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11: 3.9 Acceleration of Convergence

… ►If $s_n$ is the $n$th partial sum of a power series $f$, then $t_{n,2k}={\epsilon }_{2k}^{(n)}$ is the Padé approximant ${[(n+k)/k]}_f$ (§3.11(iv)). …

12: 18.40 Methods of Computation

… ►The question is then: how is this possible given only $F_N(z)$, rather than $F(z)$ itself? $F_N(z)$ often converges to smooth results for $z$ off the real axis for $\mathrm{\Im }z$ at a distance greater than the pole spacing of the $x_n$, this may then be followed by approximate numerical analytic continuation via fitting to lower order continued fractions (either Padé, see §3.11(iv), or pointwise continued fraction approximants, see Schlessinger (1968, Appendix)), to $F_N(z)$ and evaluating these on the real axis in regions of higher pole density that those of the approximating function. …

13: Bibliography

… ►

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

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External Links:

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

Referenced by:

§17.18.

Encodings:

BibTeX

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14: 28.8 Asymptotic Expansions for Large $q$

… ►The approximants are elementary functions, Airy functions, Bessel functions, and parabolic cylinder functions; compare §2.8. …

15: 2.8 Differential Equations with a Parameter

… ►For two coalescing turning points see Olver (1975a, 1976) and Dunster (1996a); in this case the uniform approximants are parabolic cylinder functions. … ►For a coalescing turning point and double pole see Boyd and Dunster (1986) and Dunster (1990b); in this case the uniform approximants are Bessel functions of variable order. ►For a coalescing turning point and simple pole see Nestor (1984) and Dunster (1994b); in this case the uniform approximants are Whittaker functions (§13.14(i)) with a fixed value of the second parameter. …

16: Bibliography B

… ►

Padé Approximants. 2nd edition, Encyclopedia of Mathematics and its Applications, Vol. 59, Cambridge University Press, Cambridge.

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External Links:

ISBN 0-521-45007-1, MathReview (Annie A. M. Cuyt), ZentralBlatt

Referenced by:

§3.11(iv), §3.11(iv).

Encodings:

BibTeX

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17: 2.11 Remainder Terms; Stokes Phenomenon

… ►If we permit the use of nonelementary functions as approximants, then even more powerful re-expansions become available. …

18: 18.30 Associated OP’s

… ►The $p_n^{(0)}(x)$ are also referred to as the numerator polynomials, the $p_n(x)$ then being the denominator polynomials, in that the $n$-th approximant of the continued fraction, $z\in ℂ$, …

19: 18.2 General Orthogonal Polynomials

… ►Using the terminology of §1.12(ii), the $n$-th approximant of the continued fraction …