer, J. and Yang, H. (2019), ‘Fast algorithms for Jacobi expansions via nonoscillatory phase functions’, IMA J. Numer. Anal. doi: 10.1093/imanum/drz016 Google Scholar
Bunch, J. R., Nielsen, C. P. and Sorensen, D. C. (1978), ‘Rank-one modification of the symmetric eigenproblem’, Numer. Math. 31, 31–48.CrossRefGoogle Scholar
Burns, K. J., Vasil, G. M., Oishi, J. S., Lecoanet, D. and Brown, B. P. (2020), ‘Dedalus: A flexible framework for numerical simulations with spectral methods’, Phys. Rev. Research 2, 023068.CrossRefGoogle Scholar
Chandrasekaran, S. and Gu, M. (2004), ‘A divide-and-conquer algorithm for the eigendecomposition of symmetric block-diagonal plus semiseparable matrices’, Numer. Math. 96, 723–731.CrossRefGoogle Scholar
Chen, S., Shen, J. and Wang, L.-L. (2016), ‘Generalized Jacobi functions and their applications to fractional differential equations’, Math. Comp. 85, 1603–1638.CrossRefGoogle Scholar
Clenshaw, C. W. (1955), ‘A note on the summation of Chebyshev series’, Math. Comp. 9, 118–120.CrossRefGoogle Scholar
Clenshaw, C. W. (1957), ‘The numerical solution of linear differential equations in Chebyshev series’, Math. Proc. Cambridge Philos. Soc. 53, 134–149.CrossRefGoogle Scholar
Clenshaw, C. W. and Curtis, A. R. (1960), ‘A method for numerical integration on an automatic computer’, Numer. Math. 2, 197–205.CrossRefGoogle Scholar
Cooley, J. W. and Tukey, J. W. (1965), ‘An algorithm for the machine calculation of complex Fourier series’, Math. Comp. 19, 297–301.CrossRefGoogle Scholar
Coutsias, E., Hagstrom, T. and Torres, D. (1996), ‘An efficient spectral method for ordinary differential equations with rational function coefficients’, Math. Comp. 65, 611–635.CrossRefGoogle Scholar
Crawford, C. R. (1973), ‘Reduction of a band-symmetric generalized eigenvalue problem’, Comm. Assoc. Comput. Mach. 16, 41–44.Google Scholar
Cuppen, J. J. M. (1981), ‘A divide and conquer method for the symmetric tridiagonal eigenproblem’, Numer. Math. 36, 177–195.CrossRefGoogle Scholar
Dai, F. and Xu, Y. (2013), Approximation Theory and Harmonic Analysis on Spheres and Balls, Monographs in Mathematics, Springer.CrossRefGoogle Scholar
Davis, P. and Rabinowitz, P. (1956), ‘Abscissas and weights for Gaussian quadratures of high order’, J. Res. Nat. Bur. Standards 56, 35–37.CrossRefGoogle Scholar
Davis, P. and Rabinowitz, P. (1958), ‘Additional abscissas and weights for Gaussian quadratures of high order: Values for $n=64$ , 80, and 96’, J. Res. Nat. Bur. Standards 60, 613–614.CrossRefGoogle Scholar
Bayly, B. de F. (1938), ‘(iii) Gauss’ quadratic formula with twelve ordinates’, Biometrika 30, 193–194.Google Scholar
Deift, P. (1999), Orthogonal Polynomials and Random Matrices: A Riemann–Hilbert Approach, American Mathematical Society.Google Scholar
Doha, E. H. and Abd-Elhameed, W. M. (2002), ‘Efficient spectral-Galerkin algorithms for direct solution of second-order equations using ultraspherical polynomials’, SIAM J. Sci. Comput. 24, 548–571.CrossRefGoogle Scholar
Driscoll, J. R., Healy, D. M. Jr and Rockmore, D. N. (1997), ‘Fast discrete polynomial transforms with applications to data analysis for distance transitive graphs’, SIAM J. Sci. Comput. 26, 1066–1099.CrossRefGoogle Scholar
