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Clenshaw–Curtis formula (extended)

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1: 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). …
2: 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). …
3: 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.
  • 4: 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).
  • W. F. Perger, A. Bhalla, and M. Nardin (1993) A numerical evaluator for the generalized hypergeometric series. Comput. Phys. Comm. 77 (2), pp. 249–254.
  • R. Piessens and M. Branders (1983) Modified Clenshaw-Curtis method for the computation of Bessel function integrals. BIT 23 (3), pp. 370–381.
  • A. Poquérusse and S. Alexiou (1999) Fast analytic formulas for the modified Bessel functions of imaginary order for spectral line broadening calculations. J. Quantit. Spec. and Rad. Trans. 62 (4), pp. 389–395.
  • J. L. Powell (1947) Recurrence formulas for Coulomb wave functions. Physical Rev. (2) 72 (7), pp. 626–627.
  • 5: 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.
  • 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.
  • R. Wong (1982) Quadrature formulas for oscillatory integral transforms. Numer. Math. 39 (3), pp. 351–360.
  • 6: 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.
  • 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.
  • 7: 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. …
    8: 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).

  • 9: 33.23 Methods of Computation
    Use of extended-precision arithmetic increases the radial range that yields accurate results, but eventually other methods must be employed, for example, the asymptotic expansions of §§33.11 and 33.21. … Curtis (1964a, §10) describes the use of series, radial integration, and other methods to generate the tables listed in §33.24. … A set of consistent second-order WKBJ formulas is given by Burgess (1963: in Eq. …
    10: 36.15 Methods of Computation
    Far from the bifurcation set, the leading-order asymptotic formulas of §36.11 reproduce accurately the form of the function, including the geometry of the zeros described in §36.7. … For details, see Connor and Curtis (1982) and Kirk et al. (2000). …