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1: 18.40 Methods of Computation
However, for applications in which the OP’s appear only as terms in series expansions (compare §18.18(i)) the need to compute them can be avoided altogether by use instead of Clenshaw’s algorithm (§3.11(ii)) and its straightforward generalization to OP’s other than Chebyshev. For further information see Clenshaw (1955), Gautschi (2004, §§2.1, 8.1), and Mason and Handscomb (2003, §2.4). …
2: Bibliography C
  • C. W. Clenshaw and A. R. Curtis (1960) A method for numerical integration on an automatic copmputer. Numer. Math. 2 (4), pp. 197–205.
  • C. W. Clenshaw, D. W. Lozier, F. W. J. Olver, and P. R. Turner (1986) Generalized exponential and logarithmic functions. Comput. Math. Appl. Part B 12 (5-6), pp. 1091–1101.
  • C. W. Clenshaw, G. F. Miller, and M. Woodger (1962) Algorithms for special functions. I. Numer. Math. 4, pp. 403–419.
  • C. W. Clenshaw and F. W. J. Olver (1984) Beyond floating point. J. Assoc. Comput. Mach. 31 (2), pp. 319–328.
  • C. W. Clenshaw (1955) A note on the summation of Chebyshev series. Math. Tables Aids Comput. 9 (51), pp. 118–120.
  • 3: 3.5 Quadrature
    If we add 1 and 1 to this set of x k , then the resulting closed formula is the frequently-used ClenshawCurtis formula, whose weights are positive and given by … For further information, see Mason and Handscomb (2003, Chapter 8), Davis and Rabinowitz (1984, pp. 74–92), and Clenshaw and Curtis (1960). For detailed comparisons of the ClenshawCurtis formula with Gauss quadrature (§3.5(v)), see Trefethen (2008, 2011). …
    4: Bibliography P
  • R. Piessens and M. Branders (1983) Modified Clenshaw-Curtis method for the computation of Bessel function integrals. BIT 23 (3), pp. 370–381.
  • T. Prellberg and A. L. Owczarek (1995) Stacking models of vesicles and compact clusters. J. Statist. Phys. 80 (3–4), pp. 755–779.
  • 5: Bibliography T
  • J. D. Talman (1983) LSFBTR: A subroutine for calculating spherical Bessel transforms. Comput. Phys. Comm. 30 (1), pp. 93–99.
  • L. N. Trefethen (2008) Is Gauss quadrature better than Clenshaw-Curtis?. SIAM Rev. 50 (1), pp. 67–87.
  • 6: Bibliography O
  • J. Oliver (1977) An error analysis of the modified Clenshaw method for evaluating Chebyshev and Fourier series. J. Inst. Math. Appl. 20 (3), pp. 379–391.
  • F. W. J. Olver (1967b) Bounds for the solutions of second-order linear difference equations. J. Res. Nat. Bur. Standards Sect. B 71B (4), pp. 161–166.
  • C. Osácar, J. Palacián, and M. Palacios (1995) Numerical evaluation of the dilogarithm of complex argument. Celestial Mech. Dynam. Astronom. 62 (1), pp. 93–98.
  • 7: 4.47 Approximations
    Clenshaw (1962) and Luke (1975, Chapter 3) give 20D coefficients for ln , exp , sin , cos , tan , cot , arcsin , arctan , arcsinh . …
    8: Bibliography W
  • J. Waldvogel (2006) Fast construction of the Fejér and Clenshaw-Curtis quadrature rules. BIT 46 (1), pp. 195–202.
  • C. A. Wills, J. M. Blair, and P. L. Ragde (1982) Rational Chebyshev approximations for the Bessel functions J 0 ( x ) , J 1 ( x ) , Y 0 ( x ) , Y 1 ( x ) . Math. Comp. 39 (160), pp. 617–623.
  • 9: 5.24 Software
    See also Borwein and Zucker (1992), Carmignani and Tortorici Macaluso (1985), Clenshaw et al. (1962), Cody (1991), Filho and Schwachheim (1967), and Temme (1994a). …
    10: 29.20 Methods of Computation
    The Fourier series may be summed using Clenshaw’s algorithm; see §3.11(ii). …