Driscoll, T. A., Hale, N. and Trefethen, L. N., eds (2014), Chebfun Guide, Pafnuty Publications.Google Scholar
Du, Q., Gunzburger, M., Lehoucq, R. B. and Zhou, K. (2012), ‘Analysis and approximation of nonlocal diffusion problems with volume constraints’, SIAM Rev. 54, 667–696.CrossRefGoogle Scholar
Du, Q., Gunzburger, M., Lehoucq, R. B. and Zhou, K. (2013), ‘A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws’, Math. Mod. Meth. Appl. Sci. 23, 493–540.CrossRefGoogle Scholar
Dubiner, M. (1991), ‘Spectral methods on triangles and other domains’, J. Sci. Comput. 6, 345–390.CrossRefGoogle Scholar
Dunkl, C. F. and Xu, Y. (2014), Orthogonal Polynomials of Several Variables, second edition, Encyclopedia of Mathematics and its Applications, Cambridge University Press.CrossRefGoogle Scholar
Dutt, A. and Rokhlin, V. (1993), ‘Fast Fourier transforms for nonequispaced data’, SIAM J. Sci. Comput. 14, 1368–1393.CrossRefGoogle Scholar
Edelman, A. and Sutton, B. D. (2007), ‘From random matrices to stochastic operators’, J. Statist. Phys. 127, 1121–1165.CrossRefGoogle Scholar
Elliott, D. (1982), ‘The classical collocation method for singular integral equations’, SIAM J. Numer. Anal. 19, 816–832.CrossRefGoogle Scholar
Elsner, L., Fasse, A. and Langmann, E. (1997), ‘A divide-and-conquer method for the tridiagonal generalized eigenvalue problem’, J. Comput. Appl. Math. 86, 141–148.CrossRefGoogle Scholar
Erdélyi, A. et al., eds (1953), Higher Transcendental Functions, Vol. 2, McGraw-Hill.Google Scholar
Fejér, L. (1933), ‘On the infinite sequences arising in the theories of harmonic analysis, of interpolation, and of mechanical quadratures’, Bull. Amer. Math. Soc. 39, 521–534.CrossRefGoogle Scholar
Förster, K.-J. and Petras, K. (1990), ‘On estimates for the weights in Gaussian quadrature in the ultraspherical case’, Math. Comp. 55, 243–264.CrossRefGoogle Scholar
Förster, K.-J. and Petras, K. (1993), ‘Inequalities for the zeros of ultraspherical polynomials and Bessel functions’, Z. Angew. Math. Mech. 73, 232–236.CrossRefGoogle Scholar
Fortunato, D. and Townsend, A. (2019), ‘Fast Poisson solvers for spectral methods’, IMA J. Numer. Anal. doi: 10.1093/imanum/drz034 CrossRefGoogle Scholar
Gatteschi, L. (1979), On the construction of some Gaussian quadrature rules. In Numerische Integration, Springer, pp. 138–146.CrossRefGoogle Scholar
Gauss, C. F. (1815), Methodus nova integralium valores per approximationem inveniendi. Apud Henricum Dieterich.Google Scholar
Gautschi, W. (1967), ‘Computational aspects of three-term recurrence relations’, SIAM Rev. 9, 24–82.CrossRefGoogle Scholar
Gautschi, W. (1981a), A survey of Gauss–Christoffel quadrature formulae. In E. B. Christoffel (Butzer, P. L. and Fehér, F., eds), Birkhäuser.CrossRefGoogle Scholar
Gautschi, W. (1981b), ‘Minimal solutions of three-term recurrence relations and orthogonal polynomials’, Math. Comp. 36, 547–554.CrossRefGoogle Scholar
Gautschi, W. (1982), ‘On generating orthogonal polynomials’, SIAM J. Sci. Statist. Comput. 3, 289–317.CrossRefGoogle Scholar
Gautschi, W. (1992), ‘On mean convergence of extended Lagrange interpolation’, J. Comput. Appl. Math. 43, 19–35.CrossRefGoogle Scholar
Gautschi, W. (2004), Orthogonal Polynomials: Computation and Approximation, Clarendon Press.CrossRefGoogle Scholar
