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1: 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, F. W. J. Olver, and P. R. Turner (1989) Level-Index Arithmetic: An Introductory Survey. In Numerical Analysis and Parallel Processing (Lancaster, 1987), P. R. Turner (Ed.), Lecture Notes in Math., Vol. 1397, pp. 95–168.
  • C. W. Clenshaw and F. W. J. Olver (1984) Beyond floating point. J. Assoc. Comput. Mach. 31 (2), pp. 319–328.
  • 2: Bibliography P
  • V. I. Pagurova (1965) An asymptotic formula for the incomplete gamma function. Ž. Vyčisl. Mat. i Mat. Fiz. 5, pp. 118–121 (Russian).
  • R. Piessens (1982) Automatic computation of Bessel function integrals. Comput. Phys. Comm. 25 (3), pp. 289–295.
  • R. Piessens and M. Branders (1983) Modified Clenshaw-Curtis method for the computation of Bessel function integrals. BIT 23 (3), pp. 370–381.
  • J. L. Powell (1947) Recurrence formulas for Coulomb wave functions. Physical Rev. (2) 72 (7), pp. 626–627.
  • T. Prellberg and A. L. Owczarek (1995) Stacking models of vesicles and compact clusters. J. Statist. Phys. 80 (3–4), pp. 755–779.
  • 3: 29.20 Methods of Computation
    Subsequently, formulas typified by (29.6.4) can be applied to compute the coefficients of the Fourier expansions of the corresponding Lamé functions by backward recursion followed by application of formulas typified by (29.6.5) and (29.6.6) to achieve normalization; compare §3.6. …The Fourier series may be summed using Clenshaw’s algorithm; see §3.11(ii). … §29.15(i) includes formulas for normalizing the eigenvectors. …
    4: Bibliography O
  • F. Oberhettinger (1973) Fourier Expansions. A Collection of Formulas. Academic Press, New York-London.
  • 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.
  • F. W. J. Olver (1977a) Connection formulas for second-order differential equations with multiple turning points. SIAM J. Math. Anal. 8 (1), pp. 127–154.
  • 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.
  • 5: 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 detailed comparisons of the ClenshawCurtis formula with Gauss quadrature (§3.5(v)), see Trefethen (2008, 2011). … A comparison of several methods, including an extension of the ClenshawCurtis formula3.5(iv)), is given in Evans and Webster (1999). …
    6: 18.40 Methods of Computation
    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). … Results of low ( 2 to 3 decimal digits) precision for w ( x ) are easily obtained for N 10 to 20 . …
    7: 28.34 Methods of Computation
  • (b)

    Representations for w I ( π ; a , ± q ) with limit formulas for special solutions of the recurrence relations §28.4(ii) for fixed a and q ; see Schäfke (1961a).

  • (d)

    Solution of the systems of linear algebraic equations (28.4.5)–(28.4.8) and (28.14.4), with the conditions (28.4.9)–(28.4.12) and (28.14.5), by boundary-value methods (§3.6) to determine the Fourier coefficients. Subsequently, the Fourier series can be summed with the aid of Clenshaw’s algorithm (§3.11(ii)). See Meixner and Schäfke (1954, §2.87). This procedure can be combined with §28.34(ii)(d).

  • 8: Bibliography T
  • Y. Takei (1995) On the connection formula for the first Painlevé equation—from the viewpoint of the exact WKB analysis. Sūrikaisekikenkyūsho Kōkyūroku (931), pp. 70–99.
  • J. D. Talman (1983) LSFBTR: A subroutine for calculating spherical Bessel transforms. Comput. Phys. Comm. 30 (1), pp. 93–99.
  • S. A. Teukolsky (1972) Rotating black holes: Separable wave equations for gravitational and electromagnetic perturbations. Phys. Rev. Lett. 29 (16), pp. 1114–1118.
  • P. G. Todorov (1991) Explicit formulas for the Bernoulli and Euler polynomials and numbers. Abh. Math. Sem. Univ. Hamburg 61, pp. 175–180.
  • L. N. Trefethen (2008) Is Gauss quadrature better than Clenshaw-Curtis?. SIAM Rev. 50 (1), pp. 67–87.
  • 9: 6.20 Approximations
  • Cody and Thacher (1968) provides minimax rational approximations for E 1 ( x ) , with accuracies up to 20S.

  • Cody and Thacher (1969) provides minimax rational approximations for Ei ( x ) , with accuracies up to 20S.

  • MacLeod (1996b) provides rational approximations for the sine and cosine integrals and for the auxiliary functions f and g , with accuracies up to 20S.

  • Clenshaw (1962) gives Chebyshev coefficients for E 1 ( x ) ln | x | for 4 x 4 and e x E 1 ( x ) for x 4 (20D).

  • 10: Bibliography W
  • J. Waldvogel (2006) Fast construction of the Fejér and Clenshaw-Curtis quadrature rules. BIT 46 (1), pp. 195–202.
  • P. L. Walker (2012) Reduction formulae for products of theta functions. J. Res. Nat. Inst. Standards and Technology 117, pp. 297–303.
  • R. S. Ward (1987) The Nahm equations, finite-gap potentials and Lamé functions. J. Phys. A 20 (10), pp. 2679–2683.
  • 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.
  • J. Wimp (1968) Recursion formulae for hypergeometric functions. Math. Comp. 22 (102), pp. 363–373.