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1: 11.13 Methods of Computation
A comprehensive approach is to integrate the defining inhomogeneous differential equations (11.2.7) and (11.2.9) numerically, using methods described in §3.7. …
2: Publications
  • B. V. Saunders and Q. Wang (1999) Using Numerical Grid Generation to Facilitate 3D Visualization of Complicated Mathematical Functions, Technical Report NISTIR 6413 (November 1999), National Institute of Standards and Technology. PDF
  • 3: 2.11 Remainder Terms; Stokes Phenomenon
    §2.11(i) Numerical Use of Asymptotic Expansions
    The rest of this section is devoted to general methods for increasing this accuracy. …
    §2.11(vi) Direct Numerical Transformations
    However, direct numerical transformations need to be used with care. …For example, extrapolated values may converge to an accurate value on one side of a Stokes line (§2.11(iv)), and converge to a quite inaccurate value on the other.
    4: Barry I. Schneider
    He has authored or co-authored 140 refereed papers and books and has given numerous invited talks in the US and abroad. …
    5: 5.21 Methods of Computation
    Another approach is to apply numerical quadrature (§3.5) to the integral (5.9.2), using paths of steepest descent for the contour. …
    6: 21.2 Definitions
    For numerical purposes we use the scaled Riemann theta function θ ^ ( 𝐳 | 𝛀 ) , defined by (Deconinck et al. (2004)), …
    7: Bibliography L
  • J. L. López and N. M. Temme (2010a) Asymptotics and numerics of polynomials used in Tricomi and Buchholz expansions of Kummer functions. Numer. Math. 116 (2), pp. 269–289.
  • 8: 10.47 Definitions and Basic Properties
    §10.47(iii) Numerically Satisfactory Pairs of Solutions
    For (10.47.1) numerically satisfactory pairs of solutions are given by Table 10.2.1 with the symbols J , Y , H , and ν replaced by 𝗃 , 𝗒 , 𝗁 , and n , respectively. For (10.47.2) numerically satisfactory pairs of solutions are 𝗂 n ( 1 ) ( z ) and 𝗄 n ( z ) in the right half of the z -plane, and 𝗂 n ( 1 ) ( z ) and 𝗄 n ( z ) in the left half of the z -plane. …
    9: 18.40 Methods of Computation
    See Gautschi (1983) for examples of numerically stable and unstable use of the above recursion relations, and how one can then usefully differentiate between numerical results of low and high precision, as produced thereby. …
    10: Bibliography G
  • A. Gil, J. Segura, and N. M. Temme (2003b) Computing special functions by using quadrature rules. Numer. Algorithms 33 (1-4), pp. 265–275.