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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 $p^{\prime }(20)=19!$ and $a_{19}=1+2+\mathrm{\cdots }+20=210$. … ►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

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5: 18.40 Methods of Computation

… ►These quadrature weights and abscissas will then allow construction of a convergent sequence of approximations to $w(x)$, as will be considered in the following paragraphs. … ►Results of low ($2$ to $3$ decimal digits) precision for $w(x)$ are easily obtained for $N\sim 10$ to $20$. Gautschi (2004, p. 119–120) has explored the $\epsilon \to 0^{+}$ limit via the Wynn $\epsilon$-algorithm, (3.9.11) to accelerate convergence, finding four to eight digits of precision in $w(x)$, depending smoothly on $x^{\prime }$, for $N\approx 4000$, for an example involving first numerator Legendre OP’s. … ►Convergence is $\sim O\left(N^{-2}\right)$. … ►Further, exponential convergence in $N$, via the Derivative Rule, rather than the power-law convergence of the histogram methods, is found for the inversion of Gegenbauer, Attractive, as well as Repulsive, Coulomb–Pollaczek, and Hermite weights and zeros to approximate $w(x)$ for these OP systems on $x\in [-1,1]$ and $(-\mathrm{\infty },\mathrm{\infty })$ respectively, Reinhardt (2018), and Reinhardt (2021b), Reinhardt (2021a). …

6: Bibliography

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Exact linearization of a Painlevé transcendent. Phys. Rev. Lett. 38 (20), pp. 1103–1106.

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External Links:

Document, MathReview (Mark Adler)

Referenced by:

§32.11(ii), §32.13(i), §32.13(ii), §32.13(iii), §32.5.

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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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External Links:

ISSN 0065-1036, Pdf, MathReview (L. Skula), ZentralBlatt

Referenced by:

§24.12(iii).

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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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ISSN 0034-6861, Document, MathReview, ZentralBlatt

Referenced by:

§33.22(ii).

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Repeated integrals and derivatives of $K$ Bessel functions. SIAM J. Math. Anal. 20 (1), pp. 169–175.

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ISSN 0036-1410, Document, MathReview (A. Haimovici), ZentralBlatt

Referenced by:

§10.43(iii).

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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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ISSN 0097-3165, Document, MathReview (Kenneth A. Jukes), ZentralBlatt

Referenced by:

§17.18.

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7: 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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External Links:

ISSN 0305-4470, Document, MathReview (James M. D’Archangelo), ZentralBlatt

Referenced by:

§29.19(ii).

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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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External Links:

Document

Referenced by:

§2.11(vi), §3.9(v), §3.9(vi).

Encodings:

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8: 15.19 Methods of Computation

… ►The Gauss series (15.2.1) converges for . … ►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 $z={\mathrm{e}}^{\pm \pi \mathrm{i}/3}$. 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. …

9: 8 Incomplete Gamma and Related
Functions

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10: 28 Mathieu Functions and Hill’s Equation

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