The expansion in Gegenbauer polynomials: A simple method for the fast computation of the Gegenbauer coefficients

Journal of Computational Physics

Volumn 239, Issue , 2013, Pages 112-122

  De Micheli, Enrico ^a   Viano, Giovanni Alberto ^b  

^a CNR   (Italy)

^b UNIVERSITY OF GENOA   (Italy)

Author keywords

Fourier series; Gegenbauer polynomials; Gegenbauer reconstruction; Spectral methods

Indexed keywords

COMPUTATIONAL EFFICIENCY; FAST FOURIER TRANSFORMS; FOURIER ANALYSIS; INTEGRAL EQUATIONS; NUMERICAL METHODS; ORDINARY DIFFERENTIAL EQUATIONS; POLYNOMIALS;

FAST ALGORITHMS; FAST COMPUTATION; GEGENBAUE RECONSTRUCTION; GEGENBAUER; GEGENBAUER POLYNOMIALS; INTEGRAL TRANSFORM; SIMPLE ALGORITHM; SIMPLE METHOD; SIMPLER ALGORITHMS; SPECTRAL METHODS;

FOURIER SERIES;

EID: 84873622912     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2013.01.008     Document Type: Article

References (35)

1. Handbook of Mathematical Functions 1964, Dover, New York. M. Abramowitz, I. Stegun (Eds.).
2. Archibald R., Chen K., Gelb A., Renaut R. Improving tissue segmentation of human brain MRI through preprocessing by the Gegenbauer reconstruction method. Neuroimage 2003, 20:489-502.
3. Belen'kii A.M. Uniform convergence of the Fourier-Jacobi series on the orthogonality segment. Math. Notes 1989, 46:901-906.
4. Ben-yu G. Spectral Methods and their Applications 1998, World Scientific, Singapore.
5. Ben-yu G. Gegenbauer approximation and its applications to differential equations with rough asymptotic behaviors at infinity. Appl. Numer. Math. 2001, 38:403-425.
6. Bros J., Viano G.A. Connection between the harmonic analysis on the sphere and the harmonic analysis on the one-sheeted hyperboloid: an analytic continuation viewpoint III. Forum Math. 1997, 9:165-191.
7. Calvetti D. A stochastic roundoff error analysis for the fast Fourier transform. Math. Comput. 1991, 56:755-774.
8. Canuto C., Hussaini M.Y., Quarteroni A., Zang T.A. Spectral Methods in Fluid Dynamics 1987, Springer-Verlag, Heidelberg.
9. Cooley J.W., Tukey J.W. An algorithm for the machine calculation of complex Fourier series. Math. Comput. 1965, 19:297-301.
10. Denman H.G., Howard J.E. Application of ultraspherical polynomials to nonlinear oscillations. Q. Appl. Math. 1964, 21:325-330.
11. Elliott D. The expansion of functions in ultraspherical polynomials. J. Aust. Math. Soc. 1960, 1:428-438.
12. Fulton S.R., Schubert W.H. Chebyshev spectral methods for limited-area models. Part I: Model problem analysis. Mon. Weather Rev. 1986, 115:1940-1953.
13. M. Galassi et al., GNU Scientific Library Reference Manual, third ed., ISBN 0954612078, 2009. Available from: http://www.gnu.org/software/gsl/.
14. Gautschi W. Numerical Analysis 1997, Birkhäuser, Boston.
15. Gelb A., Tanner J. Robust reprojection methods for the resolution of the Gibbs phenomenon. Appl. Comput. Harmon. Anal. 2006, 20:3-25.
16. Gelb A. A hybrid approach to spectral reconstruction of piecewise smooth functions. J. Sci. Comput. 2000, 15:293-322.
17. Golub G.H., Welsch J.H. Calculation of Gauss quadrature rules. Math. Comput. 1969, 23:221-230.
18. Gottlieb D., Orszag S.A. Numerical Analysis of Spectral Methods: Theory and Applications 1977, SIAM, Philadelphia.
19. Gottlieb D., Shu C.-W., Solomonoff A., Vandeven H. On the Gibbs phenomenon I: recovering exponential accuracy from the Fourier partial sum of a nonperiodic analytic function. J. Comput. Appl. Math. 1992, 43:81-98.
20. Gottlieb D., Shu C.-W. On the Gibbs phenomenon and its resolution. SIAM Rev. 1997, 39:644-668.
21. Haidvogel D.B., Curchister E., Iskandarani M., Hughes R., Taylor M. Global modelling of the ocean and atmosphere using the spectral element method. Atmos. Ocean 1997, 35:sup1:505-531.
22. N. Hale, A. Townsend, Fast and accurate computation of Gauss-Legendre and Gauss-Jacobi quadrature nodes and weights, Technical Report NA-12-13, University of Oxford, 2012.
23. Kahaner D., Moler C., Nash S. Numerical Methods and Software 1989, Prentice Hall, Englewood Cliffs, NJ.
24. Kaneko T., Liu B. Accumulation of round-off error in Fast Fourier Transform. J. Assoc. Comput. Mach. 1970, 17:637-654.
25. Kvernadze G. Uniform convergence of Fourier-Jacobi series. J. Approx. Theory 2002, 117:207-228.
26. Lui S.H. Numerical Analysis of Partial Differential Equations 2011, Wiley, Hoboken, NJ.
27. Mathematica Version 7.0, Wolfram Research, Inc., Champaign, IL, 2008.
28. Min M.S., Lee T.W., Fischer P.F., Gray S.K. Fourier spectral simulations and Gegenbauer reconstructions for electromagnetic waves in the presence of a metal nanoparticle. J. Comput. Phys. 2006, 213:730-747.
29. Monegato G., Scuderi L. Numerical integration of functions with boundary singularities. J. Comput. Appl. Math. 1999, 112:201-214.
30. Orszag S.A. Spectral methods for problems in complex geometries. J. Comput. Phys. 1980, 37:70-92.
31. Sinha S.C., Srinivasan P. Application of ultraspherical polynomials to non-linear autonomous systems. J. Sound Vib. 1971, 18:55-60.
32. Szegö G. Orthogonal Polynomials 1975, American Mathematical Society, Providence.
33. Vilenkin N.I. Special Functions and the Theory of Group Representations. Translation of Mathematical Monographs 1968, vol. 22. American Mathematical Society, Providence.
34. Vozovoi L., Weill A., Israeli M. Spectrally accurate solution of nonperiodic differential equations by the Fourier-Gegenbauer method. SIAM J. Numer. Anal. 1997, 34:1451-1471.
35. E.W. Weisstein, A Wolfram Web Resource, 2012. http://functions.wolfram.com/HypergeometricFunctions/GegenbauerC3General/.