Gautschi, W. and Wimp, J. (1987), ‘Computing the Hilbert transform of a Jacobi weight function’, BIT Numer. Math. 27, 203–215.CrossRefGoogle Scholar
Gawlik, H. J. (1958), Zeros of Legendre Polynomials of Orders 2–64 and Weight Coefficients of Gauss Quadrature Formulae, Armament Research and Development Establishment.Google Scholar
Gentleman, W. M. (1972), ‘Implementing Clenshaw–Curtis quadrature, II: Computing the cosine transformation’, Comm. Assoc. Comput. Mach. 15, 343–346.Google Scholar
Gil, A., Segura, J. and Temme, N. M. (2018), ‘Asymptotic approximations to the nodes and weights of Gauss–Hermite and Gauss–Laguerre quadratures’, Stud. Appl. Math. 140, 298–332.CrossRefGoogle Scholar
Gil, A., Segura, J. and Temme, N. M. (2019a), ‘Fast, reliable and unrestricted iterative computation of Gauss–Hermite and Gauss–Laguerre quadratures’, Numer. Math. 143, 649–682.CrossRefGoogle Scholar
Gil, A., Segura, J. and Temme, N. M. (2019b), ‘Noniterative computation of Gauss–Jacobi quadrature’, SIAM J. Sci. Comput. 41, A668–A693.CrossRefGoogle Scholar
Glaser, A., Liu, X. and Rokhlin, V. (2007), ‘A fast algorithm for the calculation of the roots of special functions’, SIAM J. Sci. Comput. 29, 1420–1438.CrossRefGoogle Scholar
Golub, G. H. and Loan, C. F. V. (2013), Matrix Computations, fourth edition, The Johns Hopkins University Press.Google Scholar
Golub, G. H. and Welsch, J. H. (1969), ‘Calculation of Gauss quadrature rules’, Math. Comput. 23, 221–230.CrossRefGoogle Scholar
Greengard, L. (1991), ‘Spectral integration and two-point boundary value problems’, SIAM J. Numer. Anal. 28, 1071–1080.CrossRefGoogle Scholar
Greengard, L. and Lee, J.-Y. (2004), ‘Accelerating the nonuniform fast Fourier transform’, SIAM Rev. 46, 443–454.CrossRefGoogle Scholar
Greengard, L. and Rokhlin, V. (1987), ‘A fast algorithm for particle simulations’, J. Comput. Phys. 73, 325–348.CrossRefGoogle Scholar
Gu, M. and Eisenstat, S. C. (1994), ‘A stable and efficient algorithm for the rank-one modification of the symmetric eigenproblem’, SIAM J. Matrix Anal. Appl. 15, 1266–1276.CrossRefGoogle Scholar
Gu, M. and Eisenstat, S. C. (1995), ‘A divide-and-conquer algorithm for the symmetric tridiagonal eigenproblem’, SIAM J. Matrix Anal. Appl. 16, 172–191.CrossRefGoogle Scholar
Gutierrez, R. H. and Laura, P. A. A. (1995), ‘Vibrations of non-uniform rings studied by means of the differential quadrature method’, J. Sound Vibration 185, 507–513.CrossRefGoogle Scholar
Hale, N. and Olver, S. (2018), ‘A fast and spectrally convergent algorithm for rational-order fractional integral and differential equations’, SIAM J. Sci. Comput. 40, A2456–A2491.CrossRefGoogle Scholar
Hale, N. and Townsend, A. (2013), ‘Fast and accurate computation of Gauss–Legendre and Gauss–Jacobi quadrature nodes and weights’, SIAM J. Sci. Comput. 35, A652–A674.CrossRefGoogle Scholar
Hale, N. and Townsend, A. (2014), ‘A fast, simple, and stable Chebyshev–Legendre transform using an asymptotic formula’, SIAM J. Sci. Comput. 36, A148–A167.CrossRefGoogle Scholar
Healy, D. M. Jr, Rockmore, D. N., Kostelec, P. J. and Moore, S. (2003), ‘FFTs for the $2$ -sphere: Improvements and variations’, J. Fourier Anal. Appl. 9, 341–385.CrossRefGoogle Scholar
Heinrichs, W. (1989), ‘Spectral methods with sparse matrices’, Numer. Math. 56, 25–41.CrossRefGoogle Scholar
