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31: 34.13 Methods of Computation
§34.13 Methods of Computation
β–ΊMethods of computation for 3 ⁒ j and 6 ⁒ j symbols include recursion relations, see Schulten and Gordon (1975a), Luscombe and Luban (1998), and Edmonds (1974, pp. 42–45, 48–51, 97–99); summation of single-sum expressions for these symbols, see Varshalovich et al. (1988, §§8.2.6, 9.2.1) and Fang and Shriner (1992); evaluation of the generalized hypergeometric functions of unit argument that represent these symbols, see Srinivasa Rao and Venkatesh (1978) and Srinivasa Rao (1981). β–ΊFor 9 ⁒ j symbols, methods include evaluation of the single-sum series (34.6.2), see Fang and Shriner (1992); evaluation of triple-sum series, see Varshalovich et al. (1988, §10.2.1) and Srinivasa Rao et al. (1989). A review of methods of computation is given in Srinivasa Rao and Rajeswari (1993, Chapter VII, pp. 235–265). …
32: 32.17 Methods of Computation
§32.17 Methods of Computation
β–ΊThe Painlevé equations can be integrated by Runge–Kutta methods for ordinary differential equations; see §3.7(v), Hairer et al. (2000), and Butcher (2003). …
33: Bibliography G
… β–Ί
  • B. Gabutti (1979) On high precision methods for computing integrals involving Bessel functions. Math. Comp. 33 (147), pp. 1049–1057.
    β“˜
    External Links:
    ISSN 0025-5718, FullText, MathReview (K. Jetter), ZentralBlatt
    Referenced by:
    §10.74(vii).
    Encodings:
    BibTeX
    • β–Ί
  • B. Gabutti (1980) On the generalization of a method for computing Bessel function integrals. J. Comput. Appl. Math. 6 (2), pp. 167–168.
    β“˜
    External Links:
    ISSN 0771-050X, Document, MathReview, ZentralBlatt
    Referenced by:
    §10.74(vii).
    Encodings:
    BibTeX
    • … β–Ί
  • A. Gil, J. Segura, and N. M. Temme (2014) Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.
    β“˜
    Notes:
    Includes Fortran program claiming 12S accuracy
    External Links:
    ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt
    Referenced by:
    3rd item, §10.77(ix).
    Encodings:
    BibTeX
    • … β–Ί
  • M. L. Glasser (1979) A method for evaluating certain Bessel integrals. Z. Angew. Math. Phys. 30 (4), pp. 722–723.
    β“˜
    External Links:
    ISSN 0044-2275, Document, MathReview
    Referenced by:
    §10.71(ii).
    Encodings:
    BibTeX
    • … β–Ί
  • B. Guo (1998) Spectral Methods and Their Applications. World Scientific Publishing Co. Inc., River Edge, NJ-Singapore.
    β“˜
    External Links:
    ISBN 981-02-3333-7, MathReview (K. MoszyΕ„ski), ZentralBlatt
    Referenced by:
    §18.38(i).
    Encodings:
    BibTeX
    • …
    34: 19.36 Methods of Computation
    §19.36 Methods of Computation
    β–Ί
    §19.36(i) Duplication Method
    … β–ΊFor computation of Legendre’s integral of the third kind, see Abramowitz and Stegun (1964, §§17.7 and 17.8, Examples 15, 17, 19, and 20). … β–Ί
    §19.36(iv) Other Methods
    β–ΊNumerical quadrature is slower than most methods for the standard integrals but can be useful for elliptic integrals that have complicated representations in terms of standard integrals. …
    35: 29.20 Methods of Computation
    §29.20 Methods of Computation
    … β–ΊA second approach is to solve the continued-fraction equations typified by (29.3.10) by Newton’s rule or other iterative methods; see §3.8. … β–ΊA third method is to approximate eigenvalues and Fourier coefficients of Lamé functions by eigenvalues and eigenvectors of finite matrices using the methods of §§3.2(vi) and 3.8(iv). …The numerical computations described in Jansen (1977) are based in part upon this method. β–ΊA fourth method is by asymptotic approximations by zeros of orthogonal polynomials of increasing degree. …
    36: Bibliography B
    … β–Ί
  • G. Backenstoss (1970) Pionic atoms. Annual Review of Nuclear and Particle Science 20, pp. 467–508.
