acceleration%20of%20convergence
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1: 3.8 Nonlinear Equations
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►The rule converges locally and is cubically convergent.
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►Consider and .
We have and .
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►The convergence of iterative methods
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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 … ►This transformation is accelerating if 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
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►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.
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►Method (1) can sometimes be improved by application of convergence acceleration procedures; see §3.9.
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4: 20 Theta Functions
Chapter 20 Theta Functions
…5: Bibliography
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Exact linearization of a Painlevé transcendent.
Phys. Rev. Lett. 38 (20), pp. 1103–1106.
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On the degrees of irreducible factors of higher order Bernoulli polynomials.
Acta Arith. 62 (4), pp. 329–342.
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Study of nuclear structure by electromagnetic excitation with accelerated ions.
Rev. Mod. Phys. 28, pp. 432–542.
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Repeated integrals and derivatives of Bessel functions.
SIAM J. Math. Anal. 20 (1), pp. 169–175.
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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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6: Bibliography W
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The Nahm equations, finite-gap potentials and Lamé functions.
J. Phys. A 20 (10), pp. 2679–2683.
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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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7: 15.19 Methods of Computation
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►The Gauss series (15.2.1) converges for .
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►However, by appropriate choice of the constant in (15.15.1) we can obtain an infinite series that converges on a disk containing .
Moreover, it is also possible to accelerate convergence by appropriate choice of .
►Large values of or , for example, delay convergence of the Gauss series, and may also lead to severe cancellation.
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8: 8 Incomplete Gamma and Related
Functions
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9: 28 Mathieu Functions and Hill’s Equation
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10: 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 and , 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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