Huybrechs, D. (2010), ‘On the Fourier extension of nonperiodic functions’, SIAM J. Numer. Anal. 47, 4326–4355.CrossRefGoogle Scholar
Hylleraas, E. A. (1929), ‘Neue Berechnung der Energie des Heliums im Grundzustande, sowie des tiefsten Terms von Ortho-Helium’, Z. Physik 54, 347–366.CrossRefGoogle Scholar
Ishioka, K. (2018), ‘A new recurrence formula for efficient computation of spherical harmonic transform’, J. Met. Soc. Japan 96, 241–249.CrossRefGoogle Scholar
Jacobi, C. G. J. (1826), ‘Ueber Gauss neue Methode, die Werthe der Integrale näherungsweise zu finden’, J. Reine Angew. Math. 1, 301–308.Google Scholar
Julien, K. and Watson, M. (2009), ‘Efficient multi-dimensional solution of PDEs using Chebyshev spectral methods’, J. Comput. Phys. 228, 1480–1503.CrossRefGoogle Scholar
Karniadakis, G. and Sherwin, S. (2013), Spectral/hp Element Methods for Computational Fluid Dynamics, Oxford University Press.Google Scholar
Kaufman, L. (1993), ‘An algorithm for the banded symmetric generalized matrix eigenvalue problem’, SIAM J. Matrix Anal. Appl. 14, 372–389.CrossRefGoogle Scholar
Kaufman, L. (2000), ‘Band reduction algorithms revisited’, ACM Trans. Math. Software 26, 551–567.CrossRefGoogle Scholar
Keiner, J. (2008), ‘Gegenbauer polynomials and semiseparable matrices’, Electron. Trans. Numer. Anal. 30, 26–53.Google Scholar
Keiner, J. (2009), ‘Computing with expansions in Gegenbauer polynomials’, SIAM J. Sci. Comput. 31, 2151–2171.CrossRefGoogle Scholar
Keiner, J. (2011), Fast Polynomial Transforms, Logos.Google Scholar
Klöckner, A., Barnett, A., Greengard, L. and O’Neil, M. (2013), ‘Quadrature by expansion: A new method for the evaluation of layer potentials’, J. Comput. Phys. 252, 332–349.CrossRefGoogle Scholar
Konoplev, V. P. (1961), ‘Polynomials orthogonal with respect to weight functions which are zero or infinite at isolated points of the interval of orthogonality’, Dokl. Akad. Nauk SSSR 141, 781–784.Google Scholar
Koornwinder, T. (1975), Two-variable analogues of the classical orthogonal polynomials. In  Theory and Application of Special Functions (Askey, R., ed.), pp. 435–495.CrossRefGoogle Scholar
Krall, H. L. (1938), ‘Certain differential equations for Tchebycheff polynomials’, Duke Math. J. 4, 705–718.CrossRefGoogle Scholar
Krishnapur, M., Rider, B. and Virág, B. (2016), ‘Universality of the stochastic Airy operator’, Commun. Pure Appl. Math. 69, 145–199.CrossRefGoogle Scholar
Kunis, S. and Potts, D. (2003), ‘Fast spherical Fourier algorithms’, J. Comp. Appl. Math. 161, 75–98.CrossRefGoogle Scholar
Lanczos, C. (1938), ‘Trigonometric interpolation of empirical and analytical functions’, J. Math. Phys. 17, 123–199.CrossRefGoogle Scholar
Lee, J.-Y. and Greengard, L. (1997), ‘A fast adaptive numerical method for stiff two-point boundary value problems’, SIAM J. Sci. Comput. 18, 403–429.CrossRefGoogle Scholar
Lether, F. G. (1978), ‘On the construction of Gauss–Legendre quadrature rules’, J. Comput. Appl. Math. 4, 47–52.CrossRefGoogle Scholar
Li, H. and Shen, J. (2010), ‘Optimal error estimates in Jacobi-weighted Sobolev spaces for polynomial approximations on the triangle’, Math. Comp. 79, 1621–1646.CrossRefGoogle Scholar