    β“˜
    External Links:
    Document
    Referenced by:
    §33.22(iv).
    Encodings:
    BibTeX
    • … β–Ί
  • K. L. Bell and N. S. Scott (1980) Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.
    β“˜
    Notes:
    Double-precision Fortran code.
    External Links:
    Document, Code
    Referenced by:
    1st item, 4th item.
    Encodings:
    BibTeX
    • … β–Ί
  • E. A. Bender (1974) Asymptotic methods in enumeration. SIAM Rev. 16 (4), pp. 485–515.
    β“˜
    External Links:
    Document, MathReview (Bernard Harris), ZentralBlatt
    Referenced by:
    Ch.2.
    Encodings:
    BibTeX
    • … β–Ί
  • W. G. Bickley (1935) Some solutions of the problem of forced convection. Philos. Mag. Series 7 20, pp. 322–343.
    β“˜
    External Links:
    Document, ZentralBlatt
    Referenced by:
    §10.73(iv).
    Encodings:
    BibTeX
    • … β–Ί
  • Å. Björck (1996) Numerical Methods for Least Squares Problems. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.
    β“˜
    External Links:
    ISBN 0-89871-360-9, MathReview (Hongyuan Zha), ZentralBlatt
    Referenced by:
    §3.11(v).
    Encodings:
    BibTeX
    • …
    37: Bibliography D
    … β–Ί
  • P. J. Davis and P. Rabinowitz (1984) Methods of Numerical Integration. 2nd edition, Computer Science and Applied Mathematics, Academic Press Inc., Orlando, FL.
    β“˜
    External Links:
    ISBN 0-12-206360-0, MathReview (John P. Coleman), ZentralBlatt
    Referenced by:
    §3.5(iii), §3.5(i), §3.5(i), §3.5(ii), §3.5(iii), §3.5(iv), §3.5(v), §3.5(vi), §3.5(vii), §3.5(x).
    Encodings:
    BibTeX
    • … β–Ί
  • N. G. de Bruijn (1961) Asymptotic Methods in Analysis. 2nd edition, Bibliotheca Mathematica, Vol. IV, North-Holland Publishing Co., Amsterdam.
    β“˜
    External Links:
    MathReview, ZentralBlatt
    Referenced by:
    §2.2, Ch.2, §26.7(iv), §26.7(iv), §4.13.
    Encodings:
    BibTeX
    • … β–Ί
  • A. Debosscher (1998) Unification of one-dimensional Fokker-Planck equations beyond hypergeometrics: Factorizer solution method and eigenvalue schemes. Phys. Rev. E (3) 57 (1), pp. 252–275.
    β“˜
    External Links:
    ISSN 1063-651X, Document, MathReview (Victor I. Stepanov), ZentralBlatt
    Referenced by:
    §31.17(ii).
    Encodings:
    BibTeX
    • … β–Ί
  • B. Deconinck and J. N. Kutz (2006) Computing spectra of linear operators using the Floquet-Fourier-Hill method. J. Comput. Phys. 219 (1), pp. 296–321.
    β“˜
    External Links:
    ISSN 0021-9991, MathReview, ZentralBlatt
    Referenced by:
    §22.11.
    Encodings:
    BibTeX
    • … β–Ί
  • K. Dekker and J. G. Verwer (1984) Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations. CWI Monographs, Vol. 2, North-Holland Publishing Co., Amsterdam.
    β“˜
    External Links:
    ISBN 0-444-87634-0, MathReview (A. de Castro Brzezicki), ZentralBlatt
    Referenced by:
    §3.7(v).
    Encodings:
    BibTeX
    • …
    38: Bibliography P
    … β–Ί
  • R. B. Paris (2004) Exactification of the method of steepest descents: The Bessel functions of large order and argument. Proc. Roy. Soc. London Ser. A 460, pp. 2737–2759.
    β“˜
    External Links:
    ISSN 1364-5021, Document, MathReview (F. Ursell), ZentralBlatt
    Referenced by:
    §10.20(ii).
    Encodings:
    BibTeX
    • … β–Ί
  • R. Piessens (1982) Automatic computation of Bessel function integrals. Comput. Phys. Comm. 25 (3), pp. 289–295.
    β“˜
    Notes:
    Double-precision Fortran, maximum accuracy 20S.
    External Links:
    ISSN 0010-4655, Document, Code
    Referenced by:
    6th item.
    Encodings:
    BibTeX
    • … β–Ί
  • R. Piessens and M. Branders (1983) Modified Clenshaw-Curtis method for the computation of Bessel function integrals. BIT 23 (3), pp. 370–381.
    β“˜
    External Links:
    ISSN 0006-3835, Document, MathReview (H. E. Fettis), ZentralBlatt
    Referenced by:
    §10.74(vii).
    Encodings:
    BibTeX
    • … β–Ί
  • R. Piessens and M. Branders (1985) A survey of numerical methods for the computation of Bessel function integrals. Rend. Sem. Mat. Univ. Politec. Torino (Special Issue), pp. 249–265.
    β“˜
    Notes:
    International conference on special functions: theory and computation (Turin, 1984)
    External Links:
    ISSN 0373-1243, MathReview, ZentralBlatt
    Referenced by:
    §10.74(vii), §10.74(vii).
    Encodings:
    BibTeX
    • … β–Ί
  • M. Puoskari (1988) A method for computing Bessel function integrals. J. Comput. Phys. 75 (2), pp. 334–344.
    β“˜
    External Links:
    ISSN 0021-9991, Document, MathReview (J. G. Herriot), ZentralBlatt
    Referenced by:
    §10.74(vii), §10.74(vii).
    Encodings:
    BibTeX
    • 39: Bibliography M
    … β–Ί
  • A. J. MacLeod (1996b) Rational approximations, software and test methods for sine and cosine integrals. Numer. Algorithms 12 (3-4), pp. 259–272.
    β“˜
    Notes:
    Includes Fortran code, maximum accuracy 20S.
    External Links:
    ISSN 1017-1398, Document, MathReview, ZentralBlatt
    Referenced by:
    §6.13, 4th item, §6.21(ii).
    Encodings:
    BibTeX
    • … β–Ί
  • P. L. Marston (1992) Geometrical and Catastrophe Optics Methods in Scattering. In Physical Acoustics, A. D. Pierce and R. N. Thurston (Eds.), Vol. 21, pp. 1–234.
    β“˜
    Referenced by:
    §36.14(ii).
    Encodings:
    BibTeX
    • … β–Ί
  • J. M. McNamee (2007) Numerical Methods for Roots of Polynomials. Part I. Studies in Computational Mathematics, Vol. 14, Elsevier, Amsterdam.
    β“˜
    External Links:
    ISBN 978-0-444-52729-5; 0-444-52729-X, MathReview Entry, ZentralBlatt
    Referenced by:
    §3.8(iv).
    Encodings:
    BibTeX
    • … β–Ί
  • D. S. Moak (1981) The q -analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.
    β“˜
    External Links:
    ISSN 0022-247X, Document, MathReview (R. A. Askey), ZentralBlatt
    Referenced by:
    §18.27(v).
    Encodings:
    BibTeX
    • … β–Ί
  • S. L. B. Moshier (1989) Methods and Programs for Mathematical Functions. Ellis Horwood Ltd., Chichester.
    β“˜
    Notes:
    Includes diskette
    External Links:
    ISBN 0-13-578998-2, MathReview, ZentralBlatt
    Referenced by:
    2nd item, CEPHES (free C library).
    Encodings:
    BibTeX
    • …
    40: 8 Incomplete Gamma and Related
    Functions
    …
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