A fast, simple, and stable Chebyshev Legendre transform using an asymptotic formula

SIAM Journal on Scientific Computing

Volumn 36, Issue 1, 2014, Pages

  Hale, Nicholas ^a   Townsend, Alex ^a  

^a UNIVERSITY OF OXFORD   (United Kingdom)

Author keywords

Asymptotic formula; Chebyshev; Discrete cosine transform; Legendre; Transform

Indexed keywords

CHEBYSHEV POLYNOMIALS; POLYNOMIALS;

ASYMPTOTIC FORMULAE; CHEBYSHEV; CHEBYSHEV POLYNOMIALS; LEGENDRE; LEGENDRE EXPANSION; LEGENDRE POLYNOMIALS; LEGENDRE TRANSFORMS; NUMERICAL RESULTS; SIMPLE++; WEIGHTED LINEAR COMBINATIONS;

DISCRETE COSINE TRANSFORMS;

EID: 84896973279     PISSN: 10648275     EISSN: 10957200     Source Type: Journal    
DOI: 10.1137/130932223     Document Type: Article

References (41)

1. N. Ahmed, T. Natarajan, and K. R. Rao, Discrete cosine transform, IEEE Trans. Comput., 100 (1974), pp. 90-93.
2. B. K. Alpert and V. Rokhlin, A fast algorithm for the evaluation of Legendre expansions, SIAM J. Sci. Statist. Comput., 12 (1991), pp. 158-179.
3. T. M. Apostol, Introduction to Analytic Number Theory, Undergrad. Texts Math., Springer, New York, 1976.
4. K. Atkinson and W. Han, Spherical Harmonics and Approximations on the Unit Sphere: An Introduction, Lecture Notes in Math. 2044, Springer, New York, 2012.
5. I. Bogaert, B. Michiels and J. Fostier, (1) computation of Legendre polynomials and Gauss-Legendre nodes and weights for parallel computing, J. Sci. Comput., 34 (2012), pp. C83-C101.
6. J. P. Boyd, Chebyshev and Fourier Spectral Methods, Dover Publications, New York, 2001.
7. E. De Micheli and G. A. Viano, The expansion in Gegenbauer polynomials: A simple method for the fast computation of the Gegenbauer coefficients, J. Comput. Phys., 239 (2013), pp. 112-122.
8. W. S. Don and D. Gottlieb, The Chebyshev-Legendre method: Implementing Legendre methods on Chebyshev points, SIAM J. Numer. Anal., 31 (1994), pp. 1519-1534.
9. J. R. Driscoll and D. M. Healy, Computing Fourier transforms and convolutions on the 2-sphere, Adv. in Appl. Math., 15 (1994), pp. 202-250.
10. M. Frigo and S. G. Johnson, The design and implementation of FFTW 3, Proc. IEEE, 93 (2005), pp. 216-231.
11. N. Gallagher, G. Wise, and J. Allen, A novel approach for the computation of Legendre polynomial expansions, IEEE Trans. Acoustic Speech Signal Process., 26 (1978), pp. 105-106.
12. W. M. Gentleman, Implementing Clenshaw-Curtis quadrature II: Computing the cosine transformation, Commun. ACM, 15 (1972), pp. 343-346.
13. M. Gu and S. C. Eisenstat, A divide-and-conquer algorithm for the symmetric tridiagonal eigenproblem, SIAM J. Matrix Anal. Appl., 16 (1995), pp. 172-191.
14. E. Hahn, Asymptotik bei Jacobi-Polynomen und Jacobi-Funktionen, Math. Z., 171 (1980), pp. 201-226.
15. N. Hale and A. Townsend, LEG2CHEB, http://github.com/nickhale/leg2cheb (2013).
16. N. Hale and A. Townsend, Fast and accurate computation of Gauss-Legendre and Gauss-Jacobi quadrature nodes and weights, SIAM J. Sci. Comput., 35 (2013), pp. A652-A672.
17. A. Iserles, A fast and simple algorithm for the computation of Legendre coefficients, Numer. Math., 117 (2011), pp. 529-553.
18. J. C. Mason and D. C. Handscomb, Chebyshev Polynomials, Taylor & Francis, New York, 2002.
19. M. J. Mohlenkamp, A fast transform for spherical harmonics, J. Fourier Anal. Appl., 5 (1999), pp. 159-184.
20. A. Mori, R. Suda, and M. Sugihara, An improvement on Orszag's fast algorithm for Legendre polynomial transform, Trans. Info. Process. Soc. Japan, 40 (1999), pp. 3612-3615.
21. S. Olver, A general framework for solving Riemann-Hilbert problems numerically, Numer. Math., 122 (2012), pp. 305-340.
22. F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions, Cambridge University Press, Cambridge, UK, 2010.
23. S. Olver and A. Townsend, A fast and well-conditioned spectral method, SIAM Rev., 55 (2013), pp. 462-489.
24. M. O'Neil, F. Woolfe, and V. Rokhlin, An algorithm for the rapid evaluation of special function transforms, Appl. Comput. Harmon. Anal., 28 (2010), pp. 203-226.
25. S. A. Orszag, Fast eigenfunction transforms, in Science and Computers, Academic Press, New York, 1986, pp. 13-30.
26. R. Piessens, Computation of Legendre series coefficients, Comm. ACM, 17 (1974), p. 25.
27. D. Potts, G. Steidl, and M. Tasche, Fast algorithms for discrete polynomial transforms, Math. Comp., 67 (1998), pp. 1577-1590.
28. V. Rokhlin and M. Tygert, Fast algorithms for spherical harmonic expansions, SIAM J. Sci. Comput., 27 (2006), pp. 1903-1928.
29. J. Shen, Efficient spectral-Galerkin method I. Direct solvers of second- and fourth-order equations using Legendre polynomials, SIAM J. Sci. Comput., 15 (1994), pp. 1489-1505.
30. G. W. Stewart, Afternotes Goes to Graduate School, SIAM, Philadelphia, 1998.
31. T. J. Stieltjes, Sur les polynômes de Legendre, Ann. Fac. Sci. Toulouse, 4 (1890), G1-G17.
32. R. Suda, Private communication, University of Tokyo, 2013.
33. R. Suda and M. Takami, A fast spherical harmonics transform algorithm, Math. Comput., 71 (2002), pp. 703-715.
34. G. Szego, Orthogonal Polynomials, American Mathematical Society, Providence, RI, 1939.
35. L. N. Trefethen, Approximation Theory and Approximation Practice, SIAM, Philadelphia, 2013.
36. L. N. Trefethen et al., Chebfun Version 4.2 (revision 2925), http://www.maths.ox.ac.uk/chebfun (2013).
37. M. Tygert, Fast algorithms for spherical harmonic expansions, II, J. Comput. Phys., 227 (2008), pp. 4260-4279.
38. M. Tygert, Recurrence relations and fast algorithms, Appl. Comput. Harmon. Anal., 28 (2010), pp. 121-128.
39. J. Waldvogel, Fast construction of the Fejér and Clenshaw-Curtis quadrature rules, BIT, 43 (2003), pp. 1-18.
40. Z. Wang and B. R. Hunt, The discrete W transform, Appl. Math. Comput., 16 (1985), pp. 19-48.
41. N. Yarvin and V. Rokhlin, A generalized one-dimensional fast multipole method with application to filtering of spherical harmonics, J. Comput. Phys., 147 (1998), pp. 594-609.