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1: 3.8 Nonlinear Equations
The rule converges locally and is cubically convergent. … Consider x = 20 and j = 19 . We have p ( 20 ) = 19 ! and a 19 = 1 + 2 + + 20 = 210 . … The convergence of iterative methods …
2: 3.9 Acceleration of Convergence
§3.9 Acceleration of Convergence
The transformation is accelerating if it is limit-preserving and if … This transformation is accelerating if { s n } is a linearly convergent sequence, i. … For applications to asymptotic expansions, see §2.11(vi), Olver (1997b, pp. 540–543), and Weniger (1989, 2003).
3: 17.18 Methods of Computation
Method (1) is applicable within the circles of convergence of the defining series, although it is often cumbersome owing to slowness of convergence and/or severe cancellation. … Method (1) can sometimes be improved by application of convergence acceleration procedures; see §3.9. …
4: 20 Theta Functions
Chapter 20 Theta Functions
5: Bibliography
  • M. J. Ablowitz and H. Segur (1977) Exact linearization of a Painlevé transcendent. Phys. Rev. Lett. 38 (20), pp. 1103–1106.
  • A. Adelberg (1992) On the degrees of irreducible factors of higher order Bernoulli polynomials. Acta Arith. 62 (4), pp. 329–342.
  • K. Alder, A. Bohr, T. Huus, B. Mottelson, and A. Winther (1956) Study of nuclear structure by electromagnetic excitation with accelerated ions. Rev. Mod. Phys. 28, pp. 432–542.
  • D. E. Amos (1989) Repeated integrals and derivatives of K Bessel functions. SIAM J. Math. Anal. 20 (1), pp. 169–175.
  • G. E. Andrews, I. P. Goulden, and D. M. Jackson (1986) Shanks’ convergence acceleration transform, Padé approximants and partitions. J. Combin. Theory Ser. A 43 (1), pp. 70–84.
  • 6: Bibliography W
  • R. S. Ward (1987) The Nahm equations, finite-gap potentials and Lamé functions. J. Phys. A 20 (10), pp. 2679–2683.
  • E. J. Weniger (1989) Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series. Computer Physics Reports 10 (5-6), pp. 189–371.
  • 7: 15.19 Methods of Computation
    The Gauss series (15.2.1) converges for | z | < 1 . … However, by appropriate choice of the constant z 0 in (15.15.1) we can obtain an infinite series that converges on a disk containing z = e ± π i / 3 . Moreover, it is also possible to accelerate convergence by appropriate choice of z 0 . Large values of | a | or | b | , for example, delay convergence of the Gauss series, and may also lead to severe cancellation. …
    8: 8 Incomplete Gamma and Related
    Functions
    9: 28 Mathieu Functions and Hill’s Equation
    10: 33.23 Methods of Computation
    The power-series expansions of §§33.6 and 33.19 converge for all finite values of the radii ρ and r , respectively, and may be used to compute the regular and irregular solutions. … Thompson and Barnett (1985, 1986) and Thompson (2004) use combinations of series, continued fractions, and Padé-accelerated asymptotic expansions (§3.11(iv)) for the analytic continuations of Coulomb functions. …