# acceleration of convergence

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## 10 matching pages

##### 1: 17.18 Methods of Computation

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►Method (1) can sometimes be improved by application of convergence acceleration procedures; see §3.9.
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##### 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

##### 4: 3.8 Nonlinear Equations

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3.8.3
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##### 5: 3.10 Continued Fractions

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►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).
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##### 6: Bibliography

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Shanks’ convergence acceleration transform, Padé approximants and partitions.
J. Combin. Theory Ser. A 43 (1), pp. 70–84.
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##### 7: Bibliography W

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Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series.
Computer Physics Reports 10 (5-6), pp. 189–371.
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##### 8: 3.5 Quadrature

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►Convergence acceleration schemes, for example Levin’s transformation (§3.9(v)), can be used when evaluating the series.
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##### 9: 33.23 Methods of Computation

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►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.
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►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.
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##### 10: 2.11 Remainder Terms; Stokes Phenomenon

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►Even when the series converges this is unwise: the tail needs to be majorized rigorously before the result can be guaranteed.
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►The transformations in §3.9 for summing slowly convergent series can also be very effective when applied to divergent asymptotic series.
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►Similar improvements are achievable by Aitken’s ${\mathrm{\Delta}}^{2}$-process, Wynn’s $\u03f5$-algorithm, and other acceleration transformations.
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►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.