Livermore, P. W. (2010), ‘Galerkin orthogonal polynomials’, J. Comput. Phys. 229, 2046–2060.CrossRefGoogle Scholar
Love, C. H. (1966), Abscissas and weights for Gaussian quadrature for $N=2$ to 100, and $N=125$ , 150, 175, 200. United States Department Of Commerce, National Bureau of Standards Monograph.CrossRefGoogle Scholar
Lowan, A. N., Davids, N. and Levenson, A. (1942), ‘Table of the zeros of the Legendre polynomials of order 1–16 and the weight coefficients for Gauss’ mechanical quadrature formula’, Bull. Amer. Math. Soc. 48, 739–742.CrossRefGoogle Scholar
Löwdin, P.-O. and Shull, H. (1956), ‘Natural orbitals in the quantum theory of two-electron systems’, Phys. Rev. 101, 1730–1739.CrossRefGoogle Scholar
Lozier, D. W. (1980), Numerical solution of linear difference equations. NBSIR Technical Report 80-1976, National Bureau of Standards.Google Scholar
Magnus, A. P. (1995), ‘Painlevé-type differential equations for the recurrence coefficients of semi-classical orthogonal polynomials’, J. Comput. Appl. Math. 57, 215–237.CrossRefGoogle Scholar
Martinsson, P. G. and Rokhlin, V. (2007), ‘An accelerated kernel-independent fast multipole method in one dimension’, SIAM J. Sci. Comput. 29, 1160–1178.CrossRefGoogle Scholar
Mason, J. (1993), ‘Chebyshev polynomials of the second, third and fourth kinds in approximation, indefinite integration, and integral transforms’, J. Comput. Appl. Math. 49, 169–178.CrossRefGoogle Scholar
Mohlenkamp, M. J. (1999), ‘A fast transform for spherical harmonics’, J. Fourier Anal. Appl. 5, 159–184.CrossRefGoogle Scholar
Moors, B. P. (1905), Valeur approximative d’une intégrale définie, Gauthier-Villars.Google Scholar
Mori, A., Suda, R. and Sugihara, M. (1999), ‘An improvement on Orszag’s fast algorithm for Legendre polynomial transform’, Trans. Info. Process. Soc. Japan 40, 3612–3615.Google Scholar
Muskhelishvili, N. I. (1953), Singular Integral Equations, second edition, Dover.Google Scholar
Narayan, A. and Hesthaven, J. S. (2012), ‘Computation of connection coefficients and measure modifications for orthogonal polynomials’, BIT Numer. Math. 52, 457–483.CrossRefGoogle Scholar
Nyström, E. J. (1930), ‘Über die praktische auflösung von Integralgleichungen mit Anwendungen auf Randwertaufgaben’, Acta Math. 54, 185–204.CrossRefGoogle Scholar
Olver, F. W. J. (1967), ‘Numerical solution of second-order linear difference equations’, J. Res. Nat. Bur. Standards 71B, 111–129.CrossRefGoogle Scholar
Olver, F. W. J., Lozier, D. W., Boisvert, R. F. and Clark, C. W., eds (2010), NIST Handbook of Mathematical Functions, Cambridge University Press.Google Scholar
Olver, S. (2011), ‘Computing the Hilbert transform and its inverse’, Math. Comp. 80, 1745–1767.CrossRefGoogle Scholar
Olver, S. (2012), ‘A general framework for solving Riemann–Hilbert problems numerically’, Numer. Math. 122, 305–340.CrossRefGoogle Scholar
Olver, S. and Townsend, A. (2013), ‘A fast and well-conditioned spectral method’, SIAM Rev. 55, 462–489.CrossRefGoogle Scholar
Olver, S. and Xu, Y. (2019a), Non-homogeneous wave equation on a cone. arXiv:1907.08286Google Scholar
Olver, S. and Xu, Y. (2019b), ‘Orthogonal structure on a wedge and on the boundary of a square’, Found. Comput. Math. 19, 561–589.CrossRefGoogle Scholar
Olver, S. and Xu, Y. (2020a), ‘Orthogonal polynomials in and on a quadratic surface of revolution’, Math. Comp., to appear.CrossRefGoogle Scholar
Olver, S. and Xu, Y. (2020b), ‘Orthogonal structure on a quadratic curve’, IMA J. Numer. Anal., to appear.CrossRefGoogle Scholar
Olver, S., Goretkin, G., Slevinsky, R. M. and Townsend, A. (2016), Julia package for function approximation. GitHub. https://github.com/ApproxFun/ApproxFun.jl Google Scholar
Olver, S., Townsend, A. and Vasil, G. (2019), ‘A sparse spectral method on triangles’, SIAM J. Sci. Comput. 41, A3728–A3756.CrossRefGoogle Scholar
Olver, S., Townsend, A. and Vasil, G. M. (2020), Recurrence relations for a family of orthogonal polynomials on a triangle. In Spectral and High Order Methods for Partial Differential Equations (ICOSAHOM) 2018 , Springer.CrossRefGoogle Scholar
Orszag, S. A. (1986), Fast eigenfunction transforms. Science and Computers 10, 13–30.Google Scholar
Parlett, B. N. (1974), ‘The Rayleigh quotient iteration and some generalizations for nonnormal matrices’, Math. Comp. 28, 679–693.CrossRefGoogle Scholar
Pearcey, T. (1946), ‘The structure of an electromagnetic field in the neighbourhood of a cusp of a caustic’, Philos. Mag. 37, 311–317.CrossRefGoogle Scholar
Petras, K. (1999), ‘On the computation of the Gauss–Legendre quadrature formula with a given precision’, J. Comput. Appl. Math. 112, 253–267.Google Scholar
Potts, D. and Steidl, G. (2003), ‘Fast summation at nonequispaced knots by NFFTs’, SIAM J . Sci. Comput. 24, 2013–2037.Google Scholar
Potts, D., Steidl, G. and Tasche, M. (1998), ‘Fast and stable algorithms for discrete spherical Fourier transforms’, Linear Algebra Appl. 275–276, 433–450.CrossRefGoogle Scholar
Püschel, M. and Moura, J. M. (2003), ‘The algebraic approach to the discrete cosine and sine transforms and their fast algorithms’, SIAM J. Comput. 32, 1280–1316.Google Scholar
Rayleigh, L. (1937), The Theory of Sound, Macmillan.Google Scholar
Reinecke, M. and Seljebotn, D. S. (2013), ‘Libsharp: Spherical harmonic transforms revisited’, Astronomy Astrophys. 554, A112.CrossRefGoogle Scholar
Rokhlin, V. and Tygert, M. (2006), ‘Fast algorithms for spherical harmonic expansions’, SIAM J. Sci. Comput. 27, 1903–1928.CrossRefGoogle Scholar
Ronveaux, A. and Marcellán, F. (1989), ‘Co-recursive orthogonal polynomials and fourth-order differential equation’, J. Comput. Appl. Math. 25, 105–109.CrossRefGoogle Scholar
Ronveaux, A., Zarzo, A. and Godoy, E. (1995), ‘Fourth-order differential equations satisfied by the generalized co-recursive of all classical orthogonal polynomials: A study of their distribution of zeros’, J. Comput. Appl. Math. 59, 295–328.CrossRefGoogle Scholar
Ruiz-Antolín, D. and Townsend, A. (2018), ‘A nonuniform fast Fourier transform based on low rank approximation’, SIAM J. Sci. Comput. 40, A529–A547.CrossRefGoogle Scholar
Rutishauer, H. (1962), ‘On a modification of the QD-algorithm with Graeffe-type convergence’, Z. Angew. Math. Phys. 13, 493–496.CrossRefGoogle Scholar
Schaeffer, N. (2013), ‘Efficient spherical harmonic transforms aimed at pseudospectral numerical simulations’, Geochem. Geophys. Geosyst. 14, 751–758.CrossRefGoogle Scholar
Schwarz, H. R. (1968), ‘Tridiagonalization of a symmetric band matrix’, Numer. Math. 12, 231–241.CrossRefGoogle Scholar
Shen, J. (1994), ‘Efficient spectral-Galerkin method I: Direct solvers of second- and fourth-order equations using Legendre polynomials’, SIAM J. Sci. Comput. 15, 1489–1505.CrossRefGoogle Scholar
Shull, H. and Löwdin, P.-O. (1955), ‘Role of the continuum in superposition of configurations’, J. Chem. Phys. 23, 1362–1363.Google Scholar
Slevinsky, R. M. (2016), Julia package for fast orthogonal polynomial transforms. GitHub. https://github.com/MikaelSlevinsky/FastTransforms.jl Google Scholar
Slevinsky, R. M. (2017), Conquering the pre-computation in two-dimensional harmonic polynomial transforms. arXiv:1711.07866 Google Scholar
Slevinsky, R. M. (2018a), Fast orthogonal polynomial transforms. GitHub. https://github.com/MikaelSlevinsky/FastTransforms Google Scholar
Slevinsky, R. M. (2018b), ‘On the use of Hahn’s asymptotic formula and stabilized recurrence for a fast, simple, and stable Chebyshev–Jacobi transform’, IMA J. Numer. Anal. 38, 102–124.CrossRefGoogle Scholar
Slevinsky, R. M. (2019), ‘Fast and backward stable transforms between spherical harmonic expansions and bivariate Fourier series’, Appl. Comput. Harmon. Anal. 47, 585–606.CrossRefGoogle Scholar
Slevinsky, R. M. and Olver, S. (2017), ‘A fast and well-conditioned spectral method for singular integral equations’, J. Comput. Phys. 332, 290–315.CrossRefGoogle Scholar
Slevinsky, R. M., Montanelli, H. and Du, Q. (2018), ‘A spectral method for nonlocal diffusion operators on the sphere’, J. Comput. Phys. 372, 893–911.Google Scholar
Snowball, B. and Olver, S. (2020), ‘Sparse spectral and $p$ -finite element methods for partial differential equations on disk slices and trapeziums’, Stud. Appl. Math., to appear.Google Scholar
Sommariva, A. (2013), ‘Fast construction of Fejér and Clenshaw–Curtis rules for general weight functions’, Comput. Math. Appl. 65, 682–693.CrossRefGoogle Scholar
Sonine, N. J. (1887), ‘Uber die angenäherte Berechnung der bestimmten Integrale und über die dabei vorkommenden ganzen Functionen’, Warsaw Univ. Izv. 18, 1–76.Google Scholar
Stroud, A. H. and Secrest, D. (1966), Gaussian Quadrature Formulas, Prentice Hall.Google Scholar
Suda, R. and Takami, M. (2002), ‘A fast spherical harmonics transform algorithm’, Math. Comp. 71, 703–715.CrossRefGoogle Scholar
Swarztrauber, P. N. (2003), ‘On computing the points and weights for Gauss–Legendre quadrature’, SIAM J. Sci. Comput. 24, 945–954.Google Scholar
Szegő, G. (1975), Orthogonal Polynomials, fourth edition, American Mathematical Society.Google Scholar
Tallqvist, H. (1905), ‘Grunderna af teorin fïoer sferiska funktioner jïoemte anvïoendningar inom fysiken’, Finska litteratur-sïoellskapets tryckeri.Google Scholar
Torres, D. J. and Coutsias, E. A. (1999), ‘Pseudospectral solution of the two-dimensional Navier–Stokes equations in a disk’, SIAM J. Sci. Comput. 21, 378–403.CrossRefGoogle Scholar
Townsend, A. (2015), ‘The race for high order Gauss–Legendre quadrature’, SIAM News 48, 1–3.Google Scholar
Townsend, A. and Olver, S. (2015), ‘The automatic solution of partial differential equations using a global spectral method’, J. Comput. Phys. 299, 106–123.Google Scholar
Townsend, A. and Trefethen, L. N. (2013), ‘An extension of Chebfun to two dimensions’, SIAM J. Sci. Comput. 35, C495–C518.CrossRefGoogle Scholar
Townsend, A., Trogdon, T. and Olver, S. (2016), ‘Fast computation of Gauss quadrature nodes and weights on the whole real line’, IMA J. Numer. Anal. 36, 337–358.Google Scholar
Townsend, A., Webb, M. and Olver, S. (2018), ‘Fast polynomial transforms based on Toeplitz and Hankel matrices’, Math. Comp. 87, 1913–1934.CrossRefGoogle Scholar
Trefethen, L. N. (2008), ‘Is Gauss quadrature better than Clenshaw–Curtis?’, SIAM Rev. 50, 67–87.CrossRefGoogle Scholar
Trogdon, T. and Olver, S. (2016), Riemann–Hilbert Problems, Their Numerical Solution and the Computation of Nonlinear Special Functions, SIAM.CrossRefGoogle Scholar
Tygert, M. (2008), ‘Fast algorithms for spherical harmonic expansions, II’, J. Comput. Phys. 227, 4260–4279.CrossRefGoogle Scholar
Tygert, M. (2010a), ‘Fast algorithms for spherical harmonic expansions, III’, J. Comput. Phys. 229, 6181–6192.CrossRefGoogle Scholar
Tygert, M. (2010b), ‘Recurrence relations and fast algorithms’, Appl. Comput. Harmon. Anal. 28, 121–128.CrossRefGoogle Scholar
Vasil, G. M., Burns, K. J., Lecoanet, D., Olver, S., Brown, B. P. and Oishi, J. S. (2016), ‘Tensor calculus in polar coordinates using Jacobi polynomials’, J. Comput. Phys. 325, 53–73.CrossRefGoogle Scholar
Viswanath, D. (2015), ‘Spectral integration of linear boundary value problems’, J. Comput. Appl. Math. 290, 159–173.CrossRefGoogle Scholar
Vogel, J., Xia, J., Cauley, S. and Balakrishnan, V. (2016), ‘Superfast divide-and-conquer method and perturbation analysis for structured eigenvalue solutions’, SIAM J. Sci. Comput. 38, A1358–A1382.CrossRefGoogle Scholar
Waldvogel, J. (2003), ‘Fast construction of the Fejér and Clenshaw–Curtis quadrature rules’, BIT Numer. Math. 43, 001–018.Google Scholar
Webb, M. (2017), Isospectral algorithms, Toeplitz matrices and orthogonal polynomials. PhD thesis, University of Cambridge.Google Scholar
Wegert, E. (2012), Visual Complex Functions: An Introduction with Phase Portraits, Springer Science & Business Media.CrossRefGoogle Scholar
Weniger, E. J. (2019), ‘Comment on “Fourier transform of hydrogen-type atomic orbitals”’, Canad. J. Phys. 97, 1349–1360.CrossRefGoogle Scholar
Wilber, H., Townsend, A. and Wright, Gs. B. (2017), ‘Computing with functions in spherical and polar geometries II: The disk’, SIAM J. Sci. Comput. 39, C238–C262.Google Scholar
Xu, Y. (2017), ‘Approximation and orthogonality in Sobolev spaces on a triangle’, Const. Approx. 46, 349–434.CrossRefGoogle Scholar
Yakimiw, E. (1996), ‘Accurate computation of weights in classical Gauss–Christoffel quadrature rules’, J. Comput. Phys. 129, 403–430.CrossRefGoogle Scholar
Zarzo, A., Ronveaux, A. and Godoy, E. (1993), ‘Fourth-order differential equation satisfied by the associated of any order of all classical orthogonal polynomials: A study of their distribution of zeros’, J. Comput. Appl. Math. 49, 349–359.Google Scholar
Zayernouri, M. and Karniadakis, G. E. (2014), ‘Fractional spectral collocation method’, SIAM J. Sci. Comput. 36, A40–A62.Google Scholar
Zebib, A. (1984), ‘A Chebyshev method for the solution of boundary value problems’, J. Comput. Phys. 53, 443–455.CrossRefGoogle Scholar
Zernike, F. (1934), ‘Beugungstheorie des Schneidenverfahrens und seiner verbesserten Form, der Phasenkontrastmethode’, Physica 1, 689–704.CrossRefGoogle Scholar