# acceleration of convergence

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## 10 matching pages

##### 1: 17.18 Methods of Computation
Method (1) can sometimes be improved by application of convergence acceleration procedures; see §3.9. …
##### 2: 3.9 Acceleration of Convergence
###### §3.9 Acceleration of Convergence
The transformation is accelerating if it is limit-preserving and if … It should be borne in mind that a sequence (series) transformation can be effective for one type of sequence (series) but may not accelerate convergence for another type. … For applications to asymptotic expansions, see §2.11(vi), Olver (1997b, pp. 540–543), and Weniger (1989, 2003).
##### 3: 15.19 Methods of Computation
Moreover, it is also possible to accelerate convergence by appropriate choice of $z_{0}$. …
##### 4: 3.8 Nonlinear Equations
3.8.3 $\left|z_{n+1}-\zeta\right|
##### 5: 3.10 Continued Fractions
For further information on the preceding algorithms, including convergence in the complex plane and methods for accelerating convergence, see Blanch (1964) and Lorentzen and Waadeland (1992, Chapter 3). …
##### 6: Bibliography
• 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.
• ##### 7: Bibliography W
• 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.
The power-series expansions of §§33.6 and 33.19 converge for all finite values of the radii $\rho$ 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. …
Even when the series converges this is unwise: the tail needs to be majorized rigorously before the result can be guaranteed. … The transformations in §3.9 for summing slowly convergent series can also be very effective when applied to divergent asymptotic series. … Similar improvements are achievable by Aitken’s $\Delta^{2}$-process, Wynn’s $\epsilon$-algorithm, and other acceleration transformations. … 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.