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11: 3.6 Linear Difference Equations
This is of little consequence if the wanted solution is growing in magnitude at least as fast as any other solution of (3.6.3), and the recursion process is stable. … If, as n , the wanted solution w n grows (decays) in magnitude at least as fast as any solution of the corresponding homogeneous equation, then forward (backward) recursion is stable. …
12: 3.7 Ordinary Differential Equations
If the solution w ( z ) that we are seeking grows in magnitude at least as fast as all other solutions of (3.7.1) as we pass along 𝒫 from a to b , then w ( z ) and w ( z ) may be computed in a stable manner for z = z 0 , z 1 , , z P by successive application of (3.7.5) for j = 0 , 1 , , P 1 , beginning with initial values w ( a ) and w ( a ) . … Similarly, if w ( z ) is decaying at least as fast as all other solutions along 𝒫 , then we may reverse the labeling of the z j along 𝒫 and begin with initial values w ( b ) and w ( b ) . …
13: Bibliography P
  • 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.
  • 14: Bibliography B
  • J. M. Borwein and I. J. Zucker (1992) Fast evaluation of the gamma function for small rational fractions using complete elliptic integrals of the first kind. IMA J. Numer. Anal. 12 (4), pp. 519–526.
  • R. P. Brent (1976) Fast multiple-precision evaluation of elementary functions. J. Assoc. Comput. Mach. 23 (2), pp. 242–251.
  • V. Britanak, P. C. Yip, and K. R. Rao (2007) Discrete Cosine and Sine Transforms. General Properties, Fast Algorithms and Integer Approximations. Elsevier/Academic Press, Amsterdam.
  • 15: Bibliography F
  • T. Fukushima (2010) Fast computation of incomplete elliptic integral of first kind by half argument transformation. Numer. Math. 116 (4), pp. 687–719.
  • 16: Bibliography W
  • J. Waldvogel (2006) Fast construction of the Fejér and Clenshaw-Curtis quadrature rules. BIT 46 (1), pp. 195–202.
  • 17: 19.27 Asymptotic Approximations and Expansions
    These series converge but not fast enough, given the complicated nature of their terms, to be very useful in practice. …
    18: Bibliography
  • W. L. Anderson (1982) Algorithm 588. Fast Hankel transforms using related and lagged convolutions. ACM Trans. Math. Software 8 (4), pp. 369–370.
  • 19: Bibliography D
  • J. Dexter and E. Agol (2009) A fast new public code for computing photon orbits in a Kerr spacetime. The Astrophysical Journal 696, pp. 1616–1629.
  • 20: Bibliography G
  • A. Gil, J. Segura, and N. M. Temme (2011b) Fast and accurate computation of the Weber parabolic cylinder function W ( a , x ) . IMA J. Numer. Anal. 31 (3), pp. 1194